Louisell Text

Quantum StatisticalProperties of Radiation

WILLIAM H. LOUISELLProfessor of Physks and Electrical EngineeringUniversity of Southern California

Wiley Classics Library Edition Published 1990

WILEY

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS

New York • Chichester • Brisbane • Toronto • Singapore

Preface

ANC7TETOTHE READER:This book has been electronically reproduced from digitalinformation stored at John Wiley & Sons, Inc. We arepleased that the use of this new technology will enable usto keep works of enduring scholarly value in print as longas there is a reasonable demand for them. The content ofthis book is identical to previous printings.

Copyright © 1973, by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyondthat permitted by Section 107 or 108 of the 1976 United StatesCopyright Act without the permission of the copyright owneris unlawful. Requests for permission or further informationshould be addressed to the Permissions Department, JohnWiley & Sons, inc.

Library of Congress Cataloging in Publication Data:

Louisell, William Henry, 1924—Quantum statistical properties of radiation.

"A Wiley-Interscience publication."Includes bibliographical references.1. Quantum electrodynamics. 2. Quantum statistics.

I. Title.

QC680.L65 537.6 73-547ISBN 0-471-52365-8 (pbk.)

Printed and bound in the United States of Americaby Braun-Brumfield, Inc.

10 9 8 7 6 5 4

The invention of the laser was directly responsible for a tremendous develop-ment in the field of nonequilibrium quantum statistical mechanics. Althoughmany people have made important contributions, W. E. Lamb, Jr., of YaleUniversity, H. Haken of the Technische Hochschule in Stuttgart, Germany,and M. Lax of Bell Telephone Laboratories and City College of New Yorkand their collaborators have been the trail blazers. One of the purposes ofthis book is to present some of the developments in the theory of the quantumstatistical properties of radiation in the hope they may be useful in otherareas of physics. This book is not intended to review the field, since onlyprinciples and techniques are stressed. References have been chosen only togive credit to the source or to illustrate the fundamental techniques.

Emphasis has been placed on the work of Lax,* since I have workedclosely with him. Furthermore, Hakenf has written an excellent articlewhich presents his work while LambJ in collaboration with M. O. Scully andM. Sargent is in the process of writing a book giving the Lamb schoolviewpoint. As a result, I feel that these two monumental works adequatelycover the field and no duplication here is attempted.

Little mention is made in the book of the tremendous advances of R. J.Glauber of Harvard and L. Mandel and E.. Wolf of the University of Rochesteron the coherence properties of laser light. The reader should consult theseauthors. Furthermore, no attempt has been made to cover the field ofnonlinear optics § led by Bloembergen of Harvard and Franken of Michiganto mention but a few of the major contributors.

For the sake of presenting a cohesive account of the quantum statisticalproperties of radiation from first principles, I have borrowed heavily frommaterial from my book Radiation and Noise in Quantum Electronics,

* M. Lax, Brandeis University Summer Institute of Theoretical Physics, 1966, StatisticalPhysics, Vol. 2, M. Cretien, E. P. Gross, and S. Leser (Eds.), Gordon and Breach, 1968.

t H. Haken, Handbuch der Physik, XXV 12C, Light and Matter, L. Genzel (Ed.).% W. E. Lamb, Jr., M. O. Scully, and M. Sargent, to be published.§ N. Bloembergen, Nonlinear Optics, New York: W. A. Benjamin, 1965.

VI PREFACE.

(McGraw-Hill, 1964). The publisher has very generously allowed the use ofthis material in the present work.

The book is intended for physics and electrical engineering graduatestudents interested in the field of quantum electronics, although it shouldhave interest to those in other areas such as solid-state and low temperaturewhere quantum stochastic processes are of importance. Some backgroundin quantum mechanics and statistical mechanics is necessary.

In Chapter 1,1 present the Dirac formulation of nonrelativistic quantummechanics which many readers may be able to omit. Sections 1.20 and 1.21introduce the density operator and the reduced density operator, respec-tively, which play a central role in the book.

Chapter 2 discusses some elementary quantum systems. In Part I thequantum theory of a simple harmonic oscillator is presented and the bosoncreation and annihilation operators are introduced which play a central rolein the quantum theory of radiation. The coherent state whose propertieshave been studied extensively by Glauber, Klauder, and many others isintroduced.

In Part II of Chapter 2, orbital angular momentum and Pauli spin operatorsare studied. In Part III a brief discussion of the interaction of a nonrelativisticelectron with an electric and magnetic field is given.

Chapter 3 gives a rather extensive development of operator algebra.Much of this material has never appeared in book form before. No efforthas been made at mathematical rigor. Rather the intent is to present thetechniques in a form which will make them useful. The generalization ofordering techniques to arbitrary quantum operators introduced by Laxshould prove useful in other areas of physics rather than just in the field oflaser physics. By their use one may avoid annoying quantum commutationquestions and yet retain all the quantum features of a problem. Characteristicfunctions, the Wigner distribution function, Wick's theorem, and the general-ized Wick theorem for bosons are treated. Several applications of the tech-niques are presented as well as the principle of maximum entropy introducedby Shannon in Communication theory and developed further by Jaynes andco-workers in statistical mechanics applications. Some readers may justi-fiably find criticism of this chapter due to the great amount of formalism,but it is felt that familiarity with the techniques presented may be useful inthe future in simplifying other physical problems.

The electromagnetic field is quantized in Chapter 4 in a standard pres-entation and the density operator as it applies to a radiation field is discussed.

Chapter 5 studies the interaction of radiation with matter. Various topicsinclude the absorption and emission of radiation by an atom, the Wigner-Weisskopf theory of natural linewidth and the Lamb shift, the Kramers-Heisenberg scattering cross-section with applications to Thomson and

PREFACEvii

Raman scattering as well as resonance fluorescence, the Doppler effect, andsuch.

A rather complete discussion of the quantum theory of damping usingthe density operator is presented in Chapter 6. A model for a loss mechanismis introduced which has applications in many areas of physics. The Markoffapproximation is given in both the SchrSdinger and Heisenberg pictures.Fokker-Planck equations for a damped harmonic oscillator and for atomswith linewidth are derived and discussed. The rotating wave van der Poloscillator which arises in the theory of laser linewidth is treated.

The Langevin approach to the quantum theory of damping is the subjectof Chapter 7. The fluctuating random forces present in classical Langevintheory become operators in the quantum treatment which causes uniqueproblems which are discussed. The use of generalized associated distributionfunctions introduced in Chapter 3 aids materially in the clarification of suchproblems.

To illustrate the principles of the first seven Chapters, I consider in Chapter9 the statistical properties of laser radiation. For completeness and by wayof introduction, I present a brief account of Lamb's semiclassical theory of alaser in Chapter 8. The theory which was developed earlier has also proveduseful in studying the statistical properties of optical parametric devices,and M. J. Stephens and M. O. Scully have used these techniques very profit-ably to study the statistical properties of superconducting Josephson junctions.

Many people over the years have been very helpful in making this bookpossible. In particular it is a pleasure to acknowledge help and encourage-ment from M. Lax, J. P. Gordon, and L. R. Walker of Bell TelephoneLaboratories, as well as H. Heffner of Stanford University. Also specialthanks are due to Beatrice Shube and James Gaughan of John Wiley & Sons,Inc., for their help on the manuscript.

WILLIAM H. LOUISELLUniversity of Southern CaliforniaApril 1973

Contents

Chapter 1 Dirac Formulation of Quantum Mechanics 1

1.1 Ket Vectors 5

1.2 Scalar Product; Bra Vectors 6

1.3 Linear Operators 10

1.4 Hermitian Operators 13

1.5 The Eigenvalue Problem 14

1.6 Observables, Completeness, Expansion in Eigenkets; Dirac <5Function 19

1.7 Matrices 25

1.8 Matrix Representation of Kets, Bras, and Operators 26

1.9 Transfoifmation Functions; Change of Representation;Diagonalization 30

1.10 Quantization; Example of Continuous Spectrum 34

1.11 Measurement of Observables; Probability Interpretation 43

1.12 The Heisenberg Uncertainty Principle 45I

1.13 Dynamical Behavior of a Quantum System 51

1.14 The Schrodinger Picture of Quantum Mechanics 53

1.15 The Heisenberg Picture 541.16 The Interaction Picture. Time-Dependent Perturbation

Theory, Dyson Time Ordering Operator 57

1.17 Perturbation Theory for a Heisenberg Operator 68

ix

CONTENTS

1.18 Wave Mechanics 70

1.19 The Free Particle; Change in Time of Minimum UncertaintyWave Packet

1.20 The Density Operator [9-13]; Perturbation Theory

1.21 The Reduced Density Operator

71

74

81

Chapter 2 Elementary Quantum Systems

PART I THE HARMONIC OSCILLATOR

88

2.1 The Oscillator in the Heisenberg Picture 90

2.2 The Energy-Eigenvalue Problem for the Oscillator 94

2.3 Physical Interpretation of N, a, and af I Bosons and Fermions 98

2.4 Transformation Function from N to q Representation forOscillator 102

2.5 The Coherent States [8] 104

PART II ORBITAL ANGULAR MOMENTUM; ELECTRON SPIN

2.6 Eigenvalues and Eigenvectors of Angular Momentum 110

2.7 Particle in a Central Force Field 116

2.8 Pauli Spin Operators 122

2.9 Spin Operators in the Heisenberg Picture 127

PART III ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

2.10 Hamiltonian for Electron in Electromagnetic Field 129

Chapter 3 Operator Algebra

PART I GENERAL OPERATORS

3.1 Some General Operator Theorems

132

133

CONTENTS xi

PART II BOSON CREATION AND ANNIHILATION OPERATORS

3.2 Ordered Boson Operators 138

3.3 Algebraic Properties of Boson Operators 150

3.4 Characteristic Functions [10]; The Wigner DistributionFunction 168

3.5 The Poisson Distribution

3.6 The Exponential Distribution

3.7 Generalized Wick's Theorem for Boson Operators

3.8 Wick's Theorem for Boson Operators

PART III ARBITRARY OPERATORS

176

180

182

185

3.9 Generalization of Ordering Techniques to Arbitrary QuantumOperators [14] 190

3.10 Operator Description of Independent Atoms 196

PART IV ELEMENTARY APPLICATIONS

3.11 Solution of the Schrodinger Equation by Normal Ordering;Driven Harmonic Oscillator [15] 203

3.12 Two Weakly Coupled Oscillators 205

3.13 Distribution Function for Two-Level Atom 207

3.14 Distribution Function for Harmonic Oscillator 211

3.15 Generating Function for Oscillator Eigenfunctions 213

PART V PRINCIPLE OF MAXIMUM ENTROPY

3.16 Definition of Entropy 215

3.17 Density Operator for Spin-J Particles [20] 220

Chapter 4 Quantization of the Electromagnetic field 230

4.1 Quantization of an LC Circuit with a Source 231

4.2 Quantization of a Lossless Transmission Line 235

jdi CONTENTS

4.3 Equivalence of Classical Radiation Field in Cavity to Infinite

Set of Oscillators 238

4.4 Quantization of the Radiation Field in Vacuum 246

4.5 Density of Modes 2504.6 Commutation Relations for Fields in Vacuum at Equal

Times 251

4.7 Zero-Point Field Fluctuations 256

4.8 Classical Radiation Field with Sources [7] 259

4.9 Quantization of Field with Classical Sources 261

4.10 Density Operator for Radiation Field 264

Chapter 5 Interaction of Radiation with Matter 269

5.1 Hamiltonian of an Atom in a Radiation Field 270

5.2 Absorption and Emission of Radiation by an Atom 271

5.3 Wigner-Weisskopf Theory of Natural Linewidth [2]; LambShift 285

5.4 Kramers-Heisenberg Scattering Cross-Section 296

5.5 Rayleigh Scattering 301

5.6 Thomson Scattering 303

5.7 Raman Scattering 304

5.8 Resonance Fluorescence 308

5.9 The Doppler Effect [2] 309

5.10 Propagation of Light in Vacuum [1] 314

5.11 Semiclassical Theory of Electron-Spin Resonance 318

5.12 Collision Broadening of Two-Level Spin System 323

5.13 Effect of Field Quantization on Spin Resonance [7] 323

Chapter 6 Quantum Theory of Damping—Density Operator Methods 331

6.1 Model for Loss Mechanism 332

6.2 The Markoff Approximation in the Schrodinger Picture [2-5] 336

CONTENTS xiii

6.3 The Markoff Approximation in the Heisenberg Picture [7] 360

6.4 One-Time Averages Using Associated Distribution Functions[8-10] 368

6.5 Solution of the Fokker-Planck Equation 390

6.6 Two-Time Averages, Spectra [10] 404

6.7 Rotating Wave Van der Pol Oscillator 408

Chapter 7 Quantum Theory of Damping—Langevin Approach 418

7.1 Langevin Equations of Motion for Damped Oscillator 418

7.2 Quantum Theory of Langevin Noise Sources [1] 432

7.3 Langevin Equations for a Multilevel Atom 435

7.4 Langevin Equations for N Homogeneously BroadenedThree-Level Atoms 438

7.5 Langevin Theory of Noise Sources; Associated FunctionFormulation 441

Chapter 8 Lamb's Semiclassical Theory of a Laser [1]

8.1 Modes in "Cold" Spherical Resonator

8.2 The Cavity Field Driven by Atoms

8.3 The Induced Atomic Dipole Moment

444

447

453

455

8.4 Adiabatic Elimination of the Atomic Variables: Properties ofthe Oscillator 460

Chapter 9 Statistical Properties of a Laser

9.1 The Laser Model [1^]

9.2 The Fokker-Planck Equation for a Laser

9.3 The Laser Associated Langevin Equations

9.4 Adiabatic Elimination of Atomic Variables

469

469

470

473

474

xiv CONTENTS

9.5 The Laser as a Rotating Wave van der Pol Oscillator 482

9.6 Phase and Amplitude Fluctuations: Steady-State Solution,Laser Linewidth 485

Appendix A Method of Characteristics 491

Appendix B Hanultonian for Radiation Field in Plane-Wave Repre-sentation . 494

Appendix C Momentum of Field in Cavity 496

Appendix D Properties of Transverse Delta Function 498

Appendix E Commutation Relations for D and B 501

Appendix F Heisenberg Equations of Motion for D and B 503

Appendix G Evaluation of Field Commutation Relations 505

Appendix H Evaluation of Sums in Equation 5.10.17 507

Appendix I Wiener-Khinchine Theorem 511

Appendix J Atom-Field Hamiltonian Under Dipole Approximation 514

Appendix K Properties of Fokker-PIanck Equations 518

Index 525

Quantum Statistical Properties of Radiation

1Dirac Formulation ofQuantum Mechanics

The failure of classical mechanics to account for many experimental resultssuch as the stability of atoms and matter, blackbody radiation, specific heatof solids, wave-particle duality of light and material particles, and such, ledphysicists to the realization that classical concepts were inherently inadequateto describe the physical behavior of events on an atomic scale. To explainthese phenomena, a fundamental departure from classical mechanics wasnecessary. This departure took the form of postulating, as a fundamentallaw of nature, that there is a limit to the accuracy with which a measurement(or observation) on a physical system can be made. That is, the actualmeasurement itself disturbs the system being measured in an uncontrollableway, regardless of the care, skill, or ingenuity of the experimenter. Thedisturbance produced by the measurement in turn requires modification ofthe classical concept of causality, since, in the classical sense, there is a causalconnection between the system and the measurement. This leads to a theoryin which one can predict only the probability of obtaining a certain resultwhen a measurement is made on a system rather than an exact value, as inthe classical case. This probability interpretation of the theory is fundamen-tally different from classical statistical theory. In the latter case, probability isnecessary for the practical reason that one cannot pleasure, for example, thecoordinates and momenta of 1023 gas molecules in a container, although inprinciple it would be possible to do so with complete precision. In quantummechanics, the precise measurement of both coordinates and momenta is notpossible even in principle because of the disturbance caused by a measure-ment.

If, on the other hand, no measurements are made for a certain time, thesystem follows a causal law of development during this interval. That is,when no intervening measurement is made, the state of a system at time tdevelops from the state at an earlier time t0 in a perfectly predictable andtherefore causal way.

2 DIRAC FORMULATION OF QUANTUM MECHANICS

The lack of causality and the probability interpretation of quantummechanics require for its description a type of mathematics different in manyways from the mathematics used to describe classical mechanics. Thepresentation of this mathematics will form a large part of this chapter.

Classical mechanics must be contained as a limiting case in quantummechanics because, if the disturbance caused by an observation may beneglected, classical mechanics is valid. The quantum description of a systemmust shift to a classical description in this limit, provided the quantumsystem has a classical analog. This is called the correspondence principle andrestricts the possible forms that a quantum theory may have.

In this chapter we give a simplified treatment of the Dirac formulation ofnonrelativistic quantum mechanics. We restrict ourselves to one-dimensionalproblems, for the most part, since the extension to three dimensions isfairly straightforward. For simplicity, the problem of degeneracy is also dis-cussed only briefly. In the latter part of the chapter we show how theSchrodinger formulation may be obtained as a special case of the moregeneral Dirac formulation.

No effort is made to be mathematically rigorous, and for simplicity manyof the subtle and more difficult points are omitted. The postulates of thetheory are not complete but should be sufficient to give the reader a workingacquaintance with the mathematical methods and the physical concepts in-volved in quantum mechanics. In short, this chapter is intended only as anintroduction to the Dirac formulation of quantum mechanics; for a deeperinsight, the reader is referred to any of a number of excellent books onquantum mechanics [1-7].

The Dirac formulation involves the concept of vectors (and operators) ina space that may have a finite or an infinite number of dimensions. Let usgive a simple illustration of the way in which such vectors arise in the theory.We shall consider a particle of mass m constrained to move in one dimensionin a potential V(q), where q is the coordinate of the particle which may haveany value from — oo to +00; that is, the particle may be anywhere in theone-dimensional space. According to the Schrodinger formulation of wavemechanics [4], the state of the particle at time t is described by a wavefunction in the position representation, y(q, t). If no intervening measure-ments are made, this state develops in a completely causal way from thestate at time t0, y(q, f0), according to the postulated SchrSdinger waveequation

where h is Planck's constant divided by 2-n. The probability interpretation(necessary when a measurement is made to determine the position of the

DIRAC FORMULATION OF QUANTUM MECHANICS 3

particle) of y>(q, t) is as follows: \y>(q, t)\2 dq gives the probability of findingthe particle between q and q + dq at time t when a measurement of positionis made.

We may take the Fourier transform of y>(q, t) to obtain another wavefunction*

This is called the wave function in the momentum representation, where prepresents the momentum of the particle. It is completely determined byV(?» 0 . which represents the state of the system at time t. It is thereforereasonable to say that <p(p, t) represents the same dynamical state as y>(q, t).It is just another way of describing the same state. For the momentum wavefunction the probability interpretation is that | <p(p, t)\* dp gives the probabilitythat a measurement of the momentum will yield a value between p andp + dp.

The theory can be developed in an entirely equivalent way in either theposition or the momentum representation. In fact, the representation playsa role analogous to a coordinate system in geometry. Since, in ordinarygeometry, problems may be solved by means of vectors, without the use of acoordinate system (and in more generality), it is interesting to ask if quantummechanics may be formulated without the use of a particular representation.The results would be independent of any particular representation then. Theobvious advantages of using a representation in such a formulation wouldnot be lost, however. A convenient representation could always be used tocarry out a calculation just as a coordinate system may be chosen whenvectors are used. This is the goal of the Dirac formulation of quantummechanics: to develop the theory independent of any specific representation.

To see how to go about this program, let us attemptf to give a geometricalinterpretation to the wave function \p(q) at time t to take advantage of theconcept of vectors. The coordinate q can have any value from — 00 to 4- 00,as noted earlier. For each specific value, say qu qz, q3,..., the wave functionhas a value y>(qj, y(qj, y>(q3), We may imagine that an infinite-dimensional space has a set of mutually perpendicular axes each labeled byone of the values of q (qlt q2,...), and that \p{q^ is the projection of somevector on the qt axis, y>(qj is the projection of the same vector on the q2

Kists if iJ—o

\zdq exists; that is, y>(q) [and also <p(p)] must* The Fourier transform existsbe square integrable.

t The geometrical interpretation presented here is heuristic at best and is actuallynot correct. However, it may help to give the reader some intuitive feeling for the vectorspace, called a Hilbert space, which is defined as the space of square integrable functions inconfiguration space.

DIRAC FORMULATION OF QUANTUM MECHANICS

Figure 1.1 Pictorial diagram of a ket vector and three of its coordinate representatives.

axis, and so on. The vector then represents the state of the system just as itscomponents do. This vector is not an ordinary vector since it has a complexcharacter, and we must have a special notation to designate it, just as we dofor an ordinary vector. Dirac uses the symbol | ) to designate a vector ofthis type and calls it a ket vector, or simply a ket, to distinguish it fromordinary vectors. The particular vector whose components are y>(q^v(9*)» • • • is called ket y> and written |y>). Figure 1.1 shows a very diagram-matic sketch of the vector \y>) and its "components" along the mutuallyperpendicular axes described above. Unfortunately, only three of these axescan be shown.

By way of analogy, if A is an ordinary vector and (x, y, z) represent acartesian coordinate system, A may be specified by giving its componentsalong these axes: A = (AX,AV,AX); that is, A can be represented by itscomponents. Similarly, |y> may be specified by its components along theorthogonal q axes: |y»> = [v(?i)> y(q£, • • •]• Thus A represents the vectorequally as well as its components along certain axes, and \y>) represents thestate of the system just as well as its components. The vector in this case issaid to be given in the position representation.

The vector A may also be specified by giving its components along anothercartesian coordinate system (*', y', z') rotated with respect to (x, y, z):A = (A,?, Ar, Ag'). So too |y> may be expressed in another representation:|y) = [q>(pj), 9»(pa), <p(pa), • • •]• This is called the momentum representationand is visualized roughly as the components of the same vector on a rotatedorthogonal set of axes; this is shown in Figure 1.2. The relation between theq and/> axes is given by the Fourier transform.

It should be clear that there must exist an infinite number of other equivalentrepresentations that might not have been so obvious without the intro-duction of the concept of vectors into the theory. We must now specify the

1.1 KET VECTORS

Figure 1.2 Pictorial diagram of a ket vector and three of its momentum representatives.

properties of ket vectors in more precise fashion to develop the theoryfurther.

1.1 KET VECTORS

As noted above, Dirac calls vectors designated by the symbol \a), \x), andsuch ket vectors. A general ket is denoted by | ), and the labels insidedesignate particular kets.

From the discussion above, we associate a ket vector with each state ofthe dynamical system under study. Since we shall postulate that a linearsuperposition of states of the system is also a state of the system, the ketvector space must be a linear vector space [8]. A vector space is said to belinear in the following sense. If cx and c8 are complex numbers and \a) and\b) are two kets, the linear combination

\u) = cx\a) + c2\b) (1.1.1a)

is also a ket vector, since a linear combination of two states associated with\a) and \b) is also a state of the system. If a ket depends on a parameter q',which may take on any value in a certain range, q'x <. q' ^ q'%, we maygeneralize (1.1.1a) to read

')dq\ (1.1.1b)

where c(q') is an ordinary (complex) function of q' and the vector |»> is inket space. Kets such as \u) (and |»» above are said to be linearly dependenton \a) and \b) (or \q')). If, in a certain set of ket vectors (two or more), noneof them can be expressed as a linear combination of the others, the vectorsare said to be linearly independent.


Although the classical and quantum superposition principles are different,as we shall see below, it may be stated by way of analogy that, if 1, J, and &are three mutually perpendicular unit vectors in ordinary space, any othervector may be written as a linear combination of these three; that is, anyother constant vector A may be written as A = cxl + c j + cs£. On the otherhand, 1 cannot be expressed as a linear combination of j and £ and is saidto be linearly independent of J and fc.

Another assumption in the theory is that if a state is superimposed withitself, there results not a new state vector but only the original state again;that is, when cx\a) and ct\a) are added, where cx and ct are arbitrary complexnumbers, the result is

cx\a) + ct\a) - (cx + c£\a),

and the kets cx\a), c2|o), {cx + c2)|o> all represent the same state of thesystem, with the exception of the case cx + ct = 0, which corresponds tono state at all. Thus a state is specified entirely by the direction of the ketvector. It may be concluded that +|a> and — \a) represent the same state.Therefore, there is a one-to-one correspondence between a state of a systemand a direction in ket vector space. This assumption is a departure fromclassical mechanics and shows that the classical and quantum superpositionprinciples are different.

The ket vector space may have a finite or an infinite number of dimen-sions. The dimensionality is determined by the number of linearly inde-pendent kets in the space. Since independent states of a quantum systemare represented by independent kets, the dimensionality is determined bythe number of independent states of the quantum system.

1.2 SCALAR PRODUCT; BRA VECTORS

We have introduced ket vectors in an abstract linear vector space by sayingthat their projection on a given set of orthogonal axes in an infinite-dimensionalspace gives the values of the wave function ip(q, t) in the position representa-tion at time /. This is only a pedantic crutch, but it helps to visualize theprocedure. The essential definition of kets is that a direction in ket space andevery state of the system are in one-to-one correspondence.

In the study of ordinary vector analysis, we cannot proceed very far beforedefining the scalar product of two vectors. We may define the scalar productof A and B as follows: with every two vectors A and B in the space, there isassociated a real number/, which is written

/ = A « B .

The scalar product of any two vectors is then defined, since the number toassociate with any pair of them is known. This definition may seem strange

1.2 SCALAR PRODUCT; BRA VECTORS 7

at first but a little reflection shows that it is a more general definition thanany formulas we might give for finding the number/, having been given Aand B. One such formula i s / = |A| |B| cos 0, where the first two factors arethe magnitudes of A and B, and 0 is the angle between them. But the lengthitself is defined only in terms of the scalar product of the vector with itself,and so the formula does not serve as an effective definition of a scalarproduct, although it is very useful in practice.

More generally, the scalar product of a particular vector B with all othervectors A in the space may be regarded as a way of defining B. If the set ofnumbers /(B) for all A's is given, B is determined. For three-dimensionalspace, it is sufficient to take for A the three unit vectors 1, J, and ft* which arelinearly independent, and define B by giving its scalar product with each.Thus

Bm B • I By = B • J B, = B • £,

and the three numbers Bx, By, and Bz define B.It is a postulate of the theory of ordinary vectors that the function/(B)

a linear function of B. This means that, if Bx and B2 are two vectors,

A • (Bx + B2) = A • Bj + A • B2

A • (cB) = c(A • B),

where c is a number. It is clear that the numbers/(B) may be considered afunction of B since for every A there is a number,/(B). This is what is meantby the expression a function <p(x) of a continuous variable x: with each x isassociated a number <p(x).

The scalar products defined above all involve vectors in the same space.There is an interesting example in crystallography in which the scalarproduct is used to define vectors in another space. We let a, b, c be theprimitive translation vectors in a crystal lattice in ordinary space. We maydefine the primitive translations a*, b*, c* in the reciprocal lattice space bygiving the scalar products of a*, b*, and c* with a, b, c. We have

c* • a s= 0c* . b = 0c*-c = 1.

Therefore, by defining the scalar product of a* with three independentvectors in ordinary space, we have defined a vector a* in another space.

After this lengthy introduction, we now define scalar products of ketvectors in the following way. With each ket \a) is associated a complexnumber/ (In the examples above the numbers were real but ket vectors aremore general vectors than those in ordinary space.) The set of numbersassociated with different |a)'s is a function of \a). This function must be a

a*, aa*-ba* -c

= 1= 0= 0

b*b*b*

• a•b• c

= 0= l= 0


linear function,* which means that if \at) and \a2) are two kets, the numberassociated with 1%) and \cQ is the sum of the numbers associated with \aând \az) separately, and the number associated with c\a), where c is a complexnumber, is c times the number associated with \a), that is,

(1.2.1)/(C|fll» = cfQaj).

Just as in the case of the reciprocal-lattice example above, we may visualizethe numbers/associated with all the kets in ket space as defining a vector inanother space, designated by the symbol </|. Dirac calls the vectors denotedby the symbol ( | bra vectors. We may write the scalar product of </| and\a) as

/ ( !«» = <f\a). (1.2.2)

If we give all the numbers/for each ket \a), we have defined </|. The spaceof bra vectors is different from the space of ket vectors, just as the reciprocallattice space was different from the original crystal space. The definition hereis more general, however, because/may be a complex number in (1.2.2)whereas it was real in the crystal example.

When we use the scalar product notation of (1.2.2), we may rewrite (1.2.1)as

- k » = </l*i> + <f\as)(f$c\a)) = c{f\a).

Since a bra is defined by its scalar product with a ket, (b\ = 0 if (b\a)for every ket \a). Similarly, (bx\ — <62| if (bt\a) — <62|«> for every |a>.

The sum of two bras is defined by its scalar product with \a). Thus

0

(bt$\a) - (bja)

(c{b\)\<>) - c{b\a).

(bt\a)(1.2.4)

Thus far we have defined bras only in terms of their scalar products withkets, and there is no definite relation between them. To give a connection,we make the following assumption: each ket may be associated with a singlebra in a unique way; that is, a one-to-one correspondence between kets andbras is assumed. It is therefore reasonable to give the bra the same label asthe ket with which it is associated. Thus (a\ is the bra associated with \a).Similarly, with the ket

I") = l«> + l*>. (l-2-5a)

• To be mathematically precise, one whould say that/ is a linear functional on the vectorspace.

1.2 SCALAR PRODUCT; BRA VECTORS

there is associated the bra

(u\ = (a\ + (b\,and with the ket

\v) = c\a),

where c is a complex number, there is associated the bra

(v\ = c*{a\,

(1.2.5b)

(1.2.6a)

(1.2.6b)

where c* is the complex conjugate of c. We shall not go into the reason fortaking c* instead of c but just accept it as a new assumption for simplicity[1]. It is therefore reasonable to call the bra associated with a ket its hermitianadjoint, and vice versa, and write

<«| = (|K»' |«) = ««|) f , (1.2.7)

where the dagger means that the bra is changed to its associated ket (andvice versa) and the complex conjugate of any numbers involved is taken,as in (1.2.6).

Since by assumption there is a unique correspondence between bras andkets, the direction of a bra vector may represent the state of a quantumsystem equally as well as does the direction of a ket. They are said to beduals of one another.

As yet we have not defined the length of a bra or ket. We shall considertwo kets \a) and \b) and the associated bras {a\ and {b\. From these vectorswe may form four numbers (a\b), {b\a), (a\a), and (b\b). In general, (a\b)and (b\a) will be complex, and we make the additional assumption that theyare related by

(a\b) = {b\a)*, (1.2.8)

where the asterisk means complex conjugate in the future. With this assump-tion, if we let \b) = \a), we conclude that (a\a) is real. We define the length,or norm, of \a) as (a\a), and so assumption (1.2.8) is necessary if we want thevectors to have a real norm. We make the further assumption that the lengthof a vector must be positive or zero, that is,

{a\a) £ 0. (1.2.9)

The equality holds only if \a) — 0.The assumptions (1.2.8) and (1.2.9) may be given motivation from a con-

sideration of the wave function f(q, t) and its complex conjugate y>*(q, t).We visualized ip(q, t) as components of \y>) in ket space. Likewise we mayvisualize rp*(q, t) as the components of (y>\ in bra space. We then know fromwave mechanics that the complex numbers f*(q, t)x(q, 0 a°d X*(a> 0v(?» 0


are related by

and

•r—)

/

Similar relations should hold for bras and kets since they can be intimatelyrelated to wave functions. This motivated the assumptions (1.2.8) and (1.2.9).

The concept of orthogonality is also important where vectors are concerned.In the case of bras and kets, if the scalar product {a\b) — 0, the vectors areorthogonal. In wave mechanics, y*(q) and %(q) are orthogonal if

V*(q)x(q) dq = 0. The orthogonality involved here is different from the

orthogonality of two ordinary vectors A and B. If A • B = 0, A and B are atright angles to one another. But A and B are in the same vector space. Inthe present case, {a\ and \b) are in different vector spaces. (See the crystal-lattice example treated earlier.) Nevertheless, if {a\b) — 0, it may be saidthat \a) and \b) are orthogonal as well as (a\ and (b\. When (a\b) — 0, itmay also be said that the associated quantum states of the system that theyrepresent are orthogonal.

If the norm of all vectors in the space is finite, the space is called Hilbertspace [8]. The theory must include vectors of infinite norm, as we shall seelater. The space of these vectors forms an even more general vector spacewhich is called ket or bra space. Including vectors of infinite norm requiresthe introduction of the Dirac 8 function at a later stage.

1 3 LINEAR OPERATORS

The concept of linear operators is already familiar to the reader. Forexample, if f(i) is a square integrable function of a continuous variable t,the function belongs to Hilbert space [8]. We may then define the linearoperator d\dt in this space by associating another function g(t) with/(f) andwrite

g(0 = 7/(0-at

If, with every/(r) in the space, we associate another g(t), we have defined theoperator d\dt. If, furthermore, we require that

dt

at

1.3 LINEAR OPERATORS 11

where gu g2, and g are the three functions associated wi th^ ,^ , and/re-spectively, and c is a number, then djdt is a linear operator.

We may similarly define other linear operators such as integration,multiplication by a constant, and many others and build up a whole schemeof linear operators. Clearly, such operators are needed also in vector spaceto extend its range of applicability.

We must now introduce linear operators in the space of ket and bra vectors.If with each ket \a) in the space we associate another ket \b), the associationmay be used to define an operator D which we may write in the form

l*> = D\a), (1.3.1)

where D might mean differentiation, integration, or something else. Notethe convention that an operator appears to the left of the ket on which itoperates.

We are interested only in linear operators; this means that if l^), |a2), and\a) are any three kets and c is a number, D must satisfy the relations

( ' " }D{e\a)) = c(D\a)).

Since an operator is completely defined when its effect on every ket in thespace is known, two operators Dx and D2 are equal if Dx\a) — D2\a) forevery \a). The null operator, D = 0, is defined by D\a) = 0 for every \a).The identity operator, D — I, is defined by D\a) = |a) for every \a).

At this stage we may build up an algebra of linear operators. We maydefine the sum of two operators Dx + D2 by their action on \a):

D2\a), (1.3.3)

a product

(1.3.4)

From this, if D± = Z>2, we can define powers of operators, and so on.We also have, for example,

(1.3.5)= (D2

D2) + D3)\a) = [/?! + (D2

[D^D, + D3)]\a) = Z)xZ)2|fl> + DtDJia).

The algebra of JV-dimensional square matrices is the same as the algebra oflinear operators as the reader should recognize.

The commutator of two operators Dx and Z)2 is written [Du D2] and isdefined by

[Dlt D2] = DXD2 - D2DX. (1.3.6)


In general, ^ Z ) , ?£ D%Dlt which is a property held in common withmatrices. The algebra of quantum mechanics is a noncommutative algebra.Two familiar linear operators that do not commute are Dt — x (multi-plication by x) and D2 — d/dx (differentiation). It is easily verified that, iffix) is a continuous function of x,

so that noncommutating operators are already familiar.Multiplication by a constant is a linear operation. A constant operator

commutes with all linear operators.If two operators Dx and Z)2 satisfy the equations

DtD% = DtDx = /, (1.3.7)

where / is the identity operator, then D2 is the inverse of Du and Dx is theinverse of Dt, if the inverse exists. This is written as

Dt = Dr1 A - D2~\ (1.3.8)

The inverse of a product of operators is

CDxAA)-1 = D3-*Dt-iDf\ (1.3.9)

As noted earlier, these properties of operators are common to finite squarematrices. In fact, later we shall represent operators by matrices.

We have defined the action of linear operators on kets; we must now givemeaning to their operation on a bra. We shall consider the ket

\b) = D\a).

We may take the scalar product of this ket with any bra, say (c\; this scalarproduct (c\b) = {c\(D\a)) depends linearly on \a) since D is linear. From thedefinition of a bra, the scalar product (c\b) may be considered as the scalarproduct of \a) with some bra, say {d\. Then for each (c\ there corresponds abra (d\. The bra (d\ depends linearly on (c\ so that (d\ is obtained from (c\ bythe application of a linear operator to (c\. Since this operator is uniquelydetermined by D, we may reasonably write

<d\

We adopt the convention that operators always appear to the right of brasand summarize the definition above by the relation

(c\(D\a)) = «c|Z>)|a>. (1.3.10)

It therefore is unnecessary to use the parentheses, and either side may bewritten (c\D\a). Therefore, D may first operate on (c\ and the result applied

1.4 HERMITIAN OPERATORS 13

to |o), or vice versa. The operator properties given in (1.3.2) to (1.3.5) areequally valid whether they are applied to bras or kets. Note also that{c\D\a) is a closed-bracket expression and is therefore a complex number ingeneral.

A simple example of a linear operator that occurs frequently in the quantumtheory is \a){b\ = P. We see that P may operate on a ket to give

P\c) = \a)(b\c),

which is a ket \a) multiplied by the number (b\c), and

(c\P = {c\a){b\

is a bra (b\ multiplied by the number (c\a). It is left as an exercise to showthat P satisfies the requirements of a linear operator. An example in ordinaryvector analysis that corresponds approximately to an operator such as P isthe dyadic 1J. In this case, tj • fc = 0,1 • fj = J, and so on.

Linear operators play a central role in the physical interpretation of thetheory. Following Dirac, we make the assumption that each quantity thatcan be measured for a physical system (which is called a dynamical variable)can be represented by a particular kind of linear operator, to be described inthe following section. Examples of dynamical variables associated withlinear operators are position (q), momentum (p), angular momentum L,energy {H), and such which occur in classical mechanics, as well as spinangular momentum (<x) which has no classical analog. Classically thesevariables commute with each other, but quantum-mechanically it may bepostulated that some of these operators do not commute. The commutationrelations determine the type of algebra the operators obey and mark thedeparture of quantum mechanics from classical mechanics.

1.4 HERMITIAN OPERATORS

Linear operators are, in general, complex quantities; if we let them corre-spond to dynamical variables, they would be complex. However, physically,quantities such as momentum, position, and the like give real numbers whenthey are measured. Therefore, the linear operators that represent dynamicalvariables must be restricted to real linear operators. Such operators are saidto be hennitian and are defined as follows:

The bra associated with the ket \q) = L\p), where L is a linear operator,is written

(q\ = (p\V = (L\p)Y = (\q)y.

The symbol V is called the hermitian adjoint of L; that is, the bra (q\, whichis the hermitian adjoint of \q), may be considered the result of some linearoperator acting on (p\, which is designated by V [1],


We next show thatL " = L. (1.4.1)

We let

\b) = L\p), (1.4.2)

where \p) is an arbitrary ket. Its adjoint (associated bra) is

(b\ = {p\V. (1.4.3)

If we take the adjoint again, we obtain\b) = L"\p). (1.4.4)

If we take the scalar product of an arbitrary bra (a\ with both (1.4.2) and(1.4.4) we have

{a\b) - (a\L\p) = {a\L"\p).

Since (a\ and \p) are arbitrary, (1.4.1) follows.If in (1.2.8), we let {a\ = {p\V and \a) = L\p) we have

(p\V\b) - {b\L\p)\

If a linear operator is self-adjoint,

L = L\

(1.4.5)

(1.4.6)

(1.4.7)

the operator is said to be hermitian. From (1.4.6), if L is hermitian, it mustsatisfy the equation

(p\L\b) - (b\L\p)*, (1.4.8)

for any \b) and \p). Therefore, any operator that satisfies (1.4.8) is hermitian.The following properties may be proved for any linear operator:

(cL\a)y = c*(a\L\where c is a constant,

(1.4.9)

(\a)(b\)1 =

The algebra of adjoints of operators is the same as for finite square matrices.

1.5 THE EIGENVALUE PROBLEM

Bras and ket vectors, or rather directions of bras and kets, are associatedwith states of a system, and linear hermitian operators are associated with

(L1L2|a» t = (a\L\L\

= L\L\L\\a)

1.5 THE EIGENVALUE PROBLEM 15

dynamical variables that describe the system. In the next section we show howto relate these mathematical concepts to physical measurements made onthe system. Before this, we must introduce the concept of eigenvalues ofhermitian operators.

An eigenvalue problem is a familiar one in classical mathematics as well asin classical physics. One of the simplest examples involves the solution of theequation

Lu(x) = Xu(x),

where L is known to be —d*/dx* and u(x) and A are unknown. If weadd the boundary conditions that u(0) = «(/) = 0, we find that X cantake on only a certain discrete set of eigenvalues given by AB = w2n2//2,where n = 0, ± 1 , ± 2 , . . . . The associated eigenfunctions uH(x) are un(x) =sin (irnz/l). Note that the effect of an operator L on an eigenfunction un(x)is to reproduce un(x). If L operates on an arbitrary function u(x), it will not,

' in general, reproduce u(x) times a number.We may similarly formulate an eigenvalue problem for operators in ket

(and bra) space. We let L be a linear operator and \a) a ket. If L operates on\a) and gives \a) multiplied by a number /, then \a) is an eigenket of L and /is the associated eigenvalue. This may be written

L\a) = l\a).

This is an eigenvalue problem: L is a known operator, and / and \a) areunknown, and we are asked to find kets which, when acted on by L, repro-duce the same ket times a number subject to a set of boundary conditions.It is customary to label an eigenket with its eigenvalue; with this convention,we may rewrite the eigenvalue equation as

L\l) = l\l). (1.5.1)

The eigenvalue problem may equally well be formulated in terms of bras:

{d\D = d{d\. (1.5.2)For simplicity, in this book we shall usually consider cases in which there

is only one eigenvalue for each eigenvector. If more than one independenteigenvector can be associated with a given eigenvalue, the system is saidto be degenerate. Degeneracy can be treated easily, but it complicates theformulas needlessly.

If |/) is an eigenket of L, then, by (1.5.1), any constant c times |/> is alsoan eigenket with the same eigenvalue. In line with earlier assumptions, thestates represented by |/> and c\l) are the same state.

We shall be interested usually in the solution of the eigenvalue problem forlinear hermitian operators for reasons that should become clear in the nextsection. Before attempting the solutions of any specific eigenvalue problem,


we shall prove two very important theorems valid for all linear hermitianoperators.

THEOREM 1

The eigenvalues of a linear hermitian operator are real.

PROOF

We let Z, be a linear hermitian operator. The eigenvalues of L satisfy the equation

L\t> - l\l).

If we form the scalar product of both sides with </|, we have

</|L|/> = l{l\l). (1.5.3)

If we take the complex conjugate of both sides, we obtain, by means of (1.4.6),

</|L|/>* - </|I,t|/> = /*</(/>. (1.5.4)

But since L* = L and <J|/> ?* 0, comparison of (1.5.3) and (1.5.4) shows that / = /*,and the theorem is proved. We see that </|/> = 0 only in the trivia! case, in which|/> = 0. Note that the norm {l\D is real.

THEOREM 2

Two eigenvectors of a linear hermitian operator L belonging to different eigen-values are orthogonal.

PROOF

We let /' and /' be two eigenvalues of L and 11') and \l" > be the associated eigenkets.Then ( £ , = £ * ; /'and /* are real)

L\l'> - l'\l')

{l"\L = l'{l"\.

(1.5.5)

(1.5.6)

If we form the scalar product of (1.5.5) with </'|, the scalar product of (1.5.6) with|/'>, and subtract, we find that

(/' - l"){l'\n = 0.Since /' y* /* by assumption, then </'|/*> = 0, and the theorem is proved. From(1.5.5) and (1.5.6) we see that the eigenvalues associated with eigenkets are the sameas those associated with eigenbras.

The solution of an eigenvalue problem in many cases is complicated. Weshall now solve a particularly simple one to illustrate the method. In a later

1.5 THE EIGENVALUE PROBLEM 17

chapter we discuss a physical system that may be described by this examplebut for the moment we shall consider it merely as a mathematical example.

We suppose a linear hermitian operator at that satisfies an auxiliarycondition

a/ = / , (1.5.7)

where / i s the identity operator, and we wish to solve the eigenvalue problem

a.\s) - *|J>. (1.5.8)

From Theorem 1, we know that s is real, and from Theorem 2 we know thatW > - 0 if s'^s".

To solve for the eigenvalues and eigenvectors, we multiply both sides of(1.5.8) from the left by at, use (1.5.7) and (1.5.8), and obtain

O.*[S)

orS<TZ\S)

= 0.

If we form the scalar product of this with (s\, we see that, since (s\s) ispositive and not zero, the eigenvalues of ax are given by

By Theorem 2 we know that

Since by assumption there can be no degeneracy (two eigenvalues the same),there are only two eigenvalues, and so we may rewrite (1.5.8) as

1|-1>. (1.5.9)

(1.5.10)

These are the orthogonality relations obeyed by eigenvectors belonging todifferent eigenvalues.

As we know, any eigenket multiplied by a constant is also an eigenketbelonging to the same eigenvalue. We may therefore choose a constant sothat the norm of the eigenvectors is unity as long as the norm is finite andwrite

(1.5.11)

These are the normalization conditions. Normalization does not specify thevector uniquely; we may still multiply | + 1) by exp (ioc) since <+l| will bemultiplied by exp (—/a), where a is real, and (1.5.11) will be left unchanged.Such a phase shift is of no physical significance in the theory, and we shallusually choose a = 0.

For any eigenvalue problem in which the norm of the vectors is finite, theeigenvectors may always be normalized and (1.5.10) and (1.5.11) combined

I g DIRAC FORMULATION OF QUANTUM MECHANICS

into the orthogonality relations

{l'\l") - <Vr. (1-5.12)

where di} is the Kronecker 8 defined by

jl if i\0 if 1

8< (1.5.13)

When the vectors have an infinite norm, these results have to be generalized,as we discuss later.

Anticipating future work, we shall now show that a, may be representedby a 2 x 2 matrix given by

<f, = I _ 1 0 . 5 . 1 4 )

To show this, we form the scalar products of both equations (1.5.9) with(+1| and <—1|, respectively. If we use (1.5.10) and (1.5.11), we obtain theso-called "matrix elements" of az given by

We then may group these results into a matrix such as (1.5.14), with theconvention that the rows are labeled by the eigenbras and the columns bythe eigenkets.

Any ket in the space may be expressed in terms of the eigenkets | + 1 )and | — 1). When this can be done, it is said that the eigenkets form a completeset by definition. Again we are anticipating the results of the next section.

To show that any ket \P) in the space may be expanded in terms of |+1>and | —1), we write the identity

(1 -5.15)

so that £(/ + <*z)\P) is an eigenket of az with eigenvalue +1 . It may there-fore differ from |+1) only by a constant, and we may write

! ( / + at)\P) = cj + l), (1.5.16)

where ca is a constant. Similarly, we see that the last term in (1.5.15) isgiven by

-ox)\p)) = -nw-oz)]\p>,

We consider each factor separately. On using (1.5.7), we have

1.6 OBSERVABLES, COMPLETENESS, EXPANSION IN EIGENKETS 19

so that we may write£(/-<rl)|/>) = c i8|-l>, (1.5.17)

where c2 is another constant. Thus (1.5.15) may be written by using (1.5.16)and (1.5.17) as

c8 | - l>, (1.5.18)

as originally stated. Any ket is therefore linearly dependent on the kets|+1> and | —1), and we have proved that the set {| + 1>, | — 1)} is complete.

We may also derive the so-called completeness relation in this simpleexample. We multiply (1.5.18) from the left alternatively by ( + 1 | and <—1|,use the orthonormality relations (1.5.10) and (1.5.11), and see that

(1.5.19)

If we substitute these into (1.5.18), we obtain

\P) = (I+1X+H + |-l)<Since \P) is arbitrary, this equation will be satisfied if

We may subtract these equations to obtain az:

(. | —1)<—1| ==/, (1.5.20)

which is the completeness or closure relation. We discuss the completenessrelation for general hermitian operators in the next section.

The Hilbert space in this example is two-dimensional because we con-sidered only nondegenerate eigenvalues.

If we substitute (1.5.19) in (1.5.16) and (1.5.17), we have the results

(1.5.21)

- | - 1 > < - 1 | , (1.5.22)

so that we have expressed az in terms of operators of the type \a)(a\ men-tioned near the end of Section 1.3. Eigenvalues of an operator are sometimesreferred to as its spectrum.

1.6 OBSERVABLES, COMPLETENESS, EXPANSION INEIGENKETS; DIRAC 8 FUNCTION

In the preceding section we solved a very simple eigenvalue problem byfinding the eigenvalues and eigenkets of the hermitian operator at; we showedthat the set of eigenkets was complete in that any ket in the space could beexpanded in terms of the eigenkets of az. In this section we give a physicalinterpretation to the eigenvalues. We also discuss the expansion of an


arbitrary ket in terms of eigenkets of a hennitian operator and show that theorthonormality relation (1.5.12) must be generalized when the eigenvalues ofa hennitian operator have a continuous range of values.

With every dynamical variable of a system there is associated a hennitianoperator. When a measurement of the variable is made, a real number isobtained. It is therefore reasonable to make the following physical assumptionin the theory. If the quantum system is in a particular eigenstate of L, say|/), then if we measure L, we obtain the value /. We also assume that, if wemeasure L of a system and always obtain / with certainty, the system is ineigenstate |/); that is, if we measure L for a large number of systems, eachprepared in an identical way, and always get /, then the system is in state |/>.

Furthermore, when a single measurement of L is made on a system in anarbitrary state, one of the eigenvalues of L is obtained. When the measure-ment is made on a system in an arbitrary state, the act of measurement [1]disturbs the system and causes it to jump into one of the eigenstates of themeasured quantity. If L is measured immediately a second time, the sameeigenvalue (with certainty) obtained on the first measurement is obtained here.

It is further assumed that any state of the system is linearly dependent onthe eigenstates of L; that is, the eigenstates of L form a complete set. Thosehennitian operators having a complete set of eigenvectors are calledobservables.

Proof of completeness and therefore than an operator is an observableis, in genera], impossible.41 The example of ax in the preceding section isone simple case where completeness could be proved. We therefore alwaysassume that, if a quantity can be measured, its eigenkets form a completeset.

The completeness assumption allows expansion of an arbitrary state of thesystem in terms of the eigenkets of L. We have seen this in the special casegiven in (1.5.18). An even more familiar example of completeness involves theexpansion of a certain class of periodic functions in a Fourier series of sineand cosine functions. The set of sines and cosines form a complete set withrespect to this class of functions. We work out this example later. In thecase of discrete eigenvalues of an observable, the set {|/)} is complete, andwe may expand any ket \y>) as the linear combination

\y>) = 2 c,|I>, (1.6.1)c

where the sum extends over the entire range of values that / may have. Thisrange may be finite, as in the at example of (1.5.18), or infinite, as in the

* There is, however, a spectral decomposition theorem which says that every "reasonable"hennitian operator has a decomposition into projections such as (1.S.22) (see Ref. 2,pp. 260ff).

1.6 OBSERVABLES, COMPLETENESS, EXPANSION IN EIGENKETS 21

case of a harmonic oscillator to be considered in Chapter 2. We may obtainthe value of the expansion coefficients c, (which may be complex numbers,in general) by the orthonormality relations (1.5.12). To do this, we multiplyboth sides of (1.6.1) from the left by the eigenket </'|, use (1.5.12), and have

= 5*c(I'll) = 5*c<5 / = c. (1.6 2)i i

When we put this back into (1.6.1), we obtain

(1.6.3)i

as the expansion of an arbitrary \tp) in terms of eigenkets of L. Since \y) isarbitrary, (1.6.3) is satisfied if

2 \l) V\ = I- (1-6.4)i

This is the completeness or closure relation for discrete eigenvalues. Anexample of this relation was given for the operator at in the precedingsection [see (1.5.20)].

To illustrate the orthogonality relations (1.5.12) and the expansionpostulate (1.6.1) and (1.6.3), as well as to introduce the Dirac 6 functionwhich will be needed shortly, let us consider the problem of expanding acontinuous/(a;) in terms of the complete set of "eigenfunctions"

«„(*) = n = 0, ±1, ±2 (1.6.5)

If any continuous function could be expanded in the set {un(x)}, the setwould be said to be complete with respect to the class of continuous functions.We see immediately that ujx + x0) = ujx) so that each member of theset is periodic with period x0. Therefore, a continuous function/(a;) may beexpanded only if it too is periodic with period x0. The set is complete withrespect to the class of continuous periodic functions with period x0 if anysuch function may be expanded in the form

/(») = I cnun(x) (1.6.6)

(A more general class of functions may be expanded in this way but we arenot being rigorous or exhaustive.) This expansion is the analog of (1.6.1).

The orthonormality relations are given by

f-Jo

u:<x)un{x)dx = dn.n, (1.6.7)

which are the analog of (1.5.12) when we use (1.6.5).

22 DIRAC FORMULATION OF QUANTUM MECHANICS 1.6 OBSERVABLES, COMPLETENESS, EXPANSION IN EIGENKETS 23

To obtain the expansion coefficients cB in (1.6.6), we multiply both sidesby u*-(x) and integrate from x = 0 to x = x0. We find

where we used (1.6.7). This is the analog of (1.6.2). [We are assuming thatthe integral for cn> exists and that it is legal to interchange the order ofsummation and integration in (1.6.8).] If we substitute (1.6.8) into (1.6.6)and again interchange the order of summation and integration (it is not"legal" but we do it anyway), we have

/(*) = (*°dx'f(x') | ut(x')un(x). (1.6.9)«/0 «=•—oo

This is the analog of (1.6.3). To obtain the analog of the completeness relation(1.6.4) requires the introduction of the Dirac 8 function.

Dirac defined an improper function 8(x — a;') which has the property that

8(x - x') =(0

loo

if

if(1.6.10)

Clearly, this is not a function in the ordinary sense. However, it is veryuseful, and its use can be rigorously justified by the theory of distributions.!We shall not be rigorous and merely define it by its integral properties,namely,

'x'+e

8(x-x')dx=l,J 'x

x'where e is arbitrary in size, but positive, and

J 'x'+cf

as'—f

Also it is symmetric:

f(x)8(x-x')dx=f(x').

d(x) = 8(-x).

(1.6.11)

(1.6.12)

(1.6.13)

There are many interesting representations of the 8 function. One that isquite useful is

., ,. sin ax . , „8(x) = lim . (1.6.14)a-* to TTX

It has the property that, at x = 0, 8(0) = oo, and its integral from — ooto + oo is unity. As a result, we see that

im f"«"V* = li-»oo «/—a a-

limo-»oo «/—o

lim 2 s m fca = 2TT 8(k).a-» oo fc

(1.6.15)

The reader is referred to standard quantum mechanics books [1-5] forfurther discussion of this interesting "function."

Since both sides of (1.6.9) must represent f(x), comparison with (1.6.12)shows that we must have

f u*n(x')un(x) = 8{x' - x).

This is the completeness relation analogous to (1.6.4).Let us return to the case in which the eigenvalues of the observable L are

continuous. If /' ^ /', we still have the orthogonality relation (/'|/"> = 0.We now try to generalize (1.6.1) (since by hypothesis the eigenvectors forma complete set) and expand two vectors \C) and |Z>) as

(1.6.16)

The scalar product of these is

Since </'|/") = 0 if /" 5* /', the integral

f

(1.6.17)

d\"

is zero if </'|/'> is finite. This is true since the integrand will be zero every-where except at the one point /" = /' and such an integral is zero. Since, ingeneral, (C\D) ?* 0, we can only conclude that </'|/'> = oo. But the Dirac 8function comes to the rescue. With its use the orthonormality relations(1.5.12) for continuous eigenvalues may be generalized to

</'!/"> = 8(1' — I"). (1.6.18)

From the property (1.6.11) for the 8 function, we have normalized the eigen-vectors so that

f See Ref. 2, Appendix A. J< = 1.


If we use (1.6.18) in (1.6.17) and let \C) — \D), we see that

<C|C> = [dV c*(l') cil") d(l'- I") dl"

iV £ 0 (1.6.19)= / '

by assumption. The integrals are over the range of eigenvalues of L.From (1.6.16) and (1.6.18) we may obtain the completeness relation as

follows: We form the scalar product

(cin w -n dv (1.6.20)

This is the analog of (1.6.2) in the discrete case. If we substitute this into theexpansion (1.6.16), we obtain

(1.6.21)

(1.6.22)

The d function is also useful in Fourier transforms. The expansion ofy(q) in a Fourier integral is given by

Since |C) is arbitrary, we have the completeness relation

V) dl'(l'\ = L

and the inverse transform is

If we substitute <p(p) into the integral for y>(q) and interchange the order ofintegration with respect to p and q', we have

This equation yields the identity y>(q) — y>(q) if

(1.6.26)

1.7 MATRICES 25

This is the representation of the d function given in (1.6.15), with suitablechange in notation.

Some observables have both a discrete and a continuous spectrum. Theextension of the orthonormality and completeness relations in this case isstraightforward.

Before closing this section, we discuss very briefly a function of anobservable, f(L). Uf(L) can be expanded in a power series, by repeatedapplication of the eigenvalue equation,

L\l) =

we see that

(1.6.27)

(1.6.28a)f(L)\D =For example [1], iff(L) = L%, then V\l) = I2\l). However, we shall postulatethat (1.6.28a) holds even if the function cannot be expanded in a power series.We consider the function/(L) = Lr1. Then

provided none of the eigenvalues / = 0. But this is just a requirement thatLr1 exist. Another simple example is the function/(L) = i X In this case,

and there is an ambiguity in sign. The operator VA exists and its eigenvaluesare real if the eigenvalues of L are positive. The ambiguity in sign may beremoved by choosing a sign for each eigenvalue. Usually, in practice, oneselects the positive square root.

Finally, the adjoint of (1.6.28a) is

or if G is a function of L,

1.7 MATRICES

{l\p(L) = /*( / ) ( / |

</|G(L) = <7(0</|.

(1.6.28b)

(1.6.28c)

In the following section, we give a matrix representation for ket and bravectors as well as linear operators in a space. Although a knowledge offinite matrices is assumed, we discuss here a few of the less familiar propertiesas well as extend the ideas of finite matrices to infinite matrices heuristically.

The trace of a square finite matrix is defined as the sum of the diagonalelements. Thus if A is a square finite matrix, then

(1.7.1)


where Tr is an abbreviation for the trace, and Au is the /th diagonal element.Also, the trace of a product of finite square matrices is invariant under cyclicpermutations, that is,

Tr (ABC) = Tr (BCA) = Tr (CAB) (1.7.2)

as may be proved easily provided the traces exist.The hermitian adjoint of a matrix A, written A*, is obtained by inter-

changing rows and columns and taking the complex conjugate of eachelement. If

A = A*

the matrix is said to be hermitian; that is, (/4f)w = A%A matrix A is unitary if

AA< = AU = / - > A* = A-1

tj.

(1.7.3)

(1.7.4)

where A~x is the inverse. The inverse exists if the determinant of A is notzero.

These and all the more familiar properties may be used for matrices withan infinite number of rows and columns. The rows and columns may belabeled by discrete indices, or by a set of continuous indices that extend oversome range of values, or a combination of both. For example, if q and q' canhave any value from — oo to + oo, we may write a matrix element of A withlabels Aq;q> or, equivalently, A(q;q'). When we multiply this by anothermatrix B with elements Bq.q', by analogy with the rule of matrix multiplica-tion for finite matrices, we write

CQ:Q. = f" A(q; q")B(q"; q') dq". (1.7.5)

When the integrals converge, all is well. Similarly, the trace of A is

Tr(A)=JA(q';q'')dq'.

This assumes that the integral exists. A diagonal matrix is written

A(q';q") = A(q';q')d(q'-q").

The extension to infinite matrices is straightforward.

1.8 MATRIX REPRESENTATION OF KETS, BRAS, ANDOPERATORS

It has previously been pointed out that representations play the role ofcoordinates in ordinary vector analysis. In this section we give a moreprecise meaning to these intuitive concepts. We show how to find a representa-tion and how to express operators and eigenvectors in this representation. In

1.8 MATRIX REPRESENTATION OF KETS, BRAS, AND OPERATORS 27

particular, we show that arbitrary kets and bras in the vector space may berepresented in terms of column and row vectors (more generally calledmatrices) and that operators may be represented in terms of matrices. Wehave already seen a simple example of this in (1.5.14). The advantage of usinga particular representation in solving quantum problems is the same as theadvantage of using a particular coordinate system in ordinary geometry.

We develop the theory of representations in terms of an observable Lwhich has a discrete spectrum in parallel with an observable q with a con-tinuous spectrum.

We begin by considering an observable L (or q) that satisfies the eigenvalueequations

L\l) = l\l) q\q') = q'W), (1-8.1)

where the eigenvalues / are discrete (and q continuous). Since L (and q) areobservables, the eigenkets {|/>} (and {\q')}) form a complete set. By (1.5.12)and (1.6.18) these eigenkets satisfy the orthonormality conditions

</'|O = dvr {q'W) = * « ' - q"),

and, by (1.6.4) [and (1.6.22)], the completeness relations

(\q')dq'(q'\ =J

(1.8.2)

(1.8.3)

By virtue of (1.8.3), an arbitrary ket |y> may be written

IV> = (1.8.4)f) \v)=^\q'){q'ty)dq'.

By way of analogy, if 1, j , and fe are three unit orthogonal vectors inordinary space, we may write an arbitrary vector A as

A = Axl + Ay\ + A£. (1.8.5)

This is the analog of (1.8.4). The analog of (1.8.2) is

1-J — J - t — 0;I»i = j»j = £»fe = 1, and the set {1,J,£} is complete since any vector Amay be expanded as in (1.8.5). It is said that A is expanded in the I, j , ftrepresentation. These vectors may be called basis vectors for the space.

Analogously, the set of vectors {|/)} (and {\q')}) may be regarded as aparticular set of orthogonal unit basis vectors in the sense of (1.8.2), and itis said that (1.8.4) gives the expansion of \y>) in the L (or q) representation.The numbers (l'\ip) ({q'\\p)) are the "components" or, as Dirac calls them,the representatives of \xp) in the L (or q) representation.


Just as we may select a complete set of basis vectors other than 1, ] , £ torepresent A, we may choose eigenvectors of observables other than L (or q)to use as basis vectors to represent \y>). We discuss this situation in the nextsection.

From the original definition of scalar products (see Section 1.2) we knowthat </'|v) ((q'\y>)) may be considered as a function of /' (or q') which maybe written yy [or ip(q')] since with each /' (or q') there is associateda number y r [or y{q')]. These numbers, or functions, determine \y>)uniquely (if the eigenkets have been specified) just as the numbers Ax, At,and A, specify A uniquely (when t, j , and £ have been chosen).

We may write the vector A as a column vector (or matrix) in the form

A, AJ + A J + (1.8.6)

where we label the rows by z, y, z. Similarly, we may write \f) in (1.8.4) asa matrix of one column in which we label the rows by /' (or q'):

to)i'

\V) fdq' (q'\y>M).J

(1.8.7)

Thus there is a way of representing an arbitrary ket as a column vector.*The number of rows is determined by the number of eigenvalues that /' (orq') may have. The representatives </'|y) (or (q'\y>)) are usually complex.

We may illustrate (1.8.7) with the simple example in Section 1.5. From(1.5.18) and (1.5.19) we have

•~ra-Guana (1.8.8)

as the vector representing an arbitrary ket in the space. From this, we seethat the basis vectors themselves may be represented by

l + l>R+ll+D"!

J(1.8.9)

* Representing (q'\y) as a column vector labeled by the continuous variable q' is notwell denned. However, there is no harm in visualizing this by analogy with the discretecase </>>.

1.8 MATRIX REPRESENTATION OF KETS, BRAS, AND OPERATORS 2S>

We have already pointed out earlier in the introduction that the representa-tives (q'\y>) = yiql are the Schrodinger wave functions in the coordinaterepresentation when q is the observable associated with the coordinate of aparticle constrained to move in one dimension. We go into these mattersmore fully later in the chapter.

In complete analogy with the above, an arbitrary bra (y>| may be representedby a matrix with one row and the columns labeled by /' (or q'). Thus

<W\ =J<?!«') dq' (q'\. (1.8.10)

From the general property (1.2.8), we have

<W) = (i'to)* = v* <wW) - Wto)* = v*(«')- (1.8.H)Therefore, the hermitian adjoint of a column vector is a row vector whosecorresponding elements are the complex conjugates of the row vector.

We next consider the problem of expressing any linear operator A as amatrix in the L (or q) representation. To do this, we apply the completenessrelations (1.8.3) twice and write the identities

(1.8.12)v.r

A=^dq'dq"\q'){q'\A\q"){q"\.

The numbers (l'\A\V) ((q'\A\q")) are functions of/' and /' {q' and q") (see thedefinition of a linear operator in Section 1.3), and we may write

(V\A\1") = APsr

as the matrix elements of A in the L representation and

(1.8.13)

(1.8.14)

in the (continuous) q representation. In particular, if A — L (or q), thematrix elements reduce by means of (1.8.2) and (1.8.1) to

(l'\L\l") - l'6rr (q'\q\q") = q' <*(«' - q") (1.8.15)

so that L (or q) is diagonal in its own representation. Thus (1.8.13) gives Ain the representation in which L is diagonal and (1.8.14) gives A in therepresentation in which q is diagonal. If we return once more to the exampleof Section 1.5, we see that (1.5.14) is a special case of (1.8.15); that is,

az

[<+IKL<-l|a,

I+I> <+i|<Tz||+l> (-IKI

i)

I)

-i n on

J LO - l j(1.8.16)


which is diagonal. This is called the representation in which az is diagonal,or simply the ox representation.

It has been pointed out several times previously that the algebra ofoperators is the same as the algebra of matrices; thus it is not surprising thatoperators may be represented by matrices. Similarly, the algebra of ket andbra vectors is the same as the algebra of one-column or one-row matrices (orvectors).

We have introduced a set of basis vectors as eigenvectors of a particularobservable L (or q). This is not essential, and a set of basis vectors may beintroduced quite arbitrarily; every basis chosen gives rise to a representationof vectors by column matrices and operators by square matrices. The basisvectors may always be chosen in such a way that they are orthonormal inthe sense of (1.8.2). In the next section we show how to change from one setof basis vectors to another.

1.9 TRANSFORMATION FUNCTIONS; CHANGE OFREPRESENTATION; DIAGONALIZATION

The choice of the observable used to represent state vectors and operatorsis not unique. As an example, if the system is a particle constrained to movein one dimension, either the momentum p or the coordinate q would serveas a suitable observable in representing state vectors and operators in matrixform. For some calculations, one representation may be more convenientthan the other. In this section we consider the effect on state vectors andoperators of changing from one representation to another. In the next sectionwe work out a particular example. '

This problem is completely analogous tp a coordinate rotation in ordinarygeometry. Let us illustrate this for a two-dimensional rotation. If x, y is arectangular system and if x', y' is a system rotated counterclockwise throughangle 0, we have the transformation equations

x' = cos 6x + sin Oy

y' = —sin Ox + cos By,

or, in matrix notation,

[ x'l |~ cos 0 sin 0~| pel

y'J L-sin0 cosflJU'The transformation matrix

[ cos

- s i

cos 0 sin 01

sin 0 cos 0J

(1.9.1)

(1.9.2)

(1.9.3)

1.9 TRANSFORMATION FUNCTIONS 31

satisfies the conditions that the determinant of AT, written det K, equals 1 and

(1.9.4)

where R is the transpose of K. From this we conclude that

R = K~\ (1.9.5)

where K~x is the inverse. When R = AT"1 and det K = 1, the transformationbetween x'y' and xy is a rotation. We may use (1.9.5) to invert (1.9.2), andobtain

cos 0 —sin

sin 0 cos 0(1.9.6)

Similarly, the components of an arbitrary vector A in the x', y' frame arerelated to the components in the x, y frame by

or

C3-42J-

(1.9.7)

(1.9.8)

These components are shown in Figure 1.3.Transforming from one representation to another is no more complicated

in principle than the simple example above. Let us assume that we have twoobservables L and M that satisfy the eigenvalue equations

L\V) = V\V) M\m') = m'\m'). (1.9.9)Both /' and m' may be discrete, both continuous, one discrete and the othercontinuous, or / ' as well as m' may have some discrete values and some con-tinuous. For simplicity, we assume that both /' and rri are discrete although

••x

Fi gnre 1.3 Components of a vector in two rotated coordinate systems.


any of the possibilities above may be treated in analogous fashion. Since Land M are observables,

<jn'\m") (1-9.10)

and both sets are complete so that

I = I Zlm'><m'| = J. (1.9.11)r «•'

We may therefore expand any |/'> in terms of the {\m')} or vice versa andobtain

ID = 2 \m'){m'\V) |m'> = 110</'|m'>. (1-9.12)m' I'

The numbers {m'\l') and </'K>. which by (1.2.8) are related by

<m'|/'> = </>')*, (1-9.13)

are called the transformation function between the L and M representations.Because of the completeness and orthogonality relations

( L 9 ' 1 4 )

Because of (1.9.13) and (1.9.14), the transformations (1.9.12) are calledunitary.

We may arrange the numbers (m'\l') as a matrix with the eigenvalues m'labeling the columns and /' labeling the rows.

We may also use either the L or M representation to represent anyoperator A. Thus A^- = </'|-4|/"> are the matrix elements of A in the L-representation while Am-m. == (m'\A\m") are the matrix elements in the M-representation. If we use the completeness relations, we see that

(l'\A\r)=

(m'\A\m") -

2m'.m" (1.9.15)2v.i"

Therefore, the transformation functions are needed to obtain the connectionbetween the matrix elements in the two representations. When we transformfrom one representation to the other under a unitary transformation, theoperators, according to (1.9.15), are said to transform under a similaritytransformation.

We therefore must obtain the transformation functions. Explicit examplesare given later.

1.9 TRANSFORMATION FUNCTIONS 33

Suppose that A is an observable and that we wish to solve the eigenvalueproblem

AW) = a'W), (1.9.16)

where the {|a'>} are orthogonal and complete. Assume in addition that weknow the matrix elements of A in the ^-representation. If we multiply bothsides of (1.9.16) by </'|, we obtain

(l'\AW) = «'</>')-

If we insert the completeness relation, we have

This may be rewritten as

(1.9.17)

(1.9.18)

). (1.9.19)r

This represents a set of homogeneous algebraic equations to determine thetransformation function </>') from the L-representation to the representa-tion in which A is diagonal. To have nontrivial solutions, it is necessary andsufficient that the determinant of the coefficients vanish, namely,

\\Arr - a'dri4 = 0. (1.9.20)

The values of a' that satisfy this are the eigenvalues of A. Corresponding toeach eigenvalue a', we can solve (1.9.19) for the transformation function</"|a'). Thus diagonalizing the matrix representing A is equivalent to solvingthe eigenvalue problem.

We next prove the following theorem. Two matrices may be simultaneouslydiagonalized by the same transformation if and only if they commute.

PROOF

Assume A and B are two operators which are diagonal in the M representation.Then

We then have

<m'\A\m") = Am.<m\m">

<jn'\B\m") = Bm.(m'\m").

(m'\AB\m") - J (m'\A\mm)(mm\B\m")m'

= 2 AmSm.{m'\mm){mm\m')

= Am.Bm.<nt'\m").

(1.9.21)

(1.9.22)

34

Also,

Therefore, we see that

DIRAC FORMULATION OF QUANTUM MECHANICS

"> = Bm.Am.{m'\m'1). (1.9.23)

AB = BA.

The converse is easily proved.

We have therefore developed a complete scheme for transforming vectorsand operators between two representations. We illustrate this by a simpleexample in the next section.

1.10 QUANTIZATION; EXAMPLE OF CONTINUOUSSPECTRUM

If a single measurement of an observable is made, one of its eigenvalues isobtained. The ability to solve eigenvalue problems is therefore essential torelate the theory to experiment. Thus far we have solved only one suchproblem, in Section 1.5, where the observable satisfied the algebraic equationot* = 1. In that case the Hilbert space consisted of only two eigenvectors,and the eigenvalue spectrum had only the two discrete values + 1 and —1.In this section we solve an eigenvalue problem in which the eigenvaluespectrum is continuous. This simple example demonstrates how to treatquantum-mechanically a system that has a classical analog.

Again we consider a single particle of mass m constrained to move in onedimension in a field of force. Classically, this system may be describedcompletely by a position coordinate q and a momentum p. If we specify boththese quantities at a certain time, we have specified the classical state of thesystem at this time.

To treat this system quantum-mechanically, according to the theory wehave developed thus far, we associate with each of these dynamical variables(since they are observable) a linear hermitian operator which we shall callq and p. As operators, they satisfy

p~pi q=q\ (1.10.1)

The classical hamiltonian for the system is also an observable, and wemay associate with it the hermitian operator,

2mH1 (1.10.2)

which is expressible in terms of p and q.After the operators needed to describe the physical system are enumerated,

the next step in setting up the quantum problem is to specify the algebra that

1.10 QUANTIZATION; EXAMPLE OF CONTINUOUS SPECTRUM 35

the operators must obey. This requires an additional postulate for the theory;it is given in terms of the commutation relations for p and q, namely,

' " "' " ° (110.3)

where h is Planck's constant divided by 2ir; that is, we postulate that/? and qsatisfy the commutation relations above. Classically, q and p commute sothat, to the extent they do not commute, the quantum and classical systemsdiffer. Accordingly, the classical system is quantized when the observablesq and p satisfy (1.10.3). The justification for the quantum postulate is theremarkable agreement between theory and experiment. It is possibly the mostprofound and fundamental postulate in the theory.

If ft _». o, q and p will commute so that classical mechanics should becontained in the quantum formulation in the limit as ft -*• 0. This is just thecorrespondence principle.

It is said that q and p obey a noncommutative algebra. Before proceeding,let us develop a few useful algebraic relations. If / is an integer, we mayprove by mathematical induction from (1.10.3) that (see Problem 1.1)

dp (1.10.4)

dq

From these commutator relations, it follows directly that, if F(p) and G(q)are functions that may be expanded in a power series inp and q, respectively,then

= ih^f (1.10.5a)

dq(1.10.5b)

We postulate that these are also true even if F and G cannot be expanded ina power series.

These relations may be generalized even further to the case where F(p, q)is a function of p and q:

dp.bdF

(1.10.6a)

(1.10.6b)

Extreme care must be exercised in the use of (1.10.6) since p and q do not


commute. For example, i£F(p, q) = pqp, this is not equal to p*q, and in thiscase the application of (1.10.6) gives

r- PIPdp

qp)-

In other words, the order of the factors in F(p, q) must be preserved whenusing (1.10.6).

We are now ready to solve the two eigenvalue problems

P\p')=p'\p') q\q')=q'W) (1.10.7)

for the momentum and coordinate. This involves finding the eigenvalues andeigenvectors. To do this, we shall introduce two Dirac operators, calledtranslation operators for reasons that will become apparent shortly.

We consider the operator [2]

(1.10.8)

where £ is an arbitrary real parameter. Since/' is hennitian, we see that

Q\t) = exp (^ = Q-\t) (1.10.9)

so that Q is a unitary operator if £ is real.*If we use (1.10.5a) we see that

so that

,Q] = H^r = e

dp

qQ = Qq + £Q-

(1.10.10)

If we now multiply both sides of this equation by the eigenket \q') ofq witheigenvalue q', by (1.10.7) we have

q{QW)} = iq' + £){Q\q% (i-w.ii)This says that, if \q') is an eigenket with eigenvalue q, then Q\q) is also aneigenket with eigenvalue q' + £. Now by Theorem 1, Section 1.5, the eigen-values of q must be real since q is hennitian; thus £ must be real, which weassumed in (1.10.9), but otherwise it is completely arbitrary. Therefore, theeigenvalue q' + £ may be any value from — oo to + oo. The eigenvalues ofq' have a continuous spectrum.

This is the justification for calling Q a translation operator since it trans-lates the eigenvalue from q' to q1 + £. If we multiply (1.10.10) from the left

* Eigenvalue o f f must be real; therefore, £ must be real.

1.10 QUANTIZATION; EXAMPLE OF CONTINUOUS SPECTRUM

by Q-1, we obtain

37

q + t, (1.10.12)

which shows that Q translates the operator q by an amount £.Since q' is continuous, the eigenkets may be normalized in such a way that

they satisfy the "orthononnality" relation (1.8.2),

(q'\q") = •%'-?")• (1.10.13)

Also, since q is an observable, we have the completeness relation

fj—i

\q')dq'(q'\=I. (1.10.14)

Now we have shown that Q(£)\q') is an eigenket of q with eigenvalueq' + £. It may therefore differ from the eigenket \q' + £> by a multiplicativeconstant so that we may therefore let

c(q', (1.10.15)

(1.10.16)

It also follows directly from this that

(q"\QHZ) = c*(q',

If we multiply these together and use (1.10.9), we obtain

{q"\$Q\q') = <q"\q') = c*(q\ $)c(q', £)<?" + £|?' + £>.

If we use (1.10.13) and integrate both sides over all dq", we see that

W, £)P - 1independent of q' and £. Aside from a trivial phase, we may let c = 1 so that

d-10.17)exp ( - l-^j \q') m Q(£)k'> = \q' +

In particular, if we take q' = 0 we have

where the subscript q denotes that |0)a is an eigenket of q with eigenvaluezero. If we next let £ be some arbitrary eigenvalue, sayq', then this becomes

exp ( - ')|0>« = \q'). (1.10.18)

The state with eigenvalue q' can be generated from the state with eigenvalueq' = 0 by the translation operator Q(q').


We may take the adjoint of (1.10.17) to obtain

(1.10.19)

so that

W\-Therefore, all eigenkets (and eigenbras) may be generated from one eigenket(or eigenbra). This is sufficient to determine all the properties of eigenvectorsneeded for the theory.

We may analogously introduce another unitary translation operator

Pit) - - " I T ) ' (1.10.21)

and show that the eigenvalues of/7 may have any value from — oo to + 0 0and that

= \p' + !> (1.10.22)

or

We also have

, < 0 | P V ) - p<0| exp ( -

(1.10.23)

(1.10.24)

= /7+f. (1.10.25)The eigenkets satisfy the orthonormality and completeness relations

(p'\p") = 6(p' - p") f" W) dp' (p'| = I. (1.10.26)•1—as

We have therefore solved the eigenvalue problem for both q and p. Theset \q') and the set \p') are both complete and either may be used as a set ofbasis vectors to represent arbitrary state vectors and operators. From thediscussion in the preceding section, we need to know the transformationfunction

S{p' I q') - {p'W) - <q'\pT (1.10.27)

to transform from one representation to the other. This function may becalculated quite simply by means of the translation operators Q and P.By (1.10.18) we have

But [see (1.6.28)]

(1.10.28)

(1.10.29)

1.10 QUANTIZATION; EXAMPLE OF CONTINUOUS SPECTRUM

so that

By (1.10.24) and the fact that [see (1.6.28)]

F(q)\q') = F(q')\q'),(1.10.30) may be written

39

(1.10.30)

(1.10.31)

= exp ( - l^f) ,<0|0>8 s {q'\p')*. (1-10.32)

But ,,{010), is just a constant that may be evaluated as follows: by means ofthe orthonormality relation for \p') (Eqs. 1.10.26) and the completenessrelation for \q') (Eq. 1.10.14) we have the result

£ (1.10.33)

If we substitute (1.10.32) into (1.10.33), we find that

With suitable change in notation, we see by (1.6.26) that the integral equals2irfid(p' — p") so that the constant is

We may choose the phase of p(0|0>9 so that

S(p';q') = (p'\q)1

exp I -

(1.10.34)

(q\p'f. (1.10.35)

This is the transformation function discussed in Section 1.9.We saw in the preceding section that an operator in one representation

transforms to another representation by means of (1.9.15). To apply thisresult to an operator A in the p and q representatives, we let the operatorL in (1.9.15) correspond to q and the operator M correspond top. Since theeigenvalues are continuous in this case, the sums in (1.9.15) must be replacedby integrals, and we have

W\A\q") = Trfp' rdp" {q'\p'W\A\p")<jp''\q").•>—oo J—go


If we use (1.10.35), this becomes

2irn J-ca

We now consider the special case of A -p. By (1.10.26) and (1.10.7), wehave

</> W > = / 6(P' ~ / ) • (1.1037)

If we put (1.10.37) into (1.10.36) and use (1.6.12) to carry out the integralover/>", we find that

h d i r- r»y(g' - g n , >exp dp

i a«' 2irfi J-oo L h J

where we may differentiate the integrals with respect to q' or q" inside theintegral sign. By (1.6.26) we see that (1.10.38) may be written as

i dq

But, by (1.10.13), this may be written in still another way:

~ f f

( 1 1 0 3 9 )

d-10.40)

This shows how the operator/' transforms from the/» to the -representation.We may generalize this result easily to show that, if F is a function of p,

(1.10.41){q\F{p)\q") - F(J fy

while if V is a function of 9, we have, by (1.10.31),

(q'W(q)\q")

1.10 QUANTIZATION; EXAMPLE OF CONTINUOUS SPECTRUM 41

We may similarly show how q transforms to the/>-representation. We findthat

= +7r-;0>V'>. (1.10.43)

As another application of the theory given in the preceding section, let usshow how the representatives of an arbitrary state vector \y>) are related inthe two representations. By the completeness relations, we have

f W> dp'

and similarly

•sJ2irn(1.10.45)

where we have used the transformation function (1.10.35). We know that(q'\v) = V>(q') is a function of q' and (p'\y>) = <p(p') is another function of/ . From (1.10.44) and (1.10.45) we see that yiq1) and y{p') are Fouriertransforms of each other. They express the same state vector \\p) in twodifferent representations, in line with earlier discussions. They are calledthe Schrodinger wave functions.

We may generalize (1.10.41) and (1.10.42) as follows: let us multiply(1.10.41) from the right by (q"\y>) and integrate over dq" from —00 to +00.If we use the completeness relation, (1.10.41) reduces to

(1.10.46)

where \y>) is an arbitrary state vector. A similar procedure for (1.10.42)shows that

V(q')(q'\W)= V(q')ytq'). (1.10.47)

These are very useful results.There is still another important observable for this system, namely, the

hamiltonian (1.10.2), whose eigenvalues we do not yet know. It will satisfy


the eigenvalue equation

H\E)m

(1.10.48)

Obviously we cannot solve this problem until we specify the potential V(q).[Note that the eigenvectors and eigenvalues of p and q are independent ofV(q).] Chapter 2 is devoted to a study of (1.10.48) when V(q) = $kq2, wherek is a constant. This potential corresponds to a simple harmonic oscillator.A simpler example is a free particle, in which case V(q) = 0. The eigenvalueproblem for H in this case is extremely simple but we shall postpone itssolution until Section 1.19.

We may express the energy-eigenvalue problem (1.10.48) in the coordinaterepresentation. (The eigenkets |£> are the energy representation.) To do this,we take the scalar product of both sides of (1.10.48) with an eigenbra ofq, (q'\. We then have

We may now use (1.10.46) and (1.10.47) with |y> = \E) (since |y>> was arbi-trary) and write the equation above as

— -TT* + V(q')\{q'\E) = E(q'\E). (1.10.49)2m dq'* J

The solution of this equation gives the transformation function (q'\E)between the energy and coordinate representations. It is alsivthe energyeigenfunction in the q representation or the eigenfunction of q in the energyrepresentation. Equation (1.10.49) is called the time-independent SchrSdingerwave equation. The term (q'\E) is the SchrSdinger wave function, which iswritten as

Equation (1.10.49) is a derived result from the postulates given previously.It does not require a separate postulate in the theory. All the analysis up tothis point assumes that we are looking at the system at a particular time;we shall need another postulate to tell how the system develops in time whenit is left undisturbed (see Section 1.13).

Since the hamiltonian is an observable, we conclude that the eigenkets\E) will form a complete set and satisfy the orthonormality conditions. Wecannot say in advance of specifying V(q) whether the eigenvalues will bediscrete, continuous, or some of both, or whether there will be any de-generacy. Before solving (1.10.49) for a particular V(q'), we interrupt thediscussion to give a physical interpretation of states that are not eigenstates

1.11 MEASUREMENT OF OBSERVABLES 43

of an observable. This involves a probability interpretation of the theoryalluded to in the introduction.

1.11 MEASUREMENT OF OBSERVABLES; PROBABILITYINTERPRETATION

We have given a physical interpretation to eigenstates of observables. Wehave made the assumption that if the system is in the eigenstate \l) of anobservable L, a measurement of L will yield the value /. We have alsoassumed that, when a single measurement of L is made for a system in anarbitrary state |y>>, the result will be one of the eigenvalues of L. The dis-turbance involved in the measurement will cause the system to jump intoone of its eigenstates in an uncontrollable way. We cannot predict which ofthe eigenvalues will be obtained, since the disturbance caused by the inter-action of the system with the observation mechanism will destroy the causalconnection between the measured value and the state of the system before ameasurement was made. We shall now make an assumption that makes itpossible to find the probability of obtaining a given eigenvalue of L when it ismeasured on a system in an arbitrary state \tp).

If we measure an observable L a large number (actually an infinite number)of times, each time with the system in the same state \y>), and average allthese measurements, we shall assume that the average is

(V>\L\y>)( l . l i . l )

All physically realizable states \y>) are represented by vectors of finite norm.We may generalize the assumption (1.11.1) to the case where/is a function

of L and say that the average of/(L) is

(1.11.2)

The quantum averages in (1.11.1) and (1.11.2) are ensemble averages;that is, it is assumed that there are an infinite number of identical quantumsystems, each prepared in an identical way, with no interactions betweenthem. Each system is called an element of the ensemble. Then L, or f(L), ismeasured on each element of the ensemble and the results are averaged. Theaverages are those defined by (1.11.1) and (1.11.2). It should never be over-looked that a quantum average is an ensemble average with every elementof the ensemble in state \y>). If the eigenkets off, form a complete orthonormalset, we have


if the eigenvalues are discrete. If we multiply both sides by a function of L,then

The operator |/>(/| is called a projection operator and (1.11.3) allows anyf(L) to be expressed as a weighted sum of projection operators.

If we use the completeness relation twice, we may write any operator A as

A = 2 \l)(l\A\m)(m\ » (1.11.4)l .m l .m

If we choose the function/(L) such that/(/) = 6ltV, then (1.11.3) reducesto = \nn (1.11.5)

Equation (1.5.22) is an example of the expansion (1.11.5).According to (1.11.2), the average value of f(L) given in (1.11.5) is the

probability P,< of L "having the value /'" when a measurement is made onthe system in state |y>. Therefore, by (1.11.5) and (1.2.8), we have

where we tacitly assumed (y>\y>) = 1. The wave function </'|y>> = y(O inthe L-representation is called the probability amplitude, and |y(OI* is theprobability of obtaining the value /' when a single measurement of L is madeon the system in state \ip).

We now consider the analog of the example above when the operatorL = q and has a continuous spectrum from — oo to + oo. If q\q') = q'\q') and

- q") (1.11.7a)

'ur>«ffl'<«"i, (l . iub)»

we may multiply both sides of (1.11.7b) by f{q) and obtain

/(9) = r MM) WW\ d-11-8)J—00

as the analog of (1.11.3). As a special case, we define the function/^") to be

„ f 1 if q' < q" < q' + dq' ^ JJ a\J W / — |o otherwise

For this special case, the operator/^) in (1.11.8) reduces to

/ («) = | V " W V > <*«•<«"!• O-1 1-1^

LI2 THE HEISENBERG UNCERTAINTY PRINCIPLE 45

The probability that a measurement of q will give a value between q'and q' + dq' when the system is in state \f) is the average value off(q) in(1.11.10) and is

PQ,dq'/V

rf«'. (1.11.11)Again the wave function (q'\f) — y>(q') in the q representation is the proba-bility amplitude, and \y)(q')\2dq' is the probability that a measurement of qon the system in state \y>) will give a value between q' and q' + dq'.

Two probability interpretations may be given to the transformationfunction (/'|m') = (m'|/')* from the L to the Af-representation. We assumethat /' and m' are discrete. First, we may say that |</'|m')|* gives the probabilityof obtaining the value /' when we measure L for a system in state \rri).Alternatively, we may say that it gives the probability of obtaining the valuerri when we measure M for the system in state \V). These two probabilitiesare equal. The term (l'\m') may be called the wave function for the observableM in the L-representation and it may also be called the wave function forL in the Af-representation.

The theory of measurement is important in applications of quantummechanics to measuring electromagnetic fields or detecting signals in com-munication problems. We discuss this in more detail in later chapters andhere refer the reader to the books of von Neumann [9] and Bohm [5] for avery thorough treatment of the theory of measurements. These questions areintimately connected with the uncertainty principle of Heisenberg which wederive in the next section.

1.12 THE HEISENBERG UNCERTAINTY PRINCIPLE

Let us assume an ensemble of identical noninteracting quantum systems,each in state \rp). On half the ensemble, we measure one observable A; onthe other half of the ensemble, we measure another observable B. Each halfof the ensemble has an infinite number of elements. The measurements ofA and B are called simultaneous since the state of each element of the en-semble is the same when A and B are measured.

A measurement of A on one element of the ensemble gives one of the eigen-values of A, say a, and after that measurement, the element of the ensemblejumps from state \y>) to state \a). Similarly, a measurement of B on anotherelement of the ensemble gives an eigenvalue of B, say b, and that ensembleelement jumps to state \b). The probability that a single measurement of Awill give the value a is |(a|y)|2, and the probability that a single measurementof B will give the value b is |(A|y>)P, by the previous section.


The quantum- or ensemble-average value of A and B for all these manymeasurements is

(A) = (y>\A\y>) (B) - (y>\B\y>) (1.12.1)where (y|y) = 1-

The measurements of A and B, in general, have fluctuations about theaverage value of (A) and (B). These fluctuations are not to be thought of asordinary fluctuations because the measuring instruments are not perfect.The latter type of error is assumed to be nonexistent. If we let

the mean-square deviations or fluctuations in A and B that are of quantumorigin are

(AA)* = </**> - (A)* (A5)* - <5*> - (B)\ (1.12.3)

These fluctuations will be zero if and only if the state is an eigenstate of eitherA , B, or both. In fact, this is the way an eigenstate of an observable isdefined: every measurement always gives the same eigenvalue with nofluctuations.

We now suppose that the observables A and B do not commute but satisfythe commutation relation

[A,B] = iC, (1.12.4)

where C is a constant or another observable. We shall show that, in this case,both variables cannot be simultaneously measured with complete precision(i.e., with no fluctuations) and that their mean-square deviations satisfythe inequality

(A^)*(A20*£ iKOI2, (1.12.5)where

(1.12.6)This is called the Heisenberg uncertainty relation.

Before proving this relation, let us discuss its significance briefly. We knowthat \y>) must be an eigenstate of A if we are to obtain a precise value (i.e.,with no fluctuation) when we measure A. Similarly, \ip) must be an eigen-state of B if we are to obtain a precise value when we measure B. Thatmeans that if we measure A and B simultaneously and obtain precise valuesfor both (i.e., eigenvalues for both), the state would have to be a simultaneouseigenstate for both A and B.* This implies that AA = AB = 0, which, by

* There may be some states |y> in which (y|C|y>) = 0, in which case A and B may bemeasured simultaneously for the state |y> although {A, B] — iC 9* 0. An example arisesin the case of angular momentum, to be considered later. If Lx, Ly, and Lt are the componentsof angular momentum, they satisfy [£*,£,] «• ihL, while all three components commutewith L\ the total angular momentum. The state for which Lz\y>) = 0 gives LJy) andLt\y>) different from zero; they may be measured simultaneously.

1.12 THE HEISENBERG UNCERTAINTY PRINCIPLE 47

the uncertainty principle, can be true only if \(C)\ — 0. This is possible ifC = 0, in which case A and B commute, by (1.12.4). Two observables maybe measured simultaneously with complete precision if they commute.

HA and B do not commute but satisfy (1.12.4), the mean-square fluctua-tions, sometimes called the uncertainty in the measurement, satisfy (1.12.5).

To prove (1.12.5), let us define two new variables, a and /?, by

a = A - (A) 0 = B - (B). (1.12.7)Since (A) and (B) axe numbers, it follows from (1.12.4) that a and /? satisfythe commutation relation

[a,/S] = iC. (1.12.8)Since <a> = <fi) = 0, by (1.12.7), we see that

(Aoc)2 = (AA)* = <<x*> (Afl* = (&B)2 = <F). (1.12.9)The product of (Aa)2(Aj3)2 is

(AoOW)* = <Vla2IV>Wlv>. (1.12.10)We may now use the Schwarz inequality: if \f) and \%) are any two kets,

then' \{<P\%)\* £ {9\9)<X\X\ (1.12.11)

where the equality holds if and only if

\<P) - c\X), (1.12.12)where c is a constant (see Problem 1.4). Since A and B are observables, itfollows that a = a* and /S = /9V If we let \x) = 0|y> and \<p) — <x|y>> in(1.12.11), we see that (1.12.10) satisfies the inequality

(Acc)W)*£ Kvla/SIV>P. (1-12.13)

We may always write the identity

xfi = K«/? + /fa) + K«0 ~ H « K«l + fa) + ± C,

where we have used (1.12.8). If we put this into (1.12.13), then

+ P*)\y>) + (1.12.14)

Since tup + /?a as well as C are hennitian, the numbers (y|(a/? + /?a)|y») and(C) = <v|C|y>) are real. Accordingly, (1.12.14) may be written

where (1.12.9) was used. This is the uncertainty principle (1.12.5).

48 D1RAC FORMULATION OF QUANTUM MECHANICS

For the equality to hold, by (1.12.12) and (1.12.14), where \<p) = <x|y> and

\x) = P\v)- cp\y>) (1.12.15a)

0, (1.12.15b)

where c is a constant. If the state \y>) satisfies these relations, the uncertaintyproduct AA AB — £|<C)|, its minimum possible value.

As a special application, we take the operator A — q and B = p for aparticle in one dimension. By (1.10.3), [q,p] = ih, and so (1.12.5) becomesin this special case

H. (1.12.16)

This indicates that if, fo - example, we measure p when the system is in aneigenstate ofp, say \y>) =f | /> , then </>*> = (/>>* and (Ap)2 = 0. By (1.12.16)we conclude that (Ag)2, the mean-square fluctuation for a simultaneousmeasurement of q, is infinite. That is, if we measure/? repeatedly and alwaysobtain p', we know nothing about q on a simultaneous measurement. Thesame argument applies if |y») = \q'), an eigenstate of q. Thus A^ = 0 andso A/; = oo.

These limiting cases in which \y>) is an eigenstate of either p or q are inagreement with the probability interpretation of the theory given in thepreceding section. The probability that a measurement of q will yield a valuebetween q' and q' + dq' when it is known with certainty that a measurementof p will give the value p' is

\{q'\p >|" dq' - (1.12.17)

where the transformation function (1.10.35) was used. This probability isindependent oiq'\ thus it is equally probable that the particle will be foundanywhere from — oo to +oo. That is, if Ap = 0, Aq = oo, in agreementwith the uncertainty principle.

Similarly, the probability that a measurement of the momentum will yielda value between p' and p' + dp' when it is known with certainty that ameasurement of q will give the value q' is

* — (112.18)

by (1.10.35). Again, if we know q, we cannot say what the momentum is.The measurement of q with complete precision causes such a profound dis-turbance on the system that nothing can be known about the momentum.

To summarize, if the system is in a state |y> and a single measurement of/7is made, the result is one of the eigenvalues, p', and the measurement forces

1.12 THE HEISENBERG UNCERTAINTY PRINCIPLE 49

the system into state \p'). This represents a way of "preparing" the systemin state \p'). The probability of obtaining the value /?' when one makes ameasurement of/7 is \(p'\y>)\*dp' when the system is in state |y>>. In general,the system will not be in an eigenstate of/7 or q; thus A/7 and Aq will be finiteand nonzero. It is interesting to inquire what state of the system will givethe minimum uncertainty product for Ap Aq. This state corresponds tolocalizing the particle as precisely as possible in momentum space when it islocated in the region Aq in coordinate space. To find this state, we must solve(1.12.15).

If we put (1.12.15a) and its adjoint into (1.12.15b), we find that

(c + c*)(f\p*\V) = 0.

Since </?2) = (Ap)2 ^ 0 (|y> is not an eigenstate ofp), we conclude that cis pure imaginary. We write it c — — zf, where f is real.

For a = q — (q) and 0 = p — (p), (1.12.15a) becomes

(q - (q))\T) = -Wp ~

If we take the scalar product of both sides with (q'\, an eigenbra of q, and use(1.10.46) and (1.10.47), we may write this as

\ W ~ <9>W) = (j j - , ~

where tp(q') = (q'\y>). This is a simple ordinary first-order differential equa-tion for the wave function f(q') whose solution is

(1.12.19)') = c2 exp ^ (p)q' - ±(q' -

where c2 is a constant of integration. We must still determine $ and c2. Weobtain these by requiring that

<VlV> = f" <Vl«'> dq' (q'\y>> = f" lv<<z')|2 dq' = 1, (1.12.J—00 J—00

20)

where we used the completeness relation. The second requirement is thatthe prescribed mean-square fluctuation in q be

{Aq? = {(q - (q)f)

(1.12.21)


If we use the integrals _

r (1.12.22)

2 s * '

then, when we use (1.12.19) for y>(q') in (1.12.20) and (1.12.21), we find

-*. We chooseFrom these we find that H = 2(A?)2 and |cs|2 =

the phase of ct so that it is real; thus (1.12.19) becomes

This is called a minimum uncertainty wave function in the coordinaterepresentation. Repeated measurements of q for the system in this state givethe average (q) with mean-square fluctuations (A^)2, and repeated measure-ments ofp give an average of (/>). However, the mean-square fluctuation ofp is given by (Ap)2 = #2/4(Ay)2. Since (p), (q), and (A^)2 are arbitrary, thereis a triple infinity of minimum uncertainty states. These minimum uncertaintystates will play a unique role in our later considerations of the measurementof the electromagnetic field.

From the probability interpretation of the theory, the probability that theparticle will be located between q' and q' + dq' when a measurement of qis made is, from (1.12.23),

dq i e xP ~ (1.12.24)

This is a familiar gaussian probability distribution function centered atq' = (q) which has a standard deviation of A^.

Equation (1.12.23) gives the minimum uncertainty state in the coordinaterepresentation. We may use (1.10.45) to express it in the momentumrepresentation as

<P\V> = cxp [ -L h

(1.12.25)

which is the momentum representation of the same minimum uncertaintystate. Again, |9>(/OI* is a gaussian probability distribution function in

1.13 DYNAMICAL BEHAVIOR OF A QUANTUM SYSTEM 51

momentum space centered at p' — (p) with standard deviation A/> =H/2(kq). It is uniquely determined by y>(q"), by (1.10.45).

By (1.10.44) we may visualize y{q') as a superposition of plane wavesexp (ip'q'/H) of wavelength

H2w

P'

h_

P"(1.12.26)

This is the de Broglie wavelength to be associated with a particle of mass mand explains the wave nature of particles in diffraction experiments.

In (1.10.44) <p(p') was visualized as the amplitude of each superposed planewave; for the minimum uncertainty state, these waves interfere constructivelyin a region A/> in momentum space to give \y>(q')\* a large value in a rangeAq whereas they interfere destructively outside this range to make \f(q')\2

small. Thus y>{q') represents a wave packet, and (1.12.23) is a minimum un-certainty wave packet at a fixed time. Use of a wave packet makes it possibleto localize the particle in a limited region of coordinate and momentum spaceso that waves exhibit particle-like character.

It should be noted that the results given in this section are independent ofthe field of force in which a particle is located since the motion of the systemis not yet involved.

1.13 DYNAMICAL BEHAVIOR OF A QUANTUM SYSTEM

As yet, the formulation of quantum mechanics is not complete since wehave not specified how the system behaves dynamically; that is, we have notshown how the state of the system changes in time. The theory up to thispoint has been developed for some fixed time.

When a quantum system is unperturbed by any measurements, the systemdevelops in time in a completely causal manner. It is only the disturbancecaused by the interaction of the measuring device with the system that makesthe behavior cease to be strictly causal.

To give the time development of a quantum system, we postulate theexistence of a hamiltonian H for the system and require that the state vectorfor the system |y>(0) change in accordance with the Schrodinger equation

(1.13.1)</»

where J? is to be treated as an observable of the system and must therefore behermitian.

Two cases may arise. In the first case, the system is conservative and H isexplicitly independent of time. In that case we may formally integrate


(1.13.1) and obtainU(t, to)\y>ito)),

where

U(t, t0) = exp L T ~

(1.13.2)

(1.13.3)

and |y(ro)> is the state of the system at time /0. This solution may be verifiedby differentiation and substitution back into (1.13.1). [The derivative of theoperator U(t, t0) with respect to t is defined exactly like the differentiation ofordinary functions.]

From (1.13.3) it follows that U satisfies the equation

•t &V UTT (\ 13 41in— = HU, \i.i3.i)

dt

while from (1.13.2), at t = h,U must satisfy the initial condition

U{to, to) = L (113.5)Since His hennitian, it follows from (1.13.3) that

V\t, U) = exp [ ' ^ - ^ 1 = V~\t, t0), (1.13.6)L n J

which shows that U is a unitary operator. Therefore, it may be said that thestate of the system at time / develops in a completely causal way from thestate at time r0 by a unitary transformation. Based on a geometric picture ofstate vectors, we may visualize (1.13.2) as a continuous generalized "rotation"of the state vector in ket space from an initial direction at /„ to a final directionat t. Since U-1 = U\ the norm of \y>(t)) is

and does not change. The direction changes but the norm is unchanged.Since any two functions of H commute, it is easy to see from (1.13.3) that

U satisfies the so-called group property

U(t, Q - U(t, tJU(t» tt), (1.13.8)where t > tx > tt.

The adjoint equations are

-ifl-<«<0l = <v(0ltf. <U3-9)dt

since H = H\ and(1.13.10)

1.14 THE SCHRODINGER PICTURE OF QUANTUM MECHANICS 53

The second case arises when H depends explicitly on time. We shalldiscuss this case in more detail in Section 1.16, but we should note that thesolution to (1.13.1) is not given by

= exp [ - - ) dt (1.13.11)

This follows since in general I H(t') dt' does not commute with H(t) so

that when we attempt to differentiate the exponential, the order of factorswould be ambiguous. We should note that just because two operators maycommute at one time does not ensure that they commute at two differenttimes.

When H depends on time explicitly, we may still look for a solution of theform

ly(0) - U(t, to)\v(to)). (1.13.12)

When we put this into (1.13.1), we see that U satisfies

(1.13.13)

Since |y(fo)> is completely arbitrary, it follows that U must satisfy theequation

subject to the initial conditionU«o,to)=l;

(1.13.14)

(1.13.15)

H(t) must still be hermitian. It can be shown that U must be unitary andsatisfy the group property (1.13.8) even when H is time dependent.

1.14 THE SCHRODINGER PICTURE OF QUANTUMMECHANICS

The development of quantum mechanics up to this point has been in theso-called Schrodinger picture. Let us review briefly what this implies for aparticle in one dimension. The observables (p, q, and H) were taken as her-mitian operators and were time-independent. A subscript S indicates thatoperators and vectors are in the Schrodinger picture. Thus we write Ps>4s>H8. The eigenvectors of these operators are written \p')s, \q')g, |£>sJ any ofthese may be taken as stationary (time-independent) basis vectors to representoperators or state vectors. In fact, in the SchrSdinger picture state vectorsare stationary and act like a fixed coordinate system in ordinary geometry.


At a fixed instant of time, any state is represented by a linear superpositionof a set of stationary basis vectors.

The operators obey the communication relations [qa,p8] = '*» an<*[?s» <Js\ = iPs'Psl = 0 in the Schrodinger picture.

The state vector describing the dynamical behavior of the system as afunction of time is |ys(0>- In the SchrSdinger picture it moves continuouslyaccording to the Schrodinger equation from an initial direction \y>(t0)) to afinal direction \y>8(t)) at time /.

If f(p8tq8) is a function of p8 and q8, then the quantum (ensemble)average at time t is

<f(Ps>If the system is in state |yOo)> at time /„, the state at time t, by (1.13.2), is

IVs(O> = U(t, to)\y>(to)), and the probability that the system will be in somefixed state <«| at time tt is

U to)\y>(toW- (1-14.2)

This description in which the basis vectors are stationary and the dynamicalstate vector \y>s(t)) moves is called the Schrodinger picture of quantummechanics and is the picture used up to now.

1.15 THE HEISENBERG PICTURE

In the Schrodinger picture the basis vectors (any eigenvectors of observ-ables) are visualized as a fixed set of vectors and the state vector as moving.The same system can be described equally well by letting the basis vectorsmove and the state vector remain stationary, as in classical mechanics.This mode of formulating quantum mechanics is physically equivalent to theSchrodinger picture and is called the Heisenberg picture. It is clear that, ifoperators are stationary (time-independent) in the Schrodinger picture, theymust be time-dependent in the Heisenberg picture in order that the twodescriptions be physically equivalent.

The state vectors in the two pictures are related by definition by (1.13.2)

\Va(0) = U«, Q\vH(to))> (1-15.1)

where the subscript H designates the Heisenberg picture. The vector \yH(/o)>is stationary while \y>8(0) is moving. Since U(t0, to) = l, the state vectorscoincide at t — t0 in the two pictures.

The average value of an operator A8 is [compare with (1.14.1)]

(1.15.2)

1.15 THE HEISENBERG PICTURE 55

where we have used (1.15.1) and its adjoint. We define the operator in theHeisenberg picture by

AB{t) = U*(t, to)AsU(t, t0). (1.15.3)

From this definition, we see that operators that are stationary in the SchrS-dinger picture usually depend on time in the Heisenberg picture. The trans-formation (1.15.3) is called a similarity transformation when IP = U~l.With this definition, the average values of A at time / may be written

(A) = (y>B(t0)\AB(t)\y>H(t0)). (1.15.4)

The transformation law (1.15.3) is necessary in order that the average of Abe the same in both pictures, making the two physically equivalent.

From the transformation law (1.15.3), which is valid even if As has anexplicit time dependence, the equation of motion for an observable in theHeisenberg picture may be obtained. If we differentiate both sides of (1.15.3)with respect to t and use (1.13.14), its adjoint, and the fact that WU =UlP = 1, we obtain

dt

= uUsUtfHU - tfHUUUsU ihtf ^ ^ Udt

(1.15.5)

where we used (1.15.3) and define

HBif) = lP{t, to)Hs(t)U(t, t0), (1.15.6)

which is the hamiltonian in the Heisenberg picture. Equation (1.15.5) is calledthe Heisenberg equation of motion for the observable A. If dAB\dt — 0,then AH is a constant of the motion.

As a special case of (1.15.5), we let A8 = H8. If the system is conservative,dHsjdt = 0, and V(t, t0) is given by (1.13.3). Since in this case [H8, U] = 0,we see by (1.15.6) that HH = H8. By (1.15.5), we then have

dt0, (1.15.7)

which shows that H is a constant of the motion.If A8 has no explicit time dependence and the system is conservative,

(1.15.5) reduces to

-[AB,H].ihdt ih

In this case, if AB commutes with H, AB is a constant of the motion.

(1.15.8)


Another important theorem is that commutation relations have the sameform in the two pictures. This again is obviously necessary if the two picturesare to be physically equivalent. As an example, we let A8, Bs, and C8 bethree observables in the Schrodinger picture that satisfy

[A8, B8] - iC8 (1.15.9)

If we multiply both sides from the left by W and both sides from the rightby U, we have

U*A8B8U - WB8A8U = ilPCsU,

or, inserting UU* = 1 between the As and Bs, we have

{WAsU){lPB8U) - (ir>B8U)(U*A8U) = ilPC8U,which, by (1.15.3), becomes

(1.15.10)

This is the same form as (1.15.9). In short, the Heisenberg and Schrodingerpictures are physically equivalent.

If AH is a constant of the motion, it must commute with the hamiltonian.Its eigenvalues are the same in both the Schrodinger and Heisenberg picturesand are then said to be good quantum numbers.

The Heisenberg picture is extremely useful in demonstrating the formalanalogy between a quantum system and its classical analog. We shall con-sider the example of a particle in one dimension. We shall show that in thiscase the Heisenberg-operator equations of motion are identical in form withthe classical hamiltonian equations of motion. For any system having aclassical analog, this correspondence permits a check on the validity of thetheory.

Let us consider the hamiltonian (1.10.2), which is a function of p and q.By (1.15.10) we have shown that (1.10.6) are valid in the Heisenberg picture,and so the equations of motion (1.15.5) for AB = qH orpH are, by (1.10.6),

dt ih

dHH

dpH(1.15.11)

dt

since dq8jdt = dp8fdt = 0.

1 .16 THE INTERACTION PICTURE 57

(1.15.12)

These are identical in form with the classical equations of motion inhamiltonian form:

dq_dHdt ~ dpdp= _dH_dt ~ dq'

Aside from questions of ordering of operators in (1.15.11), the classicalequations correspond to the quantum equations.

In classical mechanics, if A is a function ofp, q, and t, then A satisfies theequation of motion

dA = dAdq dAdp dAdt dq dt dp dt dt

^dAdH _dAdH_ dAdq dp dp dq dt

(1.15.13)

where we used (1.15.12). The classical Poisson bracket is defined by

so that (1.15.13) may be written

dA

dp dp dq(1.15.14)

(1.15.15)

If we compare (1.15.15) with (1.15.5), we see that we may shift from classicalmechanics to quantum mechanics by replacing the Poisson bracket by(th)'1 times the commutator bracket, that is

in(1.15.16)

It may be shown that the algebra of commutators and Poisson brackets isthe same (see Problem 1.8). This gives a clue to the motivation for thequantization postulate (1.10.3). If a system has no quantum analog, thequantization rules practically are a matter of pure intuition. The only checkis comparison with experiment (as it always is).

1.16 THE INTERACTION PICTURE; TIME-DEPENDENTPERTURBATION THEORY, DYSON TIME ORDERINGOPERATOR

There is another picture besides the two discussed in the previous sectionswhich is especially useful when the hamiltonian may be written as the sum


of two terms of the formHS mm Ho

8 + V8. (1.16.1)

We shall assume Ho8 is independent of time but Vs may depend explicitly

on time although it need not. We call this the interaction picture which isdefined by the unitary transformation U0(t, t0) from the Schrddinger pictureby means of

) (1.16.2)

where the subscript I refers to the interaction picture and Uo satisfies theequation

ihdt

with

U0(t, t0) - exp [ - i H98(t

Ul = IV*

(1.16,3)

(1.16.4)

(1.16.5)

(1-16.6)

We are assuming that the Schrddinger equation when Vs = 0 may be solvedexplicitly.

The Schrodinger equation is

dt


[Ho8 + V8]\y>8). (1.16.7)

sdt

If we use (1.16.3), the first terms on the left and right cancel. If we thenmultiply both sides from the left by C/J = Uo-

1, we obtain

ih-

where we have letdt

vz(t) = ulv8u0.

(1.16.8)

(1.16.9)

Equation (1.16.8) is the Schrddinger equation for the state vector in the inter-action picture (IP). Even if Vs is not explicitly time dependent, we see by(1.16.9) that Vj(t) will usually depend on time.

Before proceeding to obtain a formal solution of (1.16.8) let us see howoperators transform under the unitary transformation (1.16.2). The average

1.16 THE INTERACTION PICTURE 59

value of A8 is<A) = {ipsimslysit))

= (v>A.t)\VlA8U0\y>j(t)), (1.16.10)where we used (1.16.2) and its adjoint.

We therefore see that it is natural to define the operator in the IP as

Aj(t) == UlAgV0. (1.16.11)Thus (1.16.9) gives the interaction energy V8 in the IP. Thus

(A) = <?jr(OMi(OIV/(')>. (1.16.12)

We may obtain the equation of motion for Az{t) if we differentiate bothsides of (1.16.11) with respect to / and use (1.16.3) and its adjoint. We have

at

UU8HO8UO -Uf

0H08AsU0 +

But since Hos is time independent,

[Hos,t/o] = 0-^Ho

ff =

thus when we use (1.16.11), (1.16.13) becomes

ih^f = [AIt Hoz] + utiH

at

ih Uo. (1.16.13)dt

(1.16.14)

dt(1.16.15)

Note that if Vs = 0, (1.16.15) becomes simply the Heisenberg equation ofmotion.

Let us return to (1.16.8) and look for a solution of the form

\Vz(t)) = U(t, to)\y>8(to)), (1.16.16)

where by (1.16.2) and (1.16.6)

IVz('o)> = lys('o)>. (1.16.17)

That is, the two pictures coincide at t = t0. If we substitute (1.16.16) into(1.16.8) and note that \y>8(t0)) is arbitrary, U satisfies

(1.16.18)


or on integrating both sides we have

U(t, t0) = 1 + - f *V^f)U(f, h) dt', (1.16.19)ihJu

where£/(/*, t9) = 1. (1.16.20)

If we let t = /' and the dummy integration variable t' -*• t", we may rewritethis as

U(t\ to) = l + - f *Vi(f)U(f, t0) df. (1.16.21)ih Ju

If we substitute this into the integrand in (1.16.19), we obtain

U(t, *0) = 1 + — I Vj(tJ dtx + I —) f dtx Vftd *dtt Vj(tJU(ti, t0).ih Jr% \inf Ju Ju

We may proceed indefinitely with iteration and obtain00 / 1 \n f * f *x r*«-»

(1.16J23)

If the interaction energy Vz is small compared with Ho this series convergesrapidly and we have a power series solution in the perturbation Vv By(1.16.2) and (1.16.16) we have

= U0(t, to)U{t, to)\tp(to)). (1.16.24)

(1.16.22)

Let us assume that in the absence of the perturbation (V8 = 0), we maysolve the energy eigenvalue problem

H08\n) = £n°|n>,

so that for any function of Hos

/ (H os ) |n>=/ (£ B ° ) |n ) ,

where

(n\m) = dnm I | n ) < n | = l .

We may then expand |y>/(0> as

(1.16.25)

(1.16.26)

(1.16.27)

= Icn(t)\n), (1.16.28)

1.16 THE INTERACTION PICTURE

where we have let

61

cB(0 = <»|v/(0> (1.16.29a)

Therefore, by (1.16.2), (1.16.4), and (1.16.26) we have

n(t) exp [ - l- En\t - t^ |«>. (1.16.29b)

From the orthogonality relations, we have

while

exp [- i (1.16.30)

(1.16.31)is the probability of finding the system in state \m) at time t if we measure itsenergy. We used (1.16.29). If we use (1.16.16), we obtain

since by (1.4.6)= (m\U(t, g i f C O X v C t ) ! ^ , 'o)|m) (1.16.32)

(l\A\r)* = (r\A1\l). (1.16.33)

If we assume at t0 we made a measurement and determined that the systemwas in the energy eigenstate |y(fo)> — 10. when we use (1.16.23), (1.16.32)becomes

|c(m, t\i, to)\*00 / 1 v f * cti r*—1

, (1.16.34)

which gives the probability of finding the system in state \m) at time * giventhat it was in the state |/) at time /0.

In zeroth order in the interaction energy, we have

, t\i, /0) |2 = 6mi. (1.16.35)

To this approximation the system remains in its initial state. In first orderwe have

|cU)(m,r|«,f0)|2 = f I fd

1«•

fdr1Ju

f '< JIO (1.16.36)


where we used (1.16.9), (1.16.4), and (1.16.26) and have let

Wmt = Em ~Ei . (1.16.37)h

If V8 is explicitly time-independent, we may carry out the integral and(1.16.36) reduces to

If V8 is sinusoidal

(1.16.36) becomes

V8=V8i)sm((0t-<p), (1.16.39)

- 1] (1.16.40)

The factor 4 sin* £eom,(f - ro)/a)m<* in (1.16.38) is a very strongly peakedfunction of comi (see Figure 1.4). At comi = 0 its amplitude increases as(t — t9)* and decreases to zero when comi == 2ir/(f — t0). The probabilitythat Vs induces the system to make a transition between state |i) and state\m) is thus very small unless energy is conserved between the initial and finalstates. Energy conservation was not put in the theory in an ad hoc mannerbut is a derived result.

If hcomi = JEU' — E\o) > 0, the second factor in (1.16.40) is large comparedwith the first when comi s* <o. Then we have

d.16.41)

mi

Figure 1.4 Probability of finding system in state \m > at time t given it is in state |/ >initially as a function of (£«,•' - fW

1.16 THE INTERACTION PICTURE 63

which again is peaked at hwmi = hco. In this case the final energy is increasedby Hco. If (omi < 0, the first term is important and h<omi = —hco, so that thefinal energy is reduced by Hco. We shall see in Chapter 5 that these may corre-spond to the emission and absorption of a photon of electromagnetic energyby an atom.

So far we have assumed that the energy levels of the system are infinitelysharp. We show in a later chapter that they must always have some line-width. Alternately, in (1.16.39) we assumed the interaction was perfectlymonochromatic whereas in practice there will always be some bandwidthassociated with any sinusoidal perturbation. Accordingly, we shall assumethat there are a number of closely spaced energy levels which are distributedwith a density g(Em) dEm between Em and Em + dEm. We may obtain thetotal transition probability to any of these levels for which (1.16.38) isapproximately valid by summing

to*mi

(1.16.42)

Because the levels are closely spaced, we have converted the sum to an integral.Because the integrand is strongly peaked at comi 0, we may remove theslowly varying factors g(Em) and |(m|Fg |/)|2 from the integral and evaluatethem at Em = Et and extend the limits from — oo to + oo. Since

• dx = 2-na,

and dEm = H dcomi, we have

«,<Mo)l £ £ T

(1.16.43)

- '«)• (1-16.44)

We may therefore replace the highly peaked factor sin2 \ax\x* by a ^-functionin (1.16.41) and write the transition probability per second from state i tostate m as

= 7 |c(1)(m, t\i, to)\z = ^ Km\Vs\i)\*at n

)• (1-16.45)

When we sum over a range of final states, the transition probability persecond is

j 2 |c(1)(m, t\i, to)\* = ?fat m H

(1-16.46)

64 DIRAC FORMULATION OF QUANTUM MECHANICS 1.16 THE INTERACTION PICTURE 65

which is called Fermi's golden rule. Only transitions in which energy isapproximately conserved (Em ^ E{) contribute. This is one of the most usedresults of time-dependent perturbation theory.

In some instances the matrix element between the initial and final statesvanish but they do not vanish for some other state |/>. In this case, thetransition may take place in second order. If we insert a completeness relationin the /I = 2 term in (1.16.34) we obtain

\cl2\m, t\i, fo)|*frjtt Jtt r

If Vs is time-independent, this reduces to(1.16.47)

couCOmi

(1.16.48)

In this case the transition from |/) to \m) takes place through the intermediatestates |/> which have nonzero matrix elements.

The first term is large in (1.16.48) when <ami = 0, that is, when energybetween initial and final states is conserved. The second term is large whencomls^0 which need not conserve energy between initial and intermediatestates. It arises from turning on the interaction suddenly at t = t0 and isnot usually important.

We next introduce the Dyson time-ordering operator which allows us towrite the perturbation expansion for U(t, t0) in (1.16.23) in extremely compactform.

We notice first that in (1.16.23) that t0 < tn < t^ < • • • < f2 < f, < t.Furthermore, the interaction operators F/fo) - • • Vx(tn) are time-orderedin the sense that the operator at the earnest time is on the right, followed bythe operator with the next later time, and so on. Consider now two non-commuting operators Afa) and B(tJ. The time-ordering operator whenapplied to AB is defined by

W'tMfc)

if

if(1.16.49)

That is, when operators are written inside the P-operator, we treat them asc-numbers since

(1.16.50)

We then positional order them so that the earlier time is on the right, andthey again become operators. In Chapter 3 we gain more experience with the

ordering concept and technique. We might put bars over the A and B insidethe P-operator to remind us they commute:

If the operators are evaluated at the same time, the P-operator does nothing.When many operators are involved, P is generalized so that the operators

are positional ordered from right to left with tfie earliest operator on theright, the next later operator to its left and so on. In (1.16.23) since theoperators are already time-ordered, we have

J - • • Vj(tn). (1.16.52)

We next show that (1.16.23) may be written as

U(t, t0) = pjexp - |

H-(1.16.53)

We shall proceed to show that this is identical with (1.16.23) up to then = 2 term. The remaining can be shown by induction which we leave asan exercise for the reader.

The n = 0 term is obviously identical. For then = I term we have

Since only one time is involved, the P-operator does nothing and this is thesame as the n = 1 term in (1.16.23). For the n — 2 term we have

where the region of integration is shaded in Figure 1.5. We may thereforebreak this up into two integrals over the two separate triangular areas so that


-» t.Figure 1.5 Region of integration in (1.16.54).

where in the first integral we integrated over tz first and then tx and did thereverse in the second. In the first integral we see that tz < fa so that

*{ Witt)} = Vi

while in the second integral t% < tz so that

*{ Witt)} = v-so that / becomes

and the F's are now positional ordered and their operator character fullyrestored. We now note that in the second integral, we could change variablesand call t2 anything we want, say x or even tu and call tx anything else, sayy or tt, without changing the value of these definite integrals. Therefore,

so that both integrals are equal and / is then identical with the n = 2 termin (1.16.23). We therefore see that the P-operator allows us to write U in thevery compact form

V{t, t0) = Pjexp - i jj O dt'j.

We therefore have

(1.16.55)

- l- Ho(t - O ] P { « P ~ Lh J V I ( O dt j|V>H('O)>- (1.16.56)

Finally, for future reference we should like to obtain explicit equations of

1.16 THE INTERACTION PICTURE

motion for the expansion coefficients cjt) in (1.16.29). We have67

(1.16.57)

where we used (1.16.16) and (1.16.18). If we insert a completeness relationand use

this becomes(1.16.58)

rfc

f), (1.16.59)

where we again used (1.16.29) and (1.16.16). If we expand c.(t) in powers ofV, we have

c,(0 = ciO)(0 + c?\t) + cJtt(O + • • •. (1.16.60)

If we use this on both sides of (1.16.59) and equate equal powers of V weobtain

ir 00

~ = Yn

(1.16.61a)

(1.16.61b)

(1.16.61c)

Integrating (1.16.61a) we have

If we put this into (1.16.61b), we have

(1.16,62a)

(1.16.62b)


If we put this in (1.16.61c), we have

<42>(0= (T-JZ f*i<«i»yi>/—"»-'•»f A,(/iFyo^""*-*' . (i.i6.62c)

and so on. These are seen to be in agreement with the prior results.

1.17 PERTURBATION THEORY FOR A HEISENBERG OPERATOR

If a system is described by a hamiltonian

H=H0+V, (1.17.1)

and M is an arbitrary operator which is explicitly time-independent in theSchrddinger picture, then M obeys the Heisenberg equation of motion(1.15.5)

ihdt

[MB, HB] - [MB, H0B + VB)t (1.17.2)

where all operators are in the Heisenberg picture. If we consider only time-independent hamiltonians, it is evident that the total hamiltonian in theHeisenberg and Schrodinger pictures are equal. This is seen as follows.For H time-independent, if we let MB =* HB, it follows from (1.17.2) that

dt ih(1.17.3)

so that HB(t) is time-independent. Since at t — t0, both the Heisenberg andSchrddinger pictures coincide, we have HB — H8. It is not necessarily truethat H0

B{t) equals Hos since by (1.15.6)

H B(t) as

7~ " 0

unless [H8, V8] = 0, an uninteresting case. Similarly, VB(t) j* Vs but+ Vf = (Ho + Vf. We may therefore write (1.17.2) as

(1.17.4)dt ihL" v

Next we make the change of variable

MB(t) = Ufa, to)X(t)Uo(t, Q, (1.17.5)where

U0(t, t9) = e-umBt'lt-t'). (1.17.6)

If we differentiate (1.17.5) with respect to /and use (1.17.6) and its adjoint,

1.17 PERTURBATION THEORY FOR A HEISENBERG OPERATOR

we find

69

fdt dt dt dt

dtih

\[M(t),H0] + UlÛ0.in at

If we compare this with (1.17.4), we have on using (1.17.5)

= \[UtXU0,Vs].

in

(1.17.7)

(1.17.8)

If we multiply from the left on both sides by Uo and from the right by C/Jand use the fact that U9Ul = UlUv = 1, we have

(1.17.9)at

Note from (1.16.9) that

V\t - t0) = Ul(t, to)V8Uo(t, t0), (1.17.10)

when Uo is given by (1.17.6). We easily see that

Vz(t0 - 0 = U0(t, to)VsUl(t, t0). (1.17.11)

With this notation we may integrate both sides of (1.17.9) to obtain

X(t) = X(t0) + ± (\x(f). V\t0 - t')] dt', (1.17.12)in Jto _

where by (1.17.5)X(t0) = MB(t0) a Afs. (1.17.13)

We proceed to iterate (1.17.12) in the familiar way to obtain

X(t) ± \\td8, V\to - h)] dtxinJu

• (1-17.14)


We next use (1.17.5) to transform X back to the Heisenberg operator:

MH(t) = Ul(t,t0)M8U0(ttt0)

+ 71 f W , *o)[M8, K'(r0 - t,)]U0(t, *0) dtx + • • •. (1.17.15)

If we use (1.16.11) and (1.17.11) and insert U0Ul = U\Un — 1 in appropriateplaces, this becomes

MB(t) = M'(t) + 1 f\M\t), V\t - tj] dtxiHJu

+ ( ^ J J } * ' 1 / , / ' 2 « M J ( 0 . v'{t - U)l vl(t -tl)} + ---.(1.17.16)

which is the desired perturbation expansion of a Heisenberg operator.

1.18 WAVE MECHANICS

We have mentioned wave functions so often in the course of this develop-ment that it is redundant to say much about wave mechanics except toreemphasize the concepts. Wave mechanics is quantum mechanics formulatedin the Schrodinger picture in the position or momentum representation. Weshall summarize some of the results for a particle in one dimension.

The coordinate representation is determined by the eigenvalue equation(all in the Schrodinger picture)

q\q')=q'W)- (1.18.1)'The orthonormality relation

(q"\q') = d(q'-q") (1.18.2)

may be regarded as the coordinate wave function in the coordinate representa-tion. It is also the representative of the eigenket \q') in the q representation.The set {\q')} is complete, and so

1.19 THE FREE PARTICLE 71

£ \q')dq'(q'\ = L0

The state vectors |y>(0> satisfy the equation of motion

dt

The representative of \y>(t)) in the q representation is

(1.18.3)

(1.18.4)

(1.18.5)

and is the Schrodinger wave function at time t for state |y(0)- If we takethe scalar product of (1.18.4) with (q'\ and use (1.18.5), (1.10.46), and(1.10.47), we have the time-dependent Schrodinger wave equation

We could take d/dt from (q'\dldt\y>(t)) since (q'\ is stationary in the SchrS-dinger picture.

The average value of F{p,q) at time t is, by (1.11.1),

+00

' (q'\F(p, q)\q") dq"

^ £. t qj rtf, t) dq', (1.18.7)

where we have used the completeness relation (1.18.3) twice, as well as(1.10.41), (1.10.42), and (1.18.2). The order of factors in F(p, q) must bestrictly preserved.

The eigenket \p') ofp in the ^-representation is, by (1.10.35),

v> exp (1.18.8)

This is the momentum eigenfunction in the coordinate representation. Bymeans of this transformation function, the Schrodinger picture may beexpressed in the momentum representation. From the previous work this isstraightforward.

1.19 THE FREE PARTICLE; CHANGE IN TIME OFMINIMUM UNCERTAINTY WAVE PACKET

In this section, we consider a free particle in which the potential V(q) = 0and the hamiltonian becomes

H = -*- = H'. (1.19.1)2m

Since the system is conservative, His a constant of the motion. Also,

[p,H]*=0, (1.19.2)

and so p is a constant of the motion for a free particle.


The operator p satisfies the eigenvalue problem

P\P')=P'\P'), (1-19.3)

where — oo </>' < +00. If we operate on both sides of (1.19.3) with H,we have

Hp\p') = PH\p') = p'HIp'), (1.19.4)

since H and/? commute. We therefore see that H\p') is an eigenket of p witheigenvalue/?'. Thus Hip') can differ from \p') only by a constant factor whichwe call E, and so we may write

Hip') - E\p'). (1.19.5)

This is just the energy-eigenvalue problem. Since p and H commute, theeigenket of p may be a simultaneous eigenket of H. Thus we may write(1.19.5) as

Hlp',E) = E\p',E).By (1.19.1) and (1.19.3),

2m\p',E) pi

2m\p\ E) - E\p', E),

(1.19.6)

(1.19.7)

or the eigenvalues of H are

£ =-Pi2m

(1.19.8)

which are continuous and have any value from 0 to +00. Therefore, thesolution of the energy-eigenvalue problem is very simple for a free particle.We note, however, that the eigenvalues +p' and —p' give the same E;that is, the states \+p',E) and |—p',E), which correspond to a particlemoving to the right and to the left, have the same energy. This is an exampleof degeneracy.

The solution of the Schrodinger equation

ihdt

is, by (1.13.3),

In the coordinate representation we have

= (q'\ exp ( -

(1.19.9)

(1.19.10)

(1.19.11)

1.19 THE FREE PARTICLE 73

for the time development of the wave function. If we use the completenessrelation for \p'), this may be written as

where we used (1.10.35) and let

</%(0)> = ?(/>', 0). (1.19.13)

This gives the wave function in the coordinate representation when ?(/>', 0)is the initial arbitrary wave function in the ^-representation.

As a special case of (1.19.12), we let <p(p', 0) be a minimum uncertaintywave packet given by (1.12.25). As time develops, the wave packet for afree particle is given by (1.19.12):

(1.19.14)

For simplicity, we let (p)t-o — (?>*-o = 0- If we replace (A/?we may easily carry out the integral to obtain

where4(&qf + 2ihtlmY ( L 1 9 - 1

f" q'^yKq', 0)|8 dq\J—00

If we calculate the average value of q at time t, we have, by (1.18.7) and(1.19.15),

<«>* = I"" IV(«', t)W dq' = 0. (1.19.16)J-00

That is, the particle stays at q' = 0, on an average, during the course of

DIRAC FORMULATION OF QUANTUM MECHANICS74

time. However, at time t

\y<q\t)Utdq' = [Aq}l0+ " ' , (1.19.17)

so that the mean-square deviation at time / is greater than its value at t — 0by an amount proportional to r2. As time goes on, the packet spreads incoordinate space. The smaller A^ is at t = 0, the faster the packet spreads.

It is also seen that the average of the momentum at time t is, by (1.18.7)and (1.19.15),

(P)< - I"" v V , 0 T djErr) W - °» (119-18>J-oa I Oq

while the mean-square deviation of p is found directly to be

= -**1""J—c

*(q', 0 oq', t) dq'

(1.19.19)

Therefore, the uncertainty product at time / is, by (1.19.17) and (1.19.19),

f„ IV £ 7 • (1-19.20)4f]f.J 4

7 + L rf„4 L4m[A4f]f

Thus it may be concluded that, as time goes on, the minimum uncertaintywave packet ceases to be a minimum uncertainty packet. When we studyelectromagnetic fields, we shall attribute this spread to the zero-pointfluctuations of the field.

1.20 THE DENSITY OPERATOR [9-13]; PERTURBATIONTHEORY

Let M be an arbitrary operator in the SchrSdinger picture (SP). If lws(0>is the state vector at time / in the SP, then we have seen that the averagevalue of M is given by

(M) = (y>8(t)\Ms\ip8(t)). (1.20.1a)

We have also seen that we may obtain the same mean if we work in theHeisenberg picture (HP), namely,

(1.20.1b)

1.20 THE DENSITY OPERATOR [9-13]; PERTURBATION THEORY

where IVfl-('o)) gives the state of the system at t0 and

75

(1.20.2)where HT is the total hamiltonian in the SP.

We next note thatTr (|u)<0|) - <p|«>. (1.20.3)

This may be proved as follows. If {|n>} represents any complete set of statevectors, then in this representation, we may write the trace as the sum of thediagonal matrix elements

Tr (|H><P|) = 2 <»l«X»|n>. (1.20.4)n

Since (n\u) and (v\n) are simply numbers, we may rewrite this as

From the completeness relation,

the trace above reduces to (1.20.3).If we next identify M\ip) in (1.20.1) with |«) and (f\ with (»|, we may use

(1.20.3) to rewrite (1.20.1) as

TrMs\y>s(t))(y>s(t)\TT MH(t)\y>H(t0)XipH(t0)\. (1.20.5)

Up to this point we have tacitly assumed that we have made sufficientmeasurements to determine that the system is in the state \y>B(t0)) initially.In many cases it may be impractical to make enough measurements to dothis. For example, if our system consists of 1023 gas molecules in a container,we could not hope to determine exactly \y>(t0)). Suppose we only know thatit is in such a state with a probability p v where

2 ^ = 1; JVÔ. (1.20.6)

We should then average (1.20.5) over this probability distribution so that themean value of M becomes

«Af»

(1.20.7)

The first averaging was due to the statistical interpretation inherent inquantum mechanics (see Section Ll l ) whereas the second averaging is of a


classical nature and would be necessary if we were treating the systemclassically. In the event we can make enough measurements to determinethat the system is in state \y>(t0)), then

P* = < W <12a8>and (1.20.7) reduces to (1.20.5).

We may next define the density operator in the SP and HP, respectively, as

(1.20.9)

(1.20.10)

= Path) - 2JVlVff('o)><VffOo)l>

so that (1.20.7) becomes

= Tr MsP8(t)

= Tr AfH(f) pB(t0),

since we may interchange the averaging (summing) over y> and tracing (alsosumming) operations.

We may easily obtain the equation of motion for ps(t) from its definitionand the Schrodinger equation

ot

aifi -

ot

(1.20.11)= (y>8\H8.

For from (1.20.9) and (1.20.11), we have (pv is time-independent)

> Ps\>

since pv is a number and therefore commutes with H.From (1.20.9), we see that p is hennitian so that

In addition, we have that

Tr p = 2 /V

where we used (1.20.3) and (1.20.6).

2V

1.

(1.20.12)

(1.20.13)

(1.20.14)

1.20 THE DENSITY OPERATOR P-13]; PERTURBATION THEORY

We next note that if \x) is any ket then

77

<X\p\X> = IPV(X\V>XV>\X) = 2 JV 0. (1.20.15)V V

That is, the diagonal matrix elements of p are all real and positive in any rep-resentation. Since the sum of the diagonal elements equals 1 by (1.20.14) inany representation, it follows that the diagonal elements always lie between0 and 1 :

0 <, (X\P\X) ^ 1. (1.20.16)

If at t0 we know the state of the system, then (1.20.8) is satisfied, and preduces to |y(0Xv(0l- We then have

= P,

since <y|y) — 1; therefore,Trp 2 (1.20.17)

When this condition is satisfied we say the ensemble is in a pure state.For a mixed state, we do not know the state at t0 precisely and we have

T r ^ 2 < l . (1.20.18)

To show that Tr/>2 ^ 1, we note that because p is hermitian, it may bediagonalized by a unitary transformation

whereP = SPS\ (1.20.19)

5 f = S-\ (1.20.20)

Since a trace is invariant under a unitary transformation we have that

Tr p* = Tr p" = 2 {p'S < ( 2 ^ T = 1. (1-20.21)

This follows since p'* is diagonal when p' is and p'n are the diagonal elements,which by (1.20.16) lie between 0 and 1. Since for a pure state Tr p2 = 1,(1.20.18) corresponds to a mixed state.

We next rederive our time dependent perturbation theory results using thedensity operator. Assume our hamiltonian may be written as the sum

H = H0+V, (1.20.22)

where we assume we have solved the energy eigenvalue problem

H0|n> = E™\n), (1.20.23)


when the set {pi)} forms a complete orthonormal set. Then (1.20.12) becomes

(1.20.24)

(1.20.25a)

(1.20.25b)

Ot

We shall transform to the IP (interaction picture) if we let

Ps(t) = Uo(t,io)Pl(t)Vl(t,to),where

(7o(r,ro) = exp[-|ifos(t-to)],

when Ho8 is time-dependent.

If we next substitute (1.20.25) into (1.20.24) we obtain

ih^ PIUO + UoPlih^ + U0(ih~fj u\ = [Ho8 + V8, UoPlUl}.

(1.20.26)

If we use (1.20.25b) and its adjoint, we see this may be written as

[Ho8, UoPlUo] + V0(iH^Vl = [Ho8 + V8, UoPlUl],

and we see that the commutator on the left cancels the first one on the right.This leaves

Uo(ifi^) Vl = V8VoPlVl - VoPlVoVs. (1.20.27)

Since Ul = U'1, if we multiply both sides from the left by C/J and both sidesfrom the right by Uo, we obtain the equation of motion for the densityoperator in the IP, namely,

iH ^ *" [V^ ~ to)> p M l (1.20.28)

where [cf. (1.16.11)]

Vft - to) - Vl(t, to)V8Vo(t, to) (1.20.29)

is the interaction in the IP. We have not restricted V8 to be time independentin the SP.

Note that since U0(t0, t0) = 1, it follows from (1.20.25) that

= P/('o). (1.20.30)

In general, (1.20.28) cannot be solved exactly. When the interaction issmall, we may obtain an approximate solution in powers of V which will

1.20 THE DENSITY OPERATOR [9-13]; PERTURBATION THEORY

converge rapidly. On iteration we obtain

PJ(0 - P(to) + 71 f W « i - 'o), f<to)] dtt

79

- to), [V,!itt - Q, fa))] + ••-. ( 1 . 2 0 . 3 1 )

We now show that under special circumstances, we may obtain the resultsgiven in Section 1.16 of time-dependent perturbation theory, but the resultshere are more general than that presented in Section 1.16. [Why?]

From (1.16.2) we have that

while by (1.20.25) we have

and by (1.20.9), we have- U0(t, t,)Pl(t)Ul(t, ro),

If we use (1.20.32) and its adjoint, this may be written as

(1.20.32)

(1.20.33)

(1.20.34)

(1.20.35)

(1.20.36)

where pv is the probability the system is in state |y(f0)) at time t0.If we next take a diagonal matrix element of both sides of (1.20.36) in the

representation (1.20.23) we obtain

(m\Pl\m) =

- U^t, t0) 2 Pvlyz(0><Vi(0l WLU to).V

If we compare (1.20.33) with this we see that in the I.P.

= I JVIVz(0><V/(0l,

c.(0l", (1.20.37)

where the last step follows from (1.16.31). If we use (1.20.31), we have

71indtt

(1.20.38)


If we next assume that at t0 the system is in some unperturbed initial energyeigenstate \i) where i 9* m, then

pv = K.t> (1.20.39)and

(1.20.40)

By the orthogonality relations, the first two terms on the right in (1.20.38)vanish, and we have

. (1.20.41)X (mllWi - t0),

If we use (1.20.29) and the fact that

U0\m) = exp [ - -h £»»(« - »,)] |m>, (1.20.42)

(1.20.41) reduces to

to

)} + ••-, (1.20.43)

where we not only have used (1.20.29), but we have also allowed V8 to beexplicitly time-dependent and have defined coim by

hcoin = Ef - £»> = -hcomi. (1.20.44)

We next show that this is identical with (1.16.36). We have from (1.16.36)

\c{1\m, t\i, ro)|2 - \

where since V% — Vs

Jto

(m\V8(t)\i)* - (i\V8(t)\m).

), (1.20.45)

(1.20.46)

We shall reduce (1.20.45) to (1.20.43) by using the same techniques used inSection 1.16 with the Dyson time-ordering operator. We rewrite (1.20.45) as

1.21 THE REDUCED DENSITY OPERATOR

(see Figure 1.5)

81

fr Jt* Jto

= i f *dh \ \n Jt<> Ju>

fl Jtt Jto

(1.20.47)

Here we used the fact that the matrix elements are c-numbers and commute,and in the last integral we have interchanged the dummy variables of integra-tion tx and r2. This is seen to be in exact agreement with the first term in(1.20.43).

We therefore see that (1.20.37) reduces to the perturbation theory resultswhen the system is known with certainty to be in a state |i) initially butincludes the more general case when we know that it is in state |y>> initiallyonly with a certain probability pv.

1.21 THE REDUCED DENSITY OPERATOR

Let us consider two independent systems A and B which are described byhamiltonians HA and HB, respectively. Assume we have solved the energyeigenvalue problems

HA\A') = EA.\A'); HB\B') = EB.\B')

(A'\A") = dA.A.; (B'\B''}=8B-B" (1-21.1)

2\A')(A'\ = l; 2\B')(B'\ = 1.A- B'

If they are put into contact at time r0, the total hamiltonian is

Hf = HA + HB+ VAB = H0+ VAB. (1.21.2)

Note that operators of the A and B systems commute since they are inde-pendent. In particular

[HA, HB\ = 0. (1.21.3)

The density operator for the two coupled systems in the SP satisfies the


equation of motion

thte£-[HT,pAB8}. (1.21.4)

otAt t = t0, the systems are uncoupled so that the density operator factors as

P('o) = pJh)pM), (121.5)

Tr^ pA(t0) = I <A'\PA«o)\A') = 1where

(1.21.6)

TrB pB(t0) = 2 <B'\Ps(t0)\B') = 1,Tt'

when we evaluate the traces in the representations in which HA and HB arediagonal. Therefore, we have "*

p(t0) - 2 (A', B'\PA(t0)PB(t0)\A', B'),A'.B'

where

Therefore,

In the IP, by the previous section, we have

t) = U0(t, to)PABz(t)Vl (t,

0) - 2 <A'\pA«0)\A') I <B'|PB(f^ ' 7?'

, t0) = exp - -n

- t0)

(1.21.7)

(1.21.8)

(1.21.9)

(1.21.10)

(1.21.11)

(1.21.12)

(1.21.13)= U0A(t,t0)U0

B(t,t0),

where the last step follows from (1.21.3).Now suppose we are only interested in making measurements on system

A. If M is any arbitrary function of operators of system A only in the SP, wehave that

2(A',B'\M8PAB8(.t)\A',B')A'B'2A'.B'I(A'\{M82<B'\PAB

8(.t)\B'))\A'). (1.21.14)A' \ B' )

1.21 THE REDUCED DENSITY OPERATOR 83

The last step is allowed because M is only a function of/*-system operators.We may therefore write this

«M» = Tr^ {M*|TrB PAB8®]}. (1.21.15)

We therefore see that if we are interested in averages for system A only, wedo not need the full density operator pjît) but only the simpler reduceddensity operator defined by

pA8(t) - TrB PAB8(t).

If we use (1.21.10) and (1.21.13), we have

PA8(0 - Tr* {V*V

(1.21.16)

2 <B'\UoAUoBpAB

I(t)UoÛ(*B\B'). (1.21.17)

B'

But

(B'\U0B = exp - { EB.(t - to)(B'\.

HTherefore, (1.21.17) reduces to

PA8(t) = VA

B'

= UATrB

(1.21.18)

A. (1.21.19)

where pA(f) is the reduced density operator in the IP.If we now iterate (1.21.11) and trace over the 2?-system we have to second

order in V

+ \ - t0), pA(t0)PB(t0)]

- t0), PA(t0)PB(t0)}],

(1.21.20)

where we used (1.21.5). The traces over B cannot be carried out until wespecify the particular problem under discussion.

The probability of finding system A in the energy eigenstate \Af) at time


t, when at t = tQ it is described by pA(t0), is

{A'\pA\t)\A>)

= (A'\pA(t0)\Af) + ft f'dhiA'Wtj,, [V&J, pA(.t0)Ps(t0))}\A')

>, pA(t0)PB(t0)])}\A') + • • • ,

(1.21.21)to second order in Vz. If at t0, system A is in state \A{), then

PA(*O) = M W I (1.21.22)and (1.21.21) reduces to

{A'\PJ{t)\A') m \cA(f, t/t, to)\*

- 7i {tdh[\TTB{<A/\V#d\A*>pM&tMA1\VM\A')tr Jt, Jt»

+ {Af\V^\A*)PMHAt\V1M)\Af)} + ••-, ( 1 . 2 1 . 2 3 )

since (Af\A*) = 0. This is the probability of finding system A in state \Af) attime t when it was in state \A€) at t0.

In the second integral, we may interchange the order of integration as. inthe previous section and obtain

If we then interchange the dummy integration variables tt -»• tt, we maycombine the two integrals in (1.21.23) to obtain

(1.21.24)

\cJJ, tlU Q\* = f2 f * i f*dtt TrB (A'

If we next use (1.21.12), (1.21.13), and (1.21.18), this reduces to

J [ } , (1-21.25)where we have let

fow/ = £^/ - EAi (1.21.26)

If we use the cyclic property of traces and the fact that V^8 is hermitian,

PROBLEMS

we may rewrite (1.21.25) as

85

where

\c{l\f, tit, to)\* = f2 7/f (1.21.27a)

(1.21.27b)

It should be emphasized that this result assumes that we have measured theenergy of system A at t0 and found it to be EA< and therefore it is in state \A*).System B is described by pB(t0). If enough measurements are made, then itsstate would be known completely also at t0. However, we discuss in Chapter3, Part IV, how we estimate pB(t0) when an incomplete set of measurementsare made. At any rate, (1.21.27) then gives the probability of finding systemA in state \Af) at time t if we again measure its energy, but no measurementsare made on system B at time t.

PROBLEMS

1.1 Derive the commutation relations (1.10.4) by mathematical induction.1.2 Evaluate the following commutators:

(a) [q, sin (p*q)); (b) [q, sin (pqp)]; (c) [q, sin (qp*)], where p and q satisfy[q,p] = ih.

1.3 Solve the eigenvalue problemp\p') = p'\p') in the coordinate representation.Hint: Use (1.10.46).

1.4 Given any two kets | q>) and | % >, show that they satisfy the Schwarz inequality

\<<P\X>\*

Show that the equality holds if and only if | <p) = c\x> where c is a constant.1.5 Show that any function of the hermitian operator

H(t) = p sin cot + q cos cat,

where [q,p] = ih and q and/; are in the Schrodinger picture, commutes with

H(t) but that | T t f ( * ' ) * ' , tf(0|*0. Can you solve the Schrodinger

(/> sin cot + q cos <u/)|v»(0>?


1.6 If H(t) — (p + q)f(O, where/(r) is any continuous function of time and q

and/j are hermitian and satisfy [q, p] = /A, show that H(t') dt', H(t) =

0. Contrast this result with Problem 1.5. Solve the Schrodinger equation(P +q)f(t)W(») - ih[d\w{f))ldt}.

1.7 Let As, B8, C8, and Ds be observables in the Schrodinger picture that satisfythe equation

D8.

Transform this equation to the Heisenberg picture.1.8 Prove that the following properties hold for Poisson brackets as well as com-

mutator brackets.

(a) {u, v) = - {p , u}(6) {«, C) = 0(C) {«i + «2» »} = {«1. »} + («2» »}(rf) {U, »! + p j = {«, Pj} + {«, P2}

{/(g) {«, {P, W}} + {P, {*>, «}} + {*, {«, P}} - 0.

The w, p, and w are considered functions of p and 9.1.9 Prove that commutation relations in the Heisenberg and interaction pictures

have the same form.1.10 Consider two observables q and N that do not commute. The eigenvalues of q

are continuous and may have any value from — 00 to +00 while the eigen-values of//are the positive integers and 0. We writey |f'> = q'\q') and N\n) =n|n>. Formally, expand a particular \q') in terms of the set {|w)} and a particular|«> in terms of the set {\q')}. Show that the transformation function betweenthese two representations is unitary.

1.11 If |f?> and \x> are two kets with finite norm, find the trace of the operators\<p){<p\ and |2><9>|< Show that the trace (0), where O is any operator, is inde-pendent of the representation used.

1.12 Let A be a hermitian operator that satisfies an equation of the form

f(A) 0,

where at, az, and a^ are real numbers and no two are equal. Show that A is anobservable. Find the eigenvalues of A and express an arbitrary ket |y> as alinear combination of the eigenkets. [Proceed as in (1.5.15).] Express |y>, A,and the eigenkets as matrices in the ^-representation.

1.13 Consider an electron beam whose energy is 100 eV moving in the x direction.Its position in the y direction is measured by allowing it to pass through a slit0.01 mm wide in the y direction. By virtue of the uncertainty principle, whatwill be the spread in the beam after it has traveled 1 m from the slit?100 m?

REFERENCES 87

1.14 If A and B are two noncommuting operators, show that Tr (AS) = Tr (BA),provided that the traces exist.

1.15 If the perturbation in (1.16.7) has a first- and second-order term whereV = Ht + Hi and Ht is of order Hf, show that the probability amplitudes(1.16.34) up to second order are given by

, t\t, 0) - yh J

i jdtt f \ j

, t\i, 0) - i

REFERENCES

[1] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford: Clarendon,1958, Chaps 1-5.

[2] A. Messiah, Quantum Mechanics, Vol. I., New York: Intersdence, 1961, Chaps.7 and 8.

[3] E. Merzbacher, Quantum Mechanics, 2nd ed., New York: Wiley, 1970.14] L. I. Schiff, Quantum Mechanics, 3rd ed., New York: McGraw-Hill, 1968, Chaps.

1-3, 6.15] D. Bohm, Quantum Theory, Englewood Cliffs, N.J.: Prentice-Hall, 1951.[6] R. H. Dicke and J. P. Wittke, Introduction to Quantum Mechanics, Reading, Mass.:

Addison-Wesley, 1960.[7] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, New York: McGraw-

Hill, 1935.[8] See, for example, M. H. Stone, Linear Transformations in Hilbert Space, New York:

American Mathematical Society, 1932.[9] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton, NJ.:

Princeton University Press, 1955, Chaps. 5 and 6.[10] J.\onê\acawx,GesellschaftderWissenschaftenzuGdttingerMath.Phys.Nachrichten,

245-272 (1927). Also P. A. M. Dirac, Proc. Camb. Phil. Soc, 25, 62 (1929); 26, 376(1930); 27,240 (1930).

[11] R. C. Tolman, The Principles ofStatisticalMechanics, Oxford: Clarendon Press, 1938.[12] U. Fano, Rev. Mod. Phys., 29,74 (1957).[13] D. ter Haar, Rept. Progr. Phys., 24, 304 (1961).

Elementary Quantum Systems

The harmonic oscillator plays a central role in the quantum theory ofelectromagnetic fields. The quantum features of these fields have becomeincreasingly important with the recent advent of amplifiers at optical fre-quencies (lasers). The oscillator is also important in the quantumtheory of lattice vibrations in solids (phonons) as well as in quantum electro-dynamics. Our primary concern is the electromagnetic field, although we usethe quantum theory of lattice vibrations for a model of an attenuator. Forthese reasons and also because the oscillator offers a simple example of adynamical system with a classical analog to illustrate the general theory givenin the previous chapter, we devote the first part of this chapter to its study.

There is another simple dynamical system which has no classical analog,namely, the spin momentum of an electron; we study this system in Part IIof this chapter. The electron spin is of fundamental importance in many areasof physics although our primary concern will be its use in a more or lessphenomenological model for a laser. The spin also offers an illustration ofhow a quantum theory of a system with no classical analog is formulated.The rigorous treatment of electron spin is given by the relativistic formulationof quantum mechanics due to Dirac [1, 2). However, we limit ourselves tothe nonrelativistic theory of Pauli.

Part III is devoted to the interaction of an electron with an electro-magnetic field. Chapter 5 will present a more detailed discussion of theinteraction of radiation with atoms when the electromagnetic field isquantized.

We begin the study of the oscillator in the Heisenberg picture. TheHeisenberg equations of motion are identical in form with the classicalequations of motion, as noted in Section 1.15, so that the Heisenberg picture

88

ELEMENTARY QUANTUM SYSTEMS 89

shows the formal analogy between the classical and quantized oscillator. Italso offers an opportunity to introduce in a very simple way creation andannihilation operators, which are a great convenience in calculations. Theyare used in Section 2.2 to solve for the energy eigenvalues of the oscillatorand the basis vectors for the energy representation and will play a verydecisive role in the theory of the quantized radiation field later.

The following section makes an effort to give a physical interpretation ofthe creation and annihilation operators, although to do this correctly involvesthe study of symmetry properties of wave functions representing an assemblyof oscillators and boson particles. In the interest of simplicity, we must becontent with a simplified version and refer the reader to a more advancedtext for a rigorous treatment [3, 4]. In the same section we introduce opera-tors that describe fennions. The rigorous interpretation in this case involvesthe theory of second quantization [3,4]. However, the operators thatdescribe electron spin can be put into a form analogous to fermion operators,as we show in Part II of the chapter.

In Section 1.10 we solved for the eigenvalues and eigenvectors for thecoordinate and momentum of a particle in one dimension; the analysis wasindependent of the potential V(q). Therefore, the position and momentumeigenvectors for the oscillator are already known. In Section 2.4 of Part I,we give the representatives of the energy basis vectors obtained in Section 2.2in the coordinate representation. In wave-mechanics language these arecalled the oscillator energy eigenfunctions. In the final section of Pait I, weintroduce the coherent states which have proved very useful in the study oflasers.

In Section 1.19 we considered the time development of a minimum un-certainty wave packet which described a free particle. In the following chapterwe develop much more powerful mathematical methods for handling wave-packets. Minimum uncertainty wave packets for an oscillator, as noted inChapter 1, will play a very important part in our later considerations of theelectromagnetic field.

In Part II we introduce orbital angular momentum operators and solvefor their eigenvalues and eigenvectors. With an added constraint, we nextstudy electron spin. The added constraint gives a model which has no classicalanalog and is justified since it predicts a splitting of spectral lines in agree-ment with experiment. In the Dirac relativistic formulation of the theory ofan electron [1], the spin is predicted and is not put into the theory in an adhoc manner as it is in the present nonrelativistic formulation. In Section 2.8we introduce the Pauli spin operators and in Section 2.9 we give theseoperators in the HP when the electron is in a d-c magnetic field.

In Part III we give the nonrelativistic hamiltonian for an electron in anunquantized electromagnetic field.

90 ELEMENTARY QUANTUM SYSTEMS

PART I. THE HARMONIC OSCILLATOR

2.1 THE OSCILLATOR IN THE HEISENBERG PICTURE

Let us consider a classical harmonic oscillator of unit mass in one dimen-sion described by a coordinate q and momentum p. The hamiltonian is

H = (2.1.1)

where co2 is a constant related to the restoring force on the particle. Theclassical hamiltonian equations of motion forq and/> are, by (1.15.12) and(2.1.1),

dt dp

_ ? .dt '

dH

dq

(2.1.2a)

(2.1.2b)

The usual way toboth sides of (2.1.2a) with respect to / and use (2.1.2b) to eliminate dpfdt.This procedure gives

and the solution is

express A and B so

(2.1.2b). If we mul

solve such a set of coupled equations is to differentiate

dt*—co*q,

— A cos cot + B sin cot,

(2.1.3)

(2.1.4a)

where A and B are constants. To find/>(/), we substitute this in (2.1.2a) andfind that

p(t) — -coA sin cot + coB cos cot. (2.1.4b)

If, at t = 0, q has l ie value ^(0) and p is/>(0), these terms may be used tothat (2.1.4) become

n(0)q(Q) cos <of + *•—- sin cat

coij(t)

kt) — — coq(0) sin co/ + p(0) cos

(2.1.5)

There is a commonly used alternative way [5] to decouple (2.1.2a) and

multr\f2coi

iply both sides of (2.1.2a) by Vw/2 and both sides of

(2.1.2b) by ±.ijs2co\ and add both equations, we obtain the two decoupled


equationsda— — — icoadt

Here we definedt

- icoa*.

a* =y/2cZ

(coq + ip)

(coq - ip),

(2.1.6a)

(2.1.6b)

(2.1.7a)

where a* is the complex conjugate of a. We may easily solve these equationsfoip and q and obtain

(2.1.7b)

The solutions of (2.1.6a) and (2.1.6b) are given directly as

1 [*>«(0)H(2.1.8)

The introduction of a and a* has made the solution of the simple equations(2.1.2) even simpler.

We may express the hamiltonian (2.1.1) in terms of a and a* by means of(2.1.7b). After minor algebra, we obtain

H = coa*a, (2.1.9)

which also looks simpler in these variables than in terms of/> and q. In fact,we may formally obtain (2.1.6) directly from the hamiltonian (2.1.9) if wetake as the hamiltonian equations of motion

.da1 dtda^dt

dHSS " £_

da*dHda

coa(2.1.10)

The oscillator has one degree of freedom and one normal mode of oscilla-tion. Pierce calls a (or a*) the normal mode amplitude [5].


Let us turn now to the quantum treatment of the oscillator in the Heisen-berg picture. From the general theory of Chapter 1, we associate hermitianoperators with the observables q, p , and H and postulate that q and/> satisfythe commutation relation (1.10.3)

The hamiltonian is[q,p] = ih. (2.1.11)

(2.1.12)

AH these operators are in the Schrodinger picture and are independentof time. The SchrSdinger equation of motion (1.13.1) is

(2.1.13)

(2.1.14)

and, by (1.13.2) and (1.13.3), the solution is

exp ( - 1~YJ \y>H(0)),

where U is unitary. This is the transformation law (1.15.1) between statevectors in the SchrSdinger and Heisenberg pictures. Operators transformbetween the two pictures by the similarity transformation (1.15.3) so that

(2.1.15)

For a conservative system, we showed in Section 1.15 that the hamiltonianin the two pictures is the same so that we may write

'(«)]• (2.1.16)

The Heisenberg equations of motion for qB(i) a.ndpB(t) are, by (1.15.11),

dHB

dt

dt

PH

dHB 2

-= -co qB.

(2.1.17a)

(2.1.17b)

The only difference between these and the classical equations (2.1.2) is theoperator character, where qB andpB satisfy

= ih. (2.1.18)


The solution of (2.1.17) is given by (2.1.5):

9isr(0 = U ('»0)?gC/(t, 0) = q8 cos cot + — sin cotCO

pB(t) = U\t, 0)p8U(t, 0) = -coq8 sin cot + p 8 cos cot,

93

(2.1.19)

where (2.1.15) was used, and q8 a.ndp8 are the operators in the Schrodingerpicture at t = 0. They are not initial conditions as in the classical sense of(2.1.5). Since U(t, 0) = exp [-i(pf + afiqtftJM), we see by (2.1.19) theeffect of commuting U through q8 andps. In the next chapter we show othertechniques for obtaining the result of U*qU and XPpXJ.

By analogy with (2.1.7), we may introduce two convenient nonhermitianoperators a and af defined by

a = ~^== (coq + ip)

: (coq - ip)

2Hco1

(2.1.20a)

or

(2.1.20b)

For reasons that will become apparent later, a is called an annihilationoperator and a* a creation operator.

The operators a and ar, like q and p , do not commute. If we substitute(2.1.20b) into (2.1.11) and use the fact that all operators commute withthemselves, we find that a and af satisfy the commutation relation

[0 ,^1 = 1. (2.1.21)

If we use this and substitute (2.1.20b) into (2.1.12), the hamiltonian becomes

— J). (2.1.22)

This differs from (2.1.9) because a and of do not commute as they doclassically. The term h(o\2 is called the zero-point energy of the oscillator.

We shall immediately need the commutation relations

[a, afa] = a

(2.1.23)

which may be proved directly from (2.1.21).

3 =at In


The Heisenberg equations (1.15.5) apply to nonhermitian as well as tohermitian operators. Therefore, the equations of motion for aB(t) andajjr(r) become

1

(2.1.24)

^ 7Tdt in

where we used (2.1.22) and (2.1.23) which are equally valid in either theSchrodinger or Heisenberg picture. The solutions of (2.1.24) are

V\t, 0)asU(t, 0) =MO<4 = U\t, 0)a%U(t, 0)

where we used (1.15.3). Alsot

U(t, 0 ) = e x p ( - i l/ i<ot\i - —J.

(2.1.25)

(2.1.26)

Both as and a% are in the Schrodinger picture.In the future we usually designate an operator M in the Heisenberg

picture by M(i) and in the Schrodinger picture by M, instead of using Hand S subscripts, if there is no likelihood of confusion.

The operator a1 a, which is hermitian, occurs so frequently that we shall let

N = ata = Nt. (2.1.27)

It is called the number operator for reasons that will become clear later. Interms of N, we may rewrite (2.1.23) as

Na = a(N - 1) (2.1.28a)

JVfl* = flf(i\r + 1). (2.1.28b)

The hamiltonian is related to N by

N = — H--. (2.1.29)ho) 2

From these results, the-close formal analogy between the classical andquantum treatments of an oscillator can be seen.

2.2 THE ENERGY-EIGENVALUE PROBLEM FOR THEOSCILLATOR

According to the physical interpretation of quantum mechanics, theeigenvalues of the energy are the only values obtainable by an experimental

2.2 THE ENERGY-EIGENVALUE PROBLEM FOR THE OSCILLATOR 95

measurement of the energy. To compare theory and experiment, we musttherefore solve the eigenvalue equation

H\E) = E\E). (2.2.1)

Because of the simple connection between N and H given by (2.1.29), theeigenvalue problem for N,

N\ri) = n'\ri), (2.2.2)

is entirely equivalent to (2.2.1). Since H and therefore N axe observables, theeigenkets {|«'» form a complete orthonormal set of basis vectors in the N-representation.

To solve (2.2.2), let us review briefly the solution of the eigenvalue problemfor the coordinate and momentum given in Section 1.10. We found an operatorwhich, when applied to a known eigenket, would generate another eigenket.In the present case, we show that the operators a and a* generate new eigen-kets from a given eigenket just as the translation operators perform thisfunction for/? and q. Both derivations are due to Dirac.

We assume that \ri) is a known eigenket of N with eigenvalue ri whichsatisfies (2.2.2). If we operate both sides of (2.1.28a) on |«') and use (2.2.2),we see that

N{a\n')} = («' - \){a\n')}.

Similarly, by (2.1.28b) we have

N{a*\n')} = (*' + l){a'|«'>}.

This shows that, if \ri) is an eigenket of N with eigenvalue n', then a\ri) is aneigenket of N with eigenvalue ri — 1 and af\n') is an eigenket with eigen-value ri + 1. From \ri) we have generated two more eigenkets.

We may repeat the process and apply both sides of (2.1.28a) to eigenketa\ri), use the result above, and generate eigenket a*\ri) with eigenvalueri — 2. Similarly, (2.1.28b) applied to a^ti) will generate eigenket an\ri)with eigenvalue ri + 2. Obviously this process may be continued indefinitelyand an infinite set of eigenkets and eigenvalues generated from a knowneigenket; they may be listed as

\ri) a\ri) a*\n')ri ri-\ ri-2\ri) a*\ri) at2|n')

ri ri + 1 ri + 2

(2.2.3a)

(2.2.3b)

Note that, in these series, successive eigenvalues differ by 1.We next show that ri can only be zero or a positive integer. Since N is

hermitian, from the general theory of Chapter 1, ri must be real and by


assumption the norm of any vector must be greater than or equal to zero.If the norm is zero, the vector is zero. If \ri) is an eigenket, then

<« V> > 0,

since \ri) = 0 is trivial. Let us form the scalar product of (2.2.2) with {n'\.We have

{ri\N\n') = <«Va|«'> = n'{n'\n'). (2.2.4)

But this is the norm of the vector a\ri), which must be greater than or equalto zero. Since <n'|n'> > 0 and (ri\cfa\ri) ^ 0, we conclude by (2.2.4) that«' ^ 0. Therefore, the eigenvalues of JV are real and nonnegative. If «' = 0,then

o|0) = 0, (2.2.5)

since the norm is zero by (2.2.4).If n' ¥=• 0, the norm of a\ri) is given by (2.2.4). However, by (2.2.3a), if

ri is not an integer, the sequence of eigenvalues would eventually becomenegative and the norms of the associated vectors would become negative,which is not allowed. The only way to prevent this is for «' to be a positiveinteger or zero. Therefore, the eigenvalues of N are the positive integers orzero, as stated.

The norm of the vector af\n) is

(n\aa*\n) = d>d)\n) n){n\n), (2.2.6)

where we used (2.1.21). Since n > 0 and {n\n) > 0, we see that ar\n) cannever be zero.

The eigenkets generated by successive application of a and a* are notnormalized to unity as yet. They may be normalized as follows: since a\n) isan eigenket of N with eigenvalue n — 1, a\n) can differ from \n — 1) by aconstant. We therefore write

a\n) = cn\n - 1). (2.2.7)

The norm of this vector is, by (2.2.4),

{nWa\n) = n(n\n) = \cn\\n - \\n - 1).

If <n — l|n — 1> is normalized to unity and if we choose \cn\ to be sfn,then (n\n) = 1. The phase of cn is arbitrary and we choose it zero; thus(2.2.7) becomes

o|n> = V « | n - 1 > . (2.2.8)

We restrict n to be greater than zero since state | —1> has no meaning.When n = 0, (2.2.8) reduces to (2.2.5).

2.2 THE ENERGY-EIGENVALUE PROBLEM FOR THE OSCILLATOR 97

Similarly, we may use (2.2.6) and write

atyi) = V« + 1 \n + I >. (2.2.9)

If <0|0) is normalized, then all others will be normalized.We may collect these important results:

N\n) = n\n)

«|0) = 0

a\n) = <Jn \n —(2.2.10)

An extremely useful result is obtained if we use (2.2.9) and apply theoperator af to the state |0) n times. From this we generate the state |«> given by

\n)

This formula will be used repeatedly.From the general theory, the orthonormality relations are

and the completeness relation is

(2.2.11)

n - 0

(2.2.12)

(2.2.13)

Since the norm of these eigenvectors is finite, they form a complete set ofbasis vector for a Hilbert space.

From (2.2.10), their adjoints and the completeness relation it follows that

= ]?n\n)(n\o

(2.2.14)= f Jn\n)(n

o

where/is any function of a* a.We may use (2.2.10) and (2.2.12) to obtain the matrix elements of a, af, and


N in the JV-representation. They are

<n'|a|0> = 0

<n'|a|n">

{n'W\ri')

(n'\N\n")

The energy eigenvalues are

(2.2.15)

+ i), (2.2.16)

where n = 0 , 1 , 2 , . . . , oo. Classically, any positive value of energy may beobtained when the energy is measured, but quantum-mechanically onlydiscrete values may be obtained. In the limit of large n (n is called a quantumnumber), the discrete character of (2.2.16) is unnoticeable and the quantumresult becomes the classical result. Since hoz lO"84 J-sec, hco is small up tooptical frequencies where the quantum features become important.

2.3 PHYSICAL INTERPRETATION OF N, a, AND af; BOSONSAND FERMIONS

The operators q, p, and H for an oscillator have the physical significanceof position, momentum, and energy, respectively. The operators a, ar, andcfa were defined in terms of q, p, and H, and we would like to give them aphysical interpretation.

Figure 2.1 shows an energy-level diagram for a quantized oscillator.Along the vertical axis the energy as given by (2.2.16) is plotted. The hori-zontal axis has no significance. Beside each level is the corresponding eigen-state of the operator N. If the oscillator is in the energy eigenstate corre-sponding to |n>, this is indicated by a dot on the line of energy (n + i)hco.The state of the oscillator in this case is indicated by the value of n and theseparation hco.

There is no inconsistency in another interpretation for the oscillator instate \n). We may assume that the hamiltonian describes a system of nidentical noninteracting quanta, each of which is in the same dynamical

h«(n+ '/gj-h«tn-"/2)- JSl -1 n -1

|}hc.Figure 2.1 Energy-level diagram for quantizedharmonic oscillator.

2.3 PHYSICAL INTERPRETATION OF N, a, AND a* 99

state with energy ha>. We may therefore interpret the state |«) as a state withn quanta while the state |0) has no quanta and is called the vacuum state.

The rigorous justification for this interpretation is given by Dirac [1], andwe do not discuss it here. It is one of the most fundamental consequencesof quantum theory since it allows a unification of the particle and waveproperties of light. The quanta in that case are called photons, and we devotea large part of this book to a study of the quantum properties of light.

The operator N is called the number operator because a measurement ofN yields one of the eigenvalues 0 , 1 , . . . , oo which is interpreted as thenumber of quanta in the state. According to this interpretation, any numberof quanta may occupy the same dynamical state. Particles in nature havingthis property are called bosons. Light quanta (photons), elastic vibrationquanta in crystal (phonons), and a particles, among others, are bosons.

It is now easy to see why a* and a may be interpreted as creation and anni-hilation operators, respectively. From (2.2.9), if the oscillator is started instate \n) with n quanta and we operate with ar, we generate state \n + 1),with n + 1 quanta. This is shown in Figure 2.1. Therefore, af is a creationor a raising operator. Similarly, by (2.2.8), if a operates on state \n), it givesstate \n — 1) with n — 1 quanta, and a is an annihilation or loweringoperator.

The eigenvalues of N are a direct consequence of the commutation relation[a, a*] — 1. Therefore, particles that obey this commutation relation arebosons. There is another class of physical particles in nature, called fermions,which have the property that no two of them can occupy the same dynamicalstate. Electrons, protons, and neutrons are examples of fermions. The Pauliexclusion principle states that no two identical particles can occupy the. samestate; thus fermions obey the exclusion principle. The fundamental quantumpostulate [a, <z+] = 1 cannot therefore be a suitable postulate for fermions.Although the complete theory of fermions involves the theory of secondquantization [1,2], we may give an oversimplified version in which weformally indicate a different quantization procedure due to Jordan andWigner [6]; this version avoids the difficulty of having a state multiplyoccupied.

In the theory of second quantization for a system with only one dynamicalstate, there are still two operators b and 6T which are interpreted as a fermionannihilation and creation operator, respectively. These have no connectionwith the momentum or coordinate of the fermion, as their boson brothersdo. There is also an operator N — b*b = JVf which is interpreted as a numberoperator defined in terms of the hamiltonian by

H e- Eb% (2.3.1)

where E is the energy eigenvalue of the one dynamical state of the system.

100 ELEMENTARY QUANTUM SYSTEMS 2.3 PHYSICAL INTERPRETATION OF N, a, AND cfi 101

Jordan and Wigner then postulate that b and b1 obey the anticommutationrelations

<*.*>+-1( 2 '3 -2 )

rather than commutation relations, where the anticommutator of A and Bis defined by

{A, B}+ = AB + BA. (2.3.3)From (2.3.2),

fc* = *t2 = 0 . (2.3.4)

We next derive the eigenvalues of N. If we use (2.3.2) and (2.3.4), we seethat

JV* = b'btfb = b\i - bfb)b - bfb = N. (2.3.5)

This simple algebraic equation that N satisfies is sufficient to find the eigen-values of N. For if

N\n) = n\n), (2.3.6)then, by (2.3.5) and (2.3.6),

/i*|/i> = JV*|»> = N\n) = n\n). (2.3.7)

Therefore, the eigenvalues of N are

n* = n or /i = l ,0 . (2.3.8)

Since we assume that there is no degeneracy, the Hilbert space will consistof only two vectors, designated as |0) and 11).

From the general theory, the eigenkets of N satisfy the orthonormalityand completeness relations

<0|0> - 1 <l|0> = <0|l> = 0

whereHXil

N =

iV|O) •» 0.

The matrix elements of N are, by (2.3.9) and (2.3.10),

N|l> <1|N|O>1 =

L<0|N|l) <0|N|0>J "It also follows from (2.3.9) and (2.3.10) that

This is the analog of (2.2.14).

F l -ip oj

(2.3.9)

(2.3.10)

(2.3.11)

(2.3.12)

Since the eigenvalues of N are 1 and 0, the state can be occupied by onlyone particle or else be empty, in agreement with the exclusion principle.Therefore, the quantum postulate of Jordan and Wigner is sufficient toensure that a dynamical state can be occupied by only one particle at a time.

Next we must show that b and bf act as creation and annihilation operators.If we start with state |0>, with no particle present, b*\0) gives a state with oneparticle present and is therefore a creation operator. Since by (2.3.4) bn = 0,we are prevented from generating a state &t2|0) with two particles. Also,b\l) gives a state |0> with no particles present and therefore acts as an anni-hilation operator. Also, 62|1) = 0.

To show these results, we note the following commutation relations for band b1 with N = b<b:

lb'N]'=b (2313)

These may be proved directly from (2.3.2) and (2.3.4). They are identical inform with the boson creation and annihilation operators in (2.1.28). Weproceed exactly as in that case to show that

b\n) = cn\l - n) (2.3.14a)

generates an eigenket of N with eigenvalue 1 — n. In contrast to the bosoncase, however, the series stops because b2\n) = 0, by (2.3.4). The norm is

(2.3.14b)

We see that if n = 0,

b\0) = 0,

while if n = 1, since <0|0> = 1 = <1|1>,

Therefore, (2.3.14a) for n - 1 is

b\\) = |0>, (2.3.14c)

where we choose the phase of ct to be real. Similarly, we may show that

0. (2.3.15)

From (2.3.9), (2.3.14b), (2.3.14c), and (2.3.15), the matrix elements of b and


in the JV-representation are

i<0|6|0>J

ro n[p oj'

The state vectors may be represented by the matrices

_ roi rcuiyi r r

(2.3.16a)

(2.3.16b)

"i ro~i r<i|iyi m- | 1 > = = . (2.3.17)

J LiJ L<oii>J LoJWe show in Part II how spin operators can be put in a one-to-one corre-

spondence with fermion annihilation and creation operators.

2.4 TRANSFORMATION FUNCTION FROM N TO qREPRESENTATION FOR OSCILLATOR

The energy-eigenvalue problem for an oscillator was solved in Section 2.2in the iV-representation. It is sometimes useful to work in the coordinaterepresentation; this requires the transformation function (q'\n), which isthe energy wave function for the oscillator. These are the representatives ofthe energy eigenkets |R) in the coordinate representation.

There are two ways of finding the transformation function

«„(*') = W\n). (2.4.1)

First, the energy-eigenvalue problem may be written as

Hco(n + i)|«>.)|«>. (2.4.2)

If we take the scalar product with (q'\, an eigenbra of q, and use (1.10.46)and (1.10.47), then (2.4.2) reduces to

7 q'2)u (q') « + &un(q'), (2.4.3)771 + 72 aq Z

where we used (2.4.1). The transformation function is therefore the solutionof (2.4.3), the Schrodinger equation, which is square integrable; that is, the

/•«solutions must be chosen so that \un(q')\* dq' exists. The solution of

J-n(2.4.3) may be found in Schiff [7].

Another method of obtaining un(q') is to start with (2.2.5)a\0) = 0,

2.4 TRANSFORMATION FUNCTION FROM N TO q REPRESENTATION 103

and replace a in terms of/? and q by (2.1.20a). This becomes

(a>q + ip)\0) = 0.

If we take the scalar product of both sides with {q'\ and use (1.10.46) and(1.10.47), we have

where we used (2.4.1). The solution of this equation normalized so that

I \uo(q')\ dq' — 1 (2.4.4)J-00

is

This is the coordinate representation of the vacuum state and is also calledthe oscillator ground-state wave function.

Next, by (2.2.9) and (2.1.20a), we see that

1W\<oq - ip\0) - <«'|I>.

If we use (1.10.46), (1.10.47), and (2.4.5), we have

c°°It is easy to verify that IMJ2 dq' = 1.

J— CO

( 2 A 6 )

In this manner we may successively generate «8, «s The result issummarized by

= un(q') (2.4.7)

where 8 = yjmjh, and Hn(x) is the Hermite polynomial of order n. These arethe eigenkets \n) in the coordinate representation. After we have developedmore techniques in the manipulation of operators in the following chapter,we shall derive a generating function that will give un(q') more neatly.


2.5 THE COHERENT STATES [8]

So far we have used the number representation \n) which satisfies the eigen-value equation

tfa\n) = n\n) n = 0,1,2 (2.5.1)

These form a complete orthogonal set of basis vectors to describe a harmonicoscillator. We now introduce the coherent state which is extremely useful indealing with radiation problems. Actually, we show that a coherent state isjust a minimum uncertainty wave packet state.

We define the eigenvector of the nonhermitian operator a to be a coherentstate. To find it we must solve the eigenvalue problem

a|a> = a|a>. (2.5.2)

Since a is nonhermitian, we cannot use our prior theorems which say thateigenvalues are real and that eigenvectors are orthogonal and complete.

To solve (2.5.2), we use the completeness relation for the W-representationto expand |«> as

00

n-0(2.5.3)

where cn(oc) = (n|oc) is the transformation between the number and coherentstate representations. The |(n|a)|s gives the probability of finding the oscillatorwith energy nhto if a measurement is made when the oscillator is in state |oc).If we substitute (2.5.3) in (2.5.2) and use (2.2.10), we have

n - 1

S2«cn(«)l">-n-0

(2.5.4)

The first sum runs from 1 to oo since the n = 0 term gives zero. We maytherefore shift indices and let n -*• n + 1 so this becomes

n - 0(2.5.5)

If we multiply both sides from the left by (m|, since (m\n) = dnm, we obtainthe simple difference equation

(2.5.6)

2.5 THE COHERENT STATES

or

a

so that

Therefore, we see that

We choose c0 so that

I !

«•(«)

oo n

Z F=n-O^/lt!

n=»0 •s/nl m!(m\n)

Thus

w i ?|co|*exp|a|2.

and the coherent state is given by

If we use (2.2.11), we have

n - 0

10S

(2.5.7)

(2.5.8)

(2.5.9)

(2.5.10)

(2.5.11)

(2.5.12a)

WlO), (2.5.12b)where we could carry out the sum on n since the "vacuum" or ground state|0> is independent of n. We shall find both forms extremely useful.

We next note that

(2.5.13)

represents a Poisson distribution over the photon number states.


From the normalization, we easily see from (2.5.12b) that

<0|e«Vot|0> = e ' < (2.5.14)

We have tacitly been assuming the eigenvalues a to be complex since a isnonhermitian.

Next we demonstrate that the coherent states are not orthogonal. Neverthe-less, they are extremely useful as we shall see. We have by (2.5.12a) and itsadjoint that

2 ^«=o m-o y/n!

n - 0

(2.5.15)

If the states were orthogonal this would be zero for a ?* /J. From this itfollows that

|</?|«x>|2 = e-l—fi\ (2.5.16)

so that they become approximately orthogonal as |<x — /S|2 increases.The coherent states do form a complete set of states. Otherwise they would

not be very useful. In fact they form what mathematicians call an over-complete set [8].

Since as we shall show a is complex, the completeness relation is written as

J|a)(«| ^ (2.5.17)

where 1 is the identity operator. The integration is over the entire complexplane. If we let a = x + iy = re*8, then rf2a = dxdy — r dr dd.

To verify (2.5.17) we use (2.5.12a) and its adjoint in the left side of (2.5.17).This gives

If we change to polar coordinates, this becomes

f|0C><a|— = f 1"><OTI f% rfr r-r>+m f**^ r«n-n), p.5.18)J IT n.m=o irJnI ml Jo Jo

Since

d8 ei{n-m)t = 2v6nm, (2.5.19)


we have

107

where we let f = r2. But the integral equals n! so we have

fJ

|a)(a| — = f |R)<II| = 1 Q.E.D.,IT 0

(2.5.20)

which follows from the completeness relation for the number representation.We next show that a coherent state is a minimum uncertainty state. From

the relations between a, aT and q,p we see that the expectation values ofp, q,p*, and q* in state |a> are

x|(at-a)|a> = I;/^(a*-a)

-^-(a|(«t2 + a2 + aat + aMK2a> K T n f

— (a*2 + a2 + 2a*a + 1)2 co

— — <«|(aTZ + a — aaT — afa)|a>

(2.5.21)

where we used o|a) = a|a> and its adjoint. The variances are therefore

so that

(Ap)2 « <p2) - (p)2

^

2co

2 '

(2.5.22)

(2.5.23)

which is the minimum value allowed by the uncertainty principle.From (2.5.21) we see that the eigenvalues a are given by

a = + i<p)l (2.5.24)


Since (q) and (p) are real and may have any value from — oo to + <», « maylie anywhere in the complex plane.

We next obtain the transformation function from the coherent state to thecoordinate representation, (q'\a.). We have from the definition of a in terms ofq and/7

a|a> = a|a>

[<*>« + lj»]|a>. (2.5.25)

If we multiply both sides from the left by (q'\, an eigenbra of q, we see bytechniques which are now familiar that

(q'\(a>q

or

We may rewrite this as

(q'\x) LV h h

or on integrating, we obtain

{q'\a) = N exp j - — q'* + — <tq'\

where N is a constant of integration. We choose it so that

By (2.5.28) we see that

K«'l«>|2 - \N\*exp |~- - q * + ~(a + «*)«']L h V n J

where we completed the square in the exponent. Since

(2.5.26)

(2.5.27)

(2-5-28)

(2-5.29)

(2.5.30)

(2.5.31)


we obtain on integrating (2.5.30)

Therefore,

N = / ^ .

where ft is an arbitrary real phase. Thus

a*)8 +J^

109

(2.5.32)

(2.5.33)

(2.5.34)

If we use (2.5.24) and its conjugate to eliminate a and a* we obtain afterminor algebra

(2.5.35)<4'|«> = ( ~ f «p f- £ [q' - (q)f + i&q'\irhf L 2/t A

If we compare this with (1.12.23), we see that it is indeed a minimum un-certainty wave packet state but with a fixed Aq and Ap given by (2.5.22). In(1.12.23) Aq could be arbitrary but satisfied (2.5.23). Thus (2.5.34) representsa single infinity of minimum uncertainty states while (1.12.23) was a doubleinfinity of such states.

With no loss of generality let us choose the phase /i so that

This is satisfactory since then

|iV|2 = [exp [ -Kkl 2 + a2 + loci2 + o c * 2 ] ^

(2.5.36)

}A

(2.5.37)

which satisfies (2.5.32). Then we have

- j 4 (2.5.38)

This form will prove useful later as we shall see.

110

PART II.

ELEMENTARY QUANTUM SYSTEMS

ORBITAL ANGULAR MOMENTUM;ELECTRON SPIN

2.6 EIGENVALUES AND EIGENVECTORS OF ANGULARMOMENTUM

Orbital angular momentum plays an essential role in quantum mechanicsjust as it does in classical mechanics. Classically, the angular momentumabout a point 0 is defined by

1 = r x p, (2.6.1)

where r is the radius vector from 0 to the particle and p is its linear momentum.Since I is an observable, we postulate that 1 is a hermitian operator definedby (2.6.1) where r and p are the coordinate and momentum operators. Welet [qi,qitq9] be the three coordinate operators corresponding to r andiPuPt>Pa\ = p be the corresponding momentum operators. We postulate asin (1.10.3) that these operators obey the commutation relations

[?«./>*]•**«; [tf«»fr]"»0-I/Wi]. (2-6-2)

where i andy = 1, 2, or 3. This says that qx and pt commute, for example.In other words, measurements of a coordinate in one direction does notinterfere with the measurement of the momentum in an orthogonal directionas it does in the same direction.

From (2.6.1) and (2.6.2), we see that

h = q%P\ ~ (2-6.3)

If we use (2.6.2), we may easily show that

so that no additional postulates are needed to quantize 1. Note also since, forexample, qt and p3 commute as do qs and p%, we do not have to worry aboutordering of the separate factors in lu /2, and /s.

The total angular momentum is

It = If + V + /,*. (2.6.5)

We leave as an exercise to show that

P2, /J = 0. (2.6.6)

2.6 EIGENVALUES AND EIGENVECTORS OF ANGULAR MOMENTUM 111

That is, each component of the angular momentum separately commuteswith I2.

It is convenient to define two nonhermitian operators / ± by

/± = h ± ik.or

k = W- + /+)

Since lx and l2 are hermitian it follows that

In terms of/± , we see that

It is left as an exercise to show that

(2.6.7)

(2.6.8)

(2.6.9)

(2.6.10)

(2.6.11a)

(2.6.11b)

(2.6.11c)[/+, /_] = 2M3.

If we alternatively add and subtract (2.6.11c) from (2.6.10) we obtain

'+'- = I2 - V + M3 (2.6.12a)

Ll+ = I2 - /32 - M3. (2.6.12b)

We have shown that I2 commutes with lu /2, and l3 but the components donot commute with each other. We have shown that any two operators thatcommute may be simultaneously diagonalized. We may therefore obtain arepresentation in which both are diagonal; that is, we may find eigenvectorswhich are simultaneous eigenvectors of two commuting operators. Considerthe eigenvalue problems

l*\/*iv) — Ml/*; v) (2.6.13a)

1*1/*; v) = vH2\/i; v). (2.6.13b)

Since [/„ I2] = 0 we see that

l*lx\/t; v) = fiHl2\p; v) = fivh*\p, v)_/ ]2 i v\ (2.6.14)

We wish to obtain the eigenvalues p. and v by techniques similar to thoseused for the harmonic oscillator.

By (2.6.11a) we have1V± = /±12. (2.6.15)

112

By (2.6.13) we see that


(2.6.16)

This says that if \fi; v) is an eigenket of I2 with eigenvalue vh2, then l+\/t; v)and l_\[i; v) are also eigenkets with the same eigenvalue.

Consider next (2.6.11b):

It therefore follows from (2.6.13) that

it* ±

(2.6.17)

(2.6.18)

Thus if \[i; v) is an eigenvector of lt with eigenvalue fth then l+\p; v) is aneigenvector of/, with eigenvalue (jx + l)ft and L\/t; v) is an eigenvector of/,with eigenvalue Qi — l)h, both with the same v by (2.6.16). We have thusgenerated two additional eigenvectors of /, from the original whose eigen-values differ by ±h. We may obviously continue this process and obtain theinfinite sequence

\P;(2.6.19)

(jt + i)h (p. + 2)h

LI/*; v) £\fi; v)

fih (ji — l)/» (ji — 2)H

where v is unchanged.Since the norm of a vector must be greater than or equal to zero, we assume

that{fi; v\/i; v) > 0. (2.6.20)

That is, the original vector exists. Then since /_ = / | , we have

(ji; v\U+\/i; v) — (ji; v|(ls — / / — Mx)\(*', v)

= h\v - /i2- fi){ix; v\/t; v) ^ 0, (2.6.21)

where we used (2.6.12b) and (2.6.13). By (2.6.20), it follows that

v - fi* - n £ 0, (2.6.22)

where v is fixed. This tells us that for any given v if fi gets arbitrarily bigeither positively or negatively the vector l+\/i; r) would develop a negativenorm which is forbidden. We must therefore anticipate an upper and lowerbound on ft for each v. The equality is satisfied above for

(2.6.23)


which are the two bounds on fi for fixed v. Let / be the largest value p mayhave so that

(2.6.24)

(2.6.25)

(2.6.26)

or

When I, we have

since otherwise we would generate an eigenvector with eigenvalue /i = I + Iwhich would violate (2.6.22).

If we start with state |/; v) and apply /_ k times then we generate state|/ - k; v) by (2.6.19). The length of L\l - k; v) is

(/ - k; v\l+L\l - k; v)

= (l-k; v|(I2 - /32 + Hl3)\l - k; v)

= {v-(l- kf + (/ - k)}h\l - k; v\l -

If \l — k; v) ^ 0, then since v = /(/ + 1), we conclude

/(/ + 1) - (/ - Jt)2 + (/ - k) > 0.

This puts a limit on the size of k

1),o r ^max is determined by

so that"*"* *»*•

0. (2.6.27)

(2.6.28)

(2.6.29)

(2.6.30a)

(2.6.30b)

But k is a positive integer so 2/ must be a positive integer. Therefore, / mayhave only the values

/ = 0,il,|,2,f (2.6.31)

When fcmax = 21, we conclude that

/ - I / - = LI-/; (2.6.32)

Therefore fi ranges between + / and —/in unit steps. It is conventional tolet fi = m and to designate v by / since r = /(/ + 1) and write

»; I) = mH\m; I)(2.6.33)


where the eigenvalues are

m - - / , -I + 1, - / + 2 1-2,1-1,1.

The eigenvectors are orthogonal since I2 and /3 are hermitian so that

\rn , ( [m, / / = OiyOmm', \/..o.jj)

so that the matrix elements of /3 and I2 are

,31 A (2-6.36)

Let us next obtain the matrix elements of l± in the representation in which/, and I* are diagonal. We have shown that l+\nt; I) is an eigenvector of I2 witheigenvalue /(/ + 1)A2 and also an eigenvector of/3 with eigenvalue (m + \)h.Therefore, l+\m; 1) can differ from \m + 1; /) by a complex constant. We maytherefore write

l+\m;l) = Xltmh\m + ! ; /> . (2.6.37)So that

(m + Ul\l+\m;l) = ^mh. (2.6.38)

If we take the complex conjugate of both sides, we have

<m; Z|Z_|m

This relation is satisfied if

Consider next

(2.6.39)

(2.6.40)

- [1(1 + 1) - m* - (2.6.41)

where we used (2.6.37), (2.6.40), (2.6.12b), and (2.6.33). Therefore, we con-clude that

- m(m

and

l+\m; /) =

L\m; I) =m

-m m)\m

(2.6.42)

(2.6.43)

m


so the matrix elements are

<m\ Z'|Z_|m; Z) = tuj{l - m

If we use (2.6.8), we obtain the nonvanishing matrix elements

t

(2.6.44)

m

; Z) = ^ ( / _<m - 1; /

<m + 1; ZlZjIm; Z> = — i i ^ V C -

<m - 1; i|Z2|m; Z> =

(2.6.45)

-m m).

We shall write out a few of these explicitly. For / = 0, we have explicitly thenull matrices

/, = 0 I2 = 0 lx = 0 = /2.

Next for / = \, m = ± J, the matrix elements are

while the state vectors become

For / = 1, m = —1, 0, + 1 , and we have

1

0

.0

0

1

.0

0

0

0

1

0

1

0

0

- 1 .

o"1

0.

9

; I2 = 2h2

. h_

1

0

.0

"oi

.0

0

1

0

—i

0

i

0

0

I.

o"—i

0 .

(2.6.46)

(2.6.47)

(2.6.48)

116

while

l+i;


(2.6.49)

1

0

0.

; 10; i> -

0

1

.0.

; l - i ; D =

0

0

A.

The reader may easily proceed. For / = f, m — ± i , and ± | so the matricesare 4 x 4 .

We have chosen I2 and l3 as the two commuting operators to diagonalize.We could have also chosen I2 and lt or I2 and /2. We say that the 3 or z-axisis the axis of quantization when we diagonalize I2 and /3. There is obviouslynothing unique about the z-axis here. If we, for example, applied a uniform

.magnetic field in a certain direction, then it would usually be advantageous tochoose this direction as the axis of quantization.

We have shown that lt may have integral or half-integral multiples of has its eigenvalues. Simply this result arises because of the commutationrelations (2.6.4) and (2.6.6) and has nothing to do with the definitions (2.6.3)of 1 in terms of the coordinates and momentum. However, if I is to representorbital angular momentum, then the eigenvectors of I2 and /3 must have co-ordinate or momentum representatives. That is, we must be able to expressthe 1 matrices in terms of coordinate and momentum matrices. We show inthe next section that this is only possible if we restrict the eigenvalues of lz

to be an integer times h. The half-integer-values do not have a classicalanalog in that \m; /) does not have a coordinate representative when m is ahalf-integer.

We have no reason for throwing out half-integral values if we say thesecorrespond to intrinsically quantum mechanical effects which have no classicalanalog. We call such intrinsic angular momentum spin angular momentum.It turns out indeed that some particles have not only orbital angular momen-tum but in addition are born with spin angular momentum. Electrons areborn with a spin / = \ so m = ± | . Effects due to this can be measuredexperimentally and all attempts to explain these effects classically have failed.

2.7 PARTICLE IN A CENTRAL FORCE FIELD

The hamiltonian for a particle of mass p in a central force field is given by

(2.7.1)H = f P* + V(r),

where V(r) is a spherically symmetric potential, and

r2 = ?i2 •+ qf + q>*.

2.7 PARTICLE IN A CENTRAL FORCE FIELD

We would like to solve the energy eigenvalue problem

H\E> = E\E).

117

(2.7.3)

To begin we prove that each component lt of the orbital angular momen-tum as well as I2 commutes with H and are therefore constants of the motionfor spherically symmetric potentials. To show this, we first use the com-mutation relations (2.6.2) to generalize (1.10.6) to three dimensions:

(2.7.4a)

(2.7.4b)

Next if we let i,j, and k form a cyclic permutation of 1, 2, 3, then (2.6.3)may be written as

(2.7.5)

(2.7.6)

(2.7.7)

Next we shall need the identity

[AB,C] = A[B, C]+ [A,C]B,

which the reader may readily verify.We next show that

If we use (2.7.4)-(2.7.6), we have

Vi, P2] = foiPk, P2] - [Prfi, P2]

= [«i. P2]J>* - i, P2]

(2.7.2)

Next we have

since in a similar way we

But by (2.7.2)

XhlPjPk - PtPil

Vi, V(r)) =

have

to] - [(<Z,P* - I

= qj\Pk> v] •

= —ihlq,—

dr__

= 0. Q.E.D.

o,

My), V]~ [Pk, V]q,

7_ 0F 1* 3qk

9T

(2.7.8)

(2.7.9)

(2.7.10)

(2.7.11)

118

SO

and (2.7.10) becomes


dV dV dr qkdVdqk dr dqk r dr

dV!i,V(r)]=-ifi —

dr- 0.

(2.7.12)

(2.7.13)

If Kis not spherically symmetric, this result does not follow. The reader maynote that since i,j, k in (2.7.10) are a permutation of 1,2,3 it may be writtenin vector form as

P, V(q)] = -ihq x VK(q). (2.7.14)

(2.7.15)

It is now very simple to show that

[I2, V(r)] = 0.

For if we use (2.7.6), we have

P'. pi = 2 Mkrt + H<P*. V{r)) - X ihUi. V(r)] + [h, 0,

(2.7.16)

where we used (2.7.7) and (2.7.9). If we use now (2.7.1) and the results abovewe see that

Vi,H] = 0P 2 , # ] = 0, (2.7.17)

which shows that each lt and I8 are constants of the motion. However, by(2.6.4) the /, do not commute with each other although [/„ I2] = 0. There-fore, we may simultaneously diagonalize H, l3, and I2 as we shall now show[see Section 1.9]. We write the eigenvalue problems as

(2.7.18)

Since [lit H] = [/±, H] = 0, all the analysis of the previous section may betaken over directly. For example,

1+\E, m,

L\E, m,

m + l)\E, m + 1, (2.7.19)

m m)\E,

That is, l+\E, m, /) is also an eigenvector of /3,12, and H with the same E and

2.7 PARTICLE IN A CENTRAL FORCE FIELD 119

/ but with m increased by 1 only. Therefore, we have

lz\E, m, 1) = mh\E, m, I) (2.7.20)

\*\E, m, I) = /(/ + l)/i2|£, m, I) (2.7.21)

H\E, m, I) = E\E, m, I). ' (2.7.22)

We obviously cannot find the energy eigenvalues until we specify the explicitV(r).

We next wish to show that we must exclude the half-integer eigenvalues of/s when 1 is true ordinary angular momentum. This can only be done byobtaining the transformation function from the energy representation\E, m, /) to the coordinate representation since true angular momentum mustbe expressible in terms of coordinates and momenta. If we multiply (2.7.20)-(2.7.22) from the left by <q| where

1i\<li, q» «3> = tiki, qk, q'z), (2.7.23)we obtain when we use the definitions of/3,12, and H'm terms of p and q

i L dqk dq'J(2.7.24)

(2.7.25)

(2.7.26)

(2.7.27)

as the reader may verify. We have also generalized (1.10.46) and (1.10.47)to three dimensions:

In (2.7.25) we use the fact that

+ (q* +

where

ql* dqi'dqi\'

(2.7.28)

(2.7.29)


In applying (2.7.28), care must be exercised in maintaining the same orderingas appeared in F. For example,

. . . h d . h o . , ,(q I JWIPI = : - qi 7 z-, <« I-

i dq{ i dq[

(2.7.30)

We have obtained three simultaneous partial differential equations thatthe energy eigenfunctions [transformation functions from the energy tocoordinate representation] must satisfy. They are most easily solved bytransforming to polar coordinates where

qi — r sin 6 cos <p

q'2 — r sin 6 sin <p

ft'2 + ft'2 + ft'8

COS0 =. ft I _ It I _ /*i + ft + ft

(2.7.31)

tan w = - .qi

(2.7.32)

ft* = r cos 0

We have by the usual rules

d dr d dd d d cos ro . . .— = sm 0 sin 9 ; —- = - cos 0 sin <p; ——= (2.7.33)3ft* oqt r 3ft r sin 0dr

- = cos0;30 sin_0

dq*~~ r

If we use (2.7.32) and (2.7.33) and preserve the order of factors, we see that

3ft* 3ft*. / . . 3 cos 0 sin 9? 3 cos 93 3 \

— r sin 0 cos o> I sin 0 sin w 1 — -J ~ ~r~ I\ 3r r 30 r sin 0 3g?/dtp)

. . . / . n . d cos 6 cos o? 9 sin 9? 3 \ 3— r sm 0 sin »l sm 0 sin w — + —; ;—- — I = r~ •Y \ Ydr r d6 r sin Od<pJ d = mh(r, <p, 6\E, m, I). (2.7.35)i o• (2.7.37)

If we use (2.7.36) in (2.7.37), we easily obtain

Next from (2.7.35), we haveE(r, <p, 0\E. m, I). (2.7.38)

^- <r, tp, d\E, m, I) = / - im{r, <p, 6\E. m, J>

= -m2<r, <p, d\E, m, I).If we use this, (2.7.36) becomes

(2.7.39)

= - / ( / + l)(r, 6, y |£ , m, />. (2.7.40)

We then have succeeded in obtaining three equations, (2.7.35), (2.7.38), and(2.7.40), that the eigenfunctions must satisfy. The solution to (2.7.35) is

<r, <p, d\E, m, 1) = eim*f(r, d; E, m, /), (2.7.41)

where/ is as yet arbitrary but independent of q>. At this point the flag goesup. If we allow m to be a half-integer, the eigenfunction would not be singlevalued which is physically required for orbital angular momentum. The half-integer values of m cannot have a coordinate representation and thereforehave no classical analog. They do exist but correspond to spin angularmomentum.

When we restrict m to integers, / must also be an integer since m runs from—/ to + / in integer steps. Then (2.7.40) is the equation for associatedLegendre polynomials P,TO(cos d) and its solution which is finite when


0 = 0 and IT is

(r, 6, <p\E, m, /> - etm*Ptn(cos 6)R(r; E, m, I), (2.7.42)

where R is an arbitrary function of r which satisfies (2.7.38), namely,

This equation is independent of m so that R and £ will depend on the quantumnumber / as well as another which we call n. Therefore, we have

6)Rnl(r), (2.7.44)<r, 0, <p\n, m, /) = AT,

where iVis a normalization constant. Since in spherical coordinates the eigen-functions have factored, we may write

and letr, 0, <p\n, m, /> - <r|n><0, <p\m, />,

Yim(B, <p) =s (0, <p\m, I) —

where we normalized so that

sin0<*0

\m\)\) ! P ' » ( c 0 S

W, <P) = M « . - ; (2-7.46)

im( » 9>) are called spherical harmonics and are the transformation functionsfrom the /s, 1* representation to the angular ("coordinate") representation.

To proceed one must specify F(r).

2.8 PAULI SPIN OPERATORS

We have seen in Section 2.6 that the commutation relations

[/„ /,] = iMk, (2.8.1)

where i,j, and k form a cyclic permutation of I, 2, 3, and

[I2, /,] - 0 (2.8.2)

leads to integer and half-integer eigenvalues of /,. In Section 2.7 we haveshown that only integer values correspond to orbital angular momentum.If we consider an electron of charge —\e\, mass ft, and with angular momentumI, it has a magnetic moment associated with it given by [9]

m = — — 1 = — -I2u H

(2.8.3)

2.8 PAULI SPIN OPERATORS 123

in mks units. Here /? is called the Bohr magneton. The electron must be insome field of force to make it go in a curved orbit in order that 1 = r x pdoes not vanish. In this case if we put the electron in a magnetic field F, itwill have in addition to its energy

for a central field of force an amount given by

J?! = - m . F = — 1 • F.Ill

If the field is a uniform field along the z-axis, this reduces to

(2.8.4)

(2.8.5)

H i = HLL2^ (2.8.6)

Since la and Is commute with each other as well as Ho and Hx, the eigenvectorsof Ho are also eigenvectors of Hv Thus

m, 2/tm\n, m, I).

So the expectation value for H= Ho + Hx is*

<n, m, l\H\n, m,nl

Fom,

where

jffo|n, m, /> = Enl\n, m, I),

(2.8.7)

(2.8.8)

(2.8.9)

and m is an integer which runs from — /to +/ in unit integer steps. Therefore,if we measure the energy of the electron in the magnetic field, we shouldobtain the possible values given by (2.8.8). However, experimentally it isfound to have an additional amount given by

H,\e\h

spin Fos (2.8.10)

where s = ± J and it is independent of n, I, and m. To explain this, Uhlenbeckand Goudsmit [10] postulated that the electron has in addition to its orbitalangular momentum an additional nondassical spin angular momentum of

* This faamiltonian is not exactly correct, as we show in Section 2.10.


Let s correspond to the spin angular momentum of the electron so it willnot be confused with 1 which we restrict to orbital angular momentum. Forconvenience we define an operator o by means of

s = %ha. (2.8.11)

Associated with this spin momentum, we assume the electron has a magneticmoment given by

m, = — -jz s = kl*2/t'

—fta. (2.8.12)

Note that this twice as large as m, given by (2.8.3). The necessity for thisdefinition is justified from experimental measurement of the energy. Since shas no classical analog, neither does m, so we must define it to give results inagreement with experiment.

We now postulate that s or equivalently a obey the same commutationrelations as 1 since half-integer eigenvalues are included there. So we postulatethat

[<r,., <r,] « 21<rk, (2.8.13)

where / , / , k form an even permutation of 1, 2, 3. Also we require

[^,0*1 = 0. (2.8.14)These postulates lead to eigenvalues of a3 that are integer or half-integermultiples of h. To restrict the eigenvalues to ±.\h only, we postulate that theyobey the anticommutation relations

{ot, 2dit. (2.8.15)

This is all that is needed as we shall show below to restrict the eigenvalues to

From (2.8.15) we see that if i — j

o*=\, (2.8.16)

where 1 is the identity operator. Note that (2.8.14) is therefore identicallysatisfied.

If we next add (2.8.13) and (2.8.15), we find that

op, = iorfc,

where 1,7, fc are an even permutation of 1, 2, 3 so that 1 ^ j .If we define the two nonhermitian operators <r± by

(2.8.17)

(2.8.18)

2.8 PAULI SPIN OPERATORS 125

we may use (2.8.13) and (2.8.15)-(2.8.17) to derive the following commuta-tion and anticommutation relations:

(2.8.19)

(2.8.20)

(2.8.21)

(2.8.22)

(2.8.23)

(2.8.24)

(2.8.25)

(2.8.26)

[a±, a2]

[o±» o3]

{a±, ax}+

{<r±, o2}+

{<*±>»»}+

= «rs

= 1

= ±i= 0

where 1 is the identity operator. Also

00.

We look for solutions of the eigenvalue problem

whereot\oS> =

(2.8.27)

(2.8.28)

<r32 = 1. (2.8.29)

We have actually already solved this eigenvalue problem in Section 1.5. Byvirtue of the postulate (2.8.15) and therefore (2.8.16), a'3 = ± 1 . This resulthas not used (2.8.13) or (2.8.14). Yet it is included as a special case of theresults of Section 2.6 based on (2.8.13). That is, the anticommutation postulatehas picked out only one of the eigenvalues of a3 allowed by (2.8.13) and(2.8.14). Therefore, we may take over (2.6.46) and (2.6.47) directly for thiscase since s =

(2.8.30)

/0 1\ /0 -t\ (I 0\

\1 0/ \+i 0/ \0 - 1 /

1+)

Since o2 = ffi2 + a22 + <r8

2 = 3 always, there is no need to specify a2. Thea, are called the Pauli spin matrices in the representation in which a3 isdiagonal. Also

126

It is also easy to verify

Also


( 2 8 3 2 )

( 2 8 3 3 )

r_|-> = 0.

Let us now show how this explains the experimental results given by(2.8.8) and (2.8.10). The total angular momentum j is given by

j = I + s = 1 + " a

while the total magnetic moment is

in = ni| -f- in.

(2.8.34)

(2.8.35)

Since 1 is orbital and o spin momentum, 1 and o commute

[lif o,] = 0. (2.8.36)

Therefore, the total hamiltonian when we apply a magnetic field is

H = Ho + p(- + a\ • F. (2.8.37)

g

The eigenvectors are \n, m, I, s) since

H0\n, m, I, s) — Enl\n, m, I, s)

oa\n, m, I, s) = s\n, m, I, s)

/3|«, m, I, s) = mh\n, m, l,s)

1«|II, m, /, s) = /(/ + l)/«8|n, m, /, *>..

That is, since as commutes with all coordinate and momentum operators,we may simultaneously diagonalize Ho, a3, /„ and I2. Therefore, if F is uni-form along the z axis, we have

<«, m, I, s\H\n, m, I, s) - £„, + pF0(m + s), (2.8.39)

which agrees with experiment. The spectral lines are split by an amount 2/JF0

due to the spin which is seen experimentally.We have noted that the spin momentum eigenvectors do not possess a

2.9 SPIN OPERATORS IN THE HEISENBERG PICTURE

coordinate representation. We have then

m,

'|n,m,/,-l> = ( ° V\<q'|n, m, />/

Also we have by Section 1.5

(s\sf) = d.,

I+1X+H + I—1><—1| — l.

127

(2.8.40)

(2.8.41)

(2.8.42)

As in the case of the harmonic oscillator, there is another formal inter-pretation of the energy of the electron spin in a d-c magnetic field. By (2.8.26)and (2.8.27)

K , ff_}+ - / (2.8.43)c* = oi = 0.

But these are identical to the fermion operators b and M in (2.3.2). Since

| ( f f , + 1) = a+a_, (2.8.44)

then

PF0(2o+o_ - 1), (2.8.45)

which aside from a trivial constant is the same as (2.3.1). Thus we may thinkof a+ and a_ as creation and annihilation operators for a particle of spin \.Then since a+a_ is a number operator with eigenvalues 0 or 1, we have

a+a_\s) \)\s)(2.8.46)

The single "fermion" state is occupied or empty in agreement with theexclusion principle.

2.9 SPIN OPERATORS IN THE HEISENBERG PICTURE

We consider a free electron with spin in a d-c magnetic field along thez axis. The spin hamiltonian is

(2.9.1)


The Heisenberg equations of motion (1.15.5) for at, a+, and er_ are

do,(t)dt

= 0

dt ih +

<M0 1 r_ ,A

(2.9.2)

dt i/i " w h

where we used (2.8.21) and (2.9.1). The solutions are

a Jit) = = at

(2.9.3)

where the separation between the two energy levels is

ha> = 2pF0, (2.9.4)

and we have used (1.15.3) with

U(t, 0) = e-**"'". (2.9.5)

The operators oz(t), a+{i), and ajt) in (2.9.3) are in the Heisenberg picture,and oz, a+, and o\_ are in the Schrodinger picture.

The state vector at time t is

|y(r)) = g- toW2|y<0)>, (2.9.6)

while an arbitrary initial state, by the completeness relation (2.8.42), may bewritten

IV(O)> - | + 1><+1|V(O)> + |-1>(-HV(O)> 9

where cx and c2 are arbitrary constants. If we use (2.8.32), (2.9.6), and (2.9.7),the state vector at time t is

(2.9.8)

(2.9.9)

In order that <v(0lv(0> = <V(°)lv(0)) = l> ci a n d c« m u s t sati

It is left as an exercise to show that the expectation values of ax and a± for

2.10 HAMILTONIAN FOR ELECTRON IN ELECTROMAGNETIC FIELD 129

the state \y>(t)) are

<*•> = <y(t)\aMt)) - Icxi2 - |c2|2

(a+) = cfc2e<0>< {ax) = c*câi + Clcte-iat (2.9.10)

< O = c1cte-iat (av) = -icfc2eiat + iClc2*e-tot,

so that {ax) and (oy) will precess about the z axis at frequency to while (<rz)is left unchanged. This behavior is also typical of a classical magnet sus-pended on a gimble ring in a magnetic field.

PART III. ELECTRONS IN ELECTRIC ANDMAGNETIC FIELDS

In Chapter 5, we give the quantum theory of the interaction of quantizedradiation with matter. We include here a brief discussion of this interactionwhen the fields are unquantized.

2.10 HAMILTONIAN FOR ELECTRON IN ELECTROMAGNETICFIELD

We consider an electron of charge e, mass /i, and spin s in a sphericalpotential as well as an electromagnetic field. If A is the vector potentialdefined by

B = curl A = jUoF, (2.10.1)

where /i0 is the permeability of free space, the hamiltonian is (in mksunits)f

H = — (p - ekf + V(r) + /3a • F2/i

= - 1 p2 + v(r) + — A2 - — (p • A + A • p) + pa - F. (2.10.2)

From the commutation relations (2.7.4), we have

X fo» ^ ] = : I —' = 7 d i v A. (2.10.3)

Therefore, we may rewrite (2.10.2) as

H = ~- p8 + V{r) + — A2 - - A • p + i — div A + /So • F. (2.10.4)2/i 2/i ft 2/i

t The generalized canonical momentum p = fiv + eA where ft and v are the electronmass and velocity, respectively.


We are free to work in the Coulomb gauge for which

divA = 0. (2.10.5)

Consider the special case in which

A = \F0[-y, x, 0]. (2.10.6)Then


div A = 0F=[0 ,0 ,F 0 ] ,

, * . *. eFa

(2.10.7)

7t. (2.10.8)

^ + er,). (2.10.9)

Since /J = \e\hl2p and /„ = ay, — ypt, we may rewrite this as [e = —\e\]

H = i - p8 + K(r) + ^ ( * 8 + y1) +

If we compare this with (2.8.37), we see that we are in error unless the extraA* term, namely, e*Fl(x* + y2)/i/i, is negligible. However, the reader mayshow that [/3, a;2 + y2] = 0 but that [I2, z2 + V ] 5* 0, so that our resultsin Section 2.7 are not exact. In actual practice except for extremely strongfields the Fo terms are small compared with the kinetic energy and the Fo*term is small compared with the Fo term, so that the analysis in Section 2.7is approximately valid for ordinary laboratory fields.

PROBLEMS

2.1 Write the matrix elements of p and q for a harmonic oscillator in the JV-representation.

2.2 Show that the oscillator energy eigenfunction in the coordinate representationcan be written as

where 8 /2.3 If n and m are integers and a and a1 are the boson annihilation and creation

operators, respectively, show that

i- m)\

in -m) =(it - m)\

(at)n-m|0>

where cta\n) = n\n).

REFERENCES

2.4 If ? is a parameter, show by a power-series expansion that

131

2.52.62.7

2.82.9

2.10

where [a, <Jf] = 1. If <0|0> = 1, normalize the vector exp (loVerify the commutation relations (2.6.4), (2.6.6), and (2.6.11).Write the matrices for lu l2, and /3 when / = f.Find the matrix elements of <r+ and a_ in a representation in which <rx isdiagonal.Derive the relations given in (2.8.19M2.8.27).Find the expectation values for ax and <ry for the state given by (2.9.8).Solve the Schrodinger equation for the hamiltonian (2.10.9) when V(r) = 0.Find a wave packet that localizes the particle on circular trajectories

REFERENCES

[1J P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford: Clarendon,1958, Chap. 11.

[2] L. I. Schiff, Quantum Mechanics, 3rd ed., New York: McGraw-Hill, 1968.[3] See Ref. 1, Chaps. 9 and 10, and Ref. 2, Chaps. 14 and IS.[4] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, New York:

Harper & Row, 1961.[5] W. H. Louisell, Coupled Mode and Parametric Electronics, New York: WUey, 1961,

Chap. 1; see / . Appl. Phys., 33, 2435 (1962).[6] See Ref. 2, Chap. 14.[7] See Ref. 2, Chap. 2.[8] J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics, New York:

W. A. Benjamin, 1968; R. J. Glauber, Phys. Rev. 131, 2766 (1963). The coherentstate has been extensively used recently. See, for example, R. Horak, L. Mista, andJ. Perina, Phys. Lett. A {Netherlands), 35A, 400 (1971).

[9] H. Goldstein, Classical Mechanics, Reading, Mass.: Addison-Wesley, 1950, Chap. 5.[10] G. E. Uhlenbeck and S. Goudsmit, Naturwiss., 13, 953 (1925); Nature, 117, 264

(1926).

3Operator Algebra

To solve problems in quantum mechanics, we have shown that operators areinvolved that obey a noncommutative algebra, which can sometimes be quiteawkward. The first purpose of this chapter is to teach some of the techniquesinvolved in dealing with operators [1].

The only difference between operators and c-numbers (commutingvariables) is that the order in which operators are written down is significantwhile order is of no consequence with c-numbers. We therefore devote amajor part of this chapter to the study of ordered operators which allows usto associate c-number functions with operators. In this way we may transformquantum problems to equivalent "classical" or c-number problems where thetechniques of solution are usually more familiar. The equivalent "classical"problem contains all the quantum mechanical features and is therefore,strictly speaking, not a true classical problem. The major part of the chapteris devoted to the "classical" correspondence for boson operators, but welater generalize to arbitrary operators. To study this correspondence is thesecond major task of this chapter.

In Section 3.1 we begin by developing a few theorems which are valid forgeneral noncommutative operators. Section 3.2 contains a discussion ofnormal and antinormal ordered boson operators. The algebraic properties ofthese operators are discussed in the following section. In Section 3.4 weintroduce three characteristic functions for boson operators and derive theWigner distribution function which is the Fourier transform of one of thecharacteristic functions. Since the Wigner distribution function as well asother distribution functions that we shall obtain may sometimes go negativeand indeed fail to exist in certain instances, we see that our equivalent c-number representation is not a true classical probability distribution. Weare not concerned here with these difficulties, however. In Sections 3.5 and3.6, we discuss the Poisson and exponential distributions explicitly since theyplay such an important role in the statistical properties of radiation.

In Section 3.7 we derive the generalized Wick's theorem which allows usto write down practically by inspection the thermal average of the product

132

3.1 SOME GENERAL OPERATOR THEOREMS 133

of a large number of boson operators. In Section 3.8 we derive Wick'stheorem for bosons which allows us to convert time-ordered operators tonormal ordered operators. This makes the problem of taking matrix ele-ments much simpler than would otherwise be the case and also allows us tomake the transition to our associated c-number problem quite simply.

In Section 3.9 we generalize the ordering techniques to arbitrary operatorswhich allows us to transform a quantum problem to an associated c-number("classical") problem. We use this correspondence when we study thedamping of atoms, and such, in later chapters.

In Section 3.10 we present an operator description of independent atomswhich later proves quite useful.

In Part IV we give some elementary applications of the operator techniquesdeveloped. In Section 3.11 we solve the Schrodinger equation for a drivenharmonic oscillator by the normal ordering technique. In Section 3.12 wediscuss two weakly coupled oscillators while in the following two sectionswe obtain the distribution functions for a two level atom and for an oscillator.In Section 3.15 we obtain a generating function for oscillator eigenfunctions.

In Part V we discuss the principle of maximum entropy which is used toobtain the best estimate of the density operator for a quantum systemsubject only to partial knowledge about its state. We also obtain the densityoperator for spin-£ particles using this principle.

PART I. GENERAL OPERATORS

3.1 SOME GENERAL OPERATOR THEOREMS

In this section we derive several useful theorems involving two non-commuting operators A and B. We refer to functions of A or B, and wetacitly assume, without explicitly so stating in each theorem, that the func-tions may be expanded in a power series. Thus a function of B may beexpanded as

2n=0

(3.1.1)

where the cn are constant expansion coefficients. The constants are calledc-numbers. This is not the most general expansion of a function, but itsuffices for our purposes. Some of the theorems are therefore true formore general functions, but we are not concerned with this.

Also, if functions such as (3.1.1) are to have meaning when applied to

134 OPERATOR ALGEBRA

eigenvectors of B, for example, the series

(3.1.2)

where b is an eigenvalue of B and in general is a complex number, mustconverge, and F(b) must be defined for complex b. If B is hermitian, thenF(b) would have to be defined only for real b. Again, we are not concernedabout such matters.

We often use parameters that are c-numbers (not operators); we are notprecise whether these parameters are real, pure imaginary, or complex, andtacitly assume that they can be complex, with other quantities definedadequately so that the meaning will be clear. For example, if f is complexand F(A) = exp ({A), we tacitly assume that F(z) is a function defined forcomplex z.

THEOREM 1

If A and B are two noncommuting operators and £ is a parameter, then, if n is aninteger,

eSAjpig-fA _ (gMj3£-t-<*)»j (3.1.3)and

etAF(B)eriA = F(eiABe~SA). (3.1.4)

When n = 1, (3.1.3) is just an identity.

PROOF

To prove this theorem, we note first that

e&AerSA = /. (3.1.5)

We may write the right side of (3.1.3) as a product of it-factors

and by (3.1.5) all factors in the middle collapse to Bn and (3.1.3) is proved.To prove (3.1.4), we use the expansion (3.1.1) to write the left side of (3.1.4) as

where we used (3.1.3) in the last step. From the expansion (3.1.1), with the argumentreplaced by e*ABeSA, we see from the above that (3.1.4) follows.


As an application of this theorem, we let A — ipfh, B = F(q), where [q,p] = ih.

h ) \ h / |_ \ h) \ ^ / J

If f is real, we have, by (1.10.8), (1.10.9), and (1.10.12),

= ? + *, (3.1.7)

(3.1.8)

so that (3.1.6) reduces to

£,/>, q) = exp i^ - ^f) - F(q + !).

There is an interesting alternative proof for (3.1.8). If we differentiate (3.1.8)partially with respect to I, we have

•K = 4 &>,/] = ¥ , (3.1-9)

where we used (1.10.6). This gives a partial differential equation that /mus t sat-isfy. By inspection, we see that any function of the form

/(*,/».*) «) (3.1.10)

is a solution of (3.1.9). To evaluate the form of the function g, we let £ = 0 in(3.1.8) and (3.1.10) and see that

f(0,p,q)=F(q)=g(q),

so that (3.1.8) follows. We make repeated use of this technique in the remainder ofthe chapter.

THEOREM 2

If A and B are two noncommuting operators and if A'1 exists, we have

/IBM"1 = (ABA-1)", (3.1.11)

where n is an integer, and

AF(B)A~1 = FiABA-1). (3.1.12)

The proof of this theorem follows the same pattern as that of Theorem 1 and isomitted. In Theorem 1 exp ( ± $A) always exists whereas here only operators A forwhich A~x exists may be used.


An interesting case that often arises is f(B) = exp (B) so that (3.1.12) becomes

AÂ-1 = exp {ABA-1) (3.1.13)

THEOREM 3

If A and B are two fixed noncommuting operators and I is a parameter, then

B + £[A, B] + | y [A, [A, B]) + | y [A, [A, [A.Bjn + ••'. (3 .1 .14)

PROOF

We let/ ( I ) = e*ABe-tA /(0) = B,

and expand/(I) in a Maclaurin series in powers of £. We then have

dl -dS |_o

d*fd?

[A,B]

, {,A B]].

If we continue in this way, (3.1.14) follows.As one application of this theorem, we let A = ip\h, B = q, and assume £ is real.

If \p, q] = —»*> we have, from (3.1.14),

(3.1.15)

since all the remaining commutators in (3.1.14) vanish. This is just (3.1.7), which wefound by an alternative method.

As another illustration of the theorem, we let A — q*\2 and B = d/dq, where q isan ordinary variable. It is easy to show that

(3.1.16)

where we tacitly assume that both sides of (3.1.16) are to operate on some functionF(q). Also,

[A, [A,B]]=

so that the series in (3.14) breaks off after the first two terms, and we have

dq dq(3.1.17)


We again tacitly assume that we are to apply both sides of this operator equality tosome function of q, say F(q).

From (3.1.17) and Theorem 2 (Eq. 3.1.11) we find also that

d(3.1.18)

where n is an integer.

THEOREM 4

If A and B are two noncommuting operators that satisfy the conditions

[A, [A, B]] = [B, [A, B]] = 0, (3.1.19)

then

This is a special case of the Baker-Hausdorff theorem of group theory; the readerinterested in pursuing the operator formalism to a greater degree of sophisticationmay find Refs. 2 to 5 of interest. The proof of this theorem given here is due toGlauber [6].

Any two operators whose commutator is a c-number, for example, [q,p] = ihand [a, a*] = 1, satisfy the conditions of the theorem; thus it is not surprising thatthis theorem has many applications.

PROOF

To prove the theorem, we consider the operator function

(3.1.21)

where I is a (c-number) parameter. If we differentiate with respect to f, we have

d* (3.1.22)= {A + e*ABe-*A)f(J),

since exp (SA) exp (—£A) = I. We may use Theorem 3 (Eq. 3.1.14) to expand thesecond term in the parentheses. By virtue of (3.1.19), all terms after the first two in(3.1.14) vanish, and so

B + £[A, B]. (3.1.23)

(3.1.24)

Equation 3.1.22 may be written

%•. = {(A + B) + i[A,


By (3.1.19) we see that the quantity A + B commutes with [A, B], and so we maytreat these two quantities as ordinary commuting variables and integrate (3.1.24)subject to the initial condition [see (3.1.21)]

/ (0) = 1. (3.1.25)

The solution of (3.1.24) that satisfies (3.1.25) is], (3.1.26)

where the last form follows since A + B commutes with [A, B]. If we now equate(3.1.21) and (3.1.26), let £ = 1, and multiply both sides from the right by« P (—iM» Bi)> * e n (3.1.20) follows. The proof of the second form of (3.1.20) isleft as an exercise.

As an application of (3.1.20), we let A = ty and B = pq, where [q,p] =ih andA and ft are parameters. Then by (3.1.20) we have

(3.1.27)

PART II. BOSON CREATION ANDANNIHILATION OPERATORS

In later chapters we are concerned with solutions of quantum problems in-volving creation and annihilation operators for bosons. This will include,for example, the calculation of expectation values of operators that arefunctions of a and a f, since these averages permit comparisons betweentheory and experiment. We also need to solve Schrddinger equations in-volving boson operators.

One example is a driven harmonic oscillator where the hamiltonian is

H = h(o(d*a + i) + hf(t)(a + at).

The function/(f) is the driving term. Since this hamiltonian is time-dependent,the system is nonconservative. In such a simple problem as this, the onlymethod of solution at our disposal at the moment is the cumbersome iteratedsolution given in Section 1.16. Our purpose here is to make the solution ofsuch problems possible by much more powerful operator techniques. Thesetechniques involve the concepts of ordered operators.

3.2 ORDERED BOSON OPERATORS

Any function/(a, af) of the boson operators a and af which satisfy thecommutation relation [a, a1] = 1 is defined by its power series expansion in


a and at. For example, sin a W is defined by

139

«=o "' (2/ + 1)! '

More generally, / (a , a1) will usually consist of sums of terms of the form

/(a, anf(l, m , . . . , n),

where /, m , . . . are positive integers or zero. We are always free to use thecommutation relation oaf — ^ o = 1 repeatedly to rearrange the a's and atssamong themselves. This will yield different forms for/(a, a1) but they willalways be equal. For example, iff— ad1, then

/ (a , af) = 00* = a*a + 1.

The two functions have different forms, but they are equal. Suppose now werepeatedly use the commutation relation in all terms i n / s o that all a's inevery term of the sum are to the right of all a^s. The function then is saidto be in normal order, and we may write it as

/(a, (3.2.1)

We put a superscript n on the operator function to remind us that it is innormal order and since we have faithfully used the commutation relation toput/into normal order , /= / ( n ) . The/J"' are expansion coefficients whichare independent of a and ar.

As another alternative we may commute all a's to the left so that the samefunction/(a, af) can also be written as

/(a, at) = (3.2.2)

In this case the function is said to be in antinormal order which we indicateby a superscript a. In general,/£' 9*/£ } although

/(a, at) = /<»>(a, a*) = / (o)(a, at). (3.2.3)

These orderings are possible for all functions which may be expanded in apower series. However, in all but a few trivial cases, the ordering will be avery tedious procedure. In the next section, we develop some very usefultechniques for obtaining the ordered operators by indirect means in a numberof useful cases.

Since the normal and antinormal forms of a function which can be expandedin a power series are unique, we can establish a one-to-one correspondencebetween either/<n)(a, a1) or/(a'(a, at) and ordinary functions fM(a, a*) or


/< o )(a, a*) of a complex variable a which will prove extremely useful. Thiscorrespondence may be accomplished as follows.

We define an operator JV~X which transforms the operator functionfln)(a, af) to an ordinary function/(n> (a, a*) of the complex variable a byreplacing a by a and a* by a*. That is,

A/—i<nw-n\ r,*lnm ex o A \

%/V \O O j = OC ft , (j.Z.HJ

where / and m are integers. The operator is defined to be linear so that

jr-i{cfi»Xa, at)} = <:/<»>(«, «*), (3.2.5)

where c is any complex number, and

^{cl} - c, (3.2.7)where / is the identity operator. From this definition, it follows from (3.2.1)that

< W = /(»>(a, «•). (3.2.8)

We have used a bar over the function to indicate that we have an ordinaryfunction, and the superscript n indicates the function is associated with thenormally ordered form of the operator fw(a, a*). Since the normal form ofthe operator is unique, there is a one-to-one correspondence between/<»>(«, J) and/(»>(oc, a*).

To obtain/<n '(a, a*) we see that we first put f(a, cfl) into normal orderand then replace a by a and a* by a*.

We may also define the normal ordering operator Jf by

= a*mal (3.2.9a)

- cf[n)(a, af) (3.2.9b)

ri»»(«, a*) +/iB)(a, a*)} - /<»>(a, a^ + A»\a, a+) (3.2.9c)= cl (3.2.9d)

= 1. (3.2.9e)

The normal ordering operator Jf replaces a* by af, a by a with all flt's tothe left of all a's.

In a similar way we may define the operator jrf~l which is applied to anti-normally ordered operators by means of

jtf-l{ald*m} = a'a*M (3.2.10a)

l - c / U f c a * ) (3.2.10b)

} = /io)(a, a*) + /2(o)(a, a^ (3.2.10c)

) = c. (3.2.10d)


We define s/ as the inverse of s/-1:

141

Therefore, when si is applied to a function of a and a* we have

^'(a, a*)

te)(a, a*)} = cf(a\a,

c} = c/.

(3.2.11)

(3.2.11a)

(3.2.1 lb)

(fl> flt)

(3.2.1 Id)

Again st replaces a by a, a* by of and puts all op's to the right of all a's.To gain further insight into the ordinary associated functions/<n>(a, a*)

and /<o)(a, a*), let us see how they are connected with the coherent staterepresentation given in Section 2.5.

THEOREM 1*

The associated normal function/(n) (a, a) is given by the diagonal matrix elementof/(a, a*) in the coherent state representation:

/<«>(<*, a*) = <«|/(fl, «t)|B> = T r [|«><a|/(fl, at)], (3.2.12)

where a|a> = a|a).The proof follows easily. From Sec. 2.5 we know that since a|a> = a|a> and

<a|at = a*<a|, then

(3.2.13)

<a|a> = 1.

If we therefore take the diagonal matrix element of both sides of (3.2.1) and use(3.2.13) we obtain

where in the last step we used (3.2.8). This proves the first half of the theorem.Since Tr|»><p|= (v\u), the last form follows if we let |«> = |a>and| = <»|/(a, «t).

* Theorems referred to in this section are those in the section unless otherwise noted.


THEOREM 2

We may put a function/(a, a*) into normal form by means of

/(a , a t ) | a > } . jrff â + _£_ , (3.2.14)

where/(a + (9/ da.*, a*) is obtained by replacing a by a + (9/9a*) and af by a*in the original power series expansion of the function /(a , a*). Note that if / isalready in normal form

/ ( B ) «+•=! . •(3.2.15)

since all the terms [a + (9/9a*)] will appear on the right and 9/9a* - 1 = 0 . Actu-ally this theorem merely replaces commutation relations by differentiations and isjust as tedious to apply in general. It will prove useful in certain special situations.

To prove the theorem we shall need the following lemma.

LEMMA

If |«> is a coherent state vector, then

|cc><<x|a= L+JL\ |a><« | . (3.2.16)

PROOF

By (2.5.12b) and its adjoint, we have

|<x)<a| - *-«*«e«"t|O>CO|e"*8.

Then we see that |<x)<«|a may be written as

|ce><<x|a - e—« ~

(3.2.17)

(3.2.18)

Q.E.D.

3.2 ORDERED BOSON OPERATORS 143

By repeated application of the lemma, we see that

|a><a|al = (a + ~ ) »|a><«|. (3.2.19)

Similarly, since <a|at = a*<a|, we see that

|aXa|at»» = a*"«|a><<x|. (3.2.20)We also note that

Tr|a><«| -<a|a> = l. (3.2.21)

Consider now a function expanded in a power series in arbitrary order:

f(a, at) = 2 • • • at'a"* • • • a V • • •. (3.2.22)

Then if we take diagonal matrix elements of both sides in the coherent state repre-sentation, we have by Theorem 1

/<»>(a, a*) = <a|/(a, at)|a> = Tr {|a><a|/(a, at)}

= Tr{|aXa|2*--« t I«m- '

If we repeatedly use (3.1.19) and (3.1.20), we may move the projection operator|a)<a| through each term as follows:

*, a*) = Tr {^ ' ' • a^laXala* •

( B V

where in the last step we used (3.2.21). If we compare this with the original expansionof the function (3.2.22), we see that we obtain the normal associated function byreplacing at everywhere by a* and a by a + (9/d«*) and let it operate on unity.The theorem then follows if we apply the operator ./f* to both sides. Wehavealreadynoted that this method of putting a function into normal order is completelyequivalent to using the commutation relations repeatedly.

THEOREM 3

Any function/(a, at) which is suitably behaved may be represented by the inte-gral

— |aXa|/<«>(a, a*)

(«, a*)},

where/(a>(«, «*) is the antinormal associated function.

(3.2.23)


This theorem states that the antinormal ordering operator is equivalent to the

integral operator I d2a|«><a|/ir. Glauber [7] has called / ( 0 ) (a, a*)/*- the /"-repre-

sentation of the operator/(fl, o t).

The proof of this theorem follows from the completeness relation (2.5.17)

d2*— |«><a| = 1.

7T

(3.2.24)

From (3.2.2) we have

- f — 2/<;V|«X«|flt«

f — |a><a| %/<?*'**' - f— |«><a|/<«»(«, a*).

Here we have inserted the completeness relation between aT and au, and used(3.2.13) and the associated antinormal function which proves the first part of thetheorem. Since

/(a,

and the second half of the theorem follows.

(a, a*)},

THEOREM 4

The antinormal form of/(a, a*) may be obtained by

/<°>(a,«f) - J / { / ( « , « • - -IV1J - J ~ | o t ) < a | / (" • a * ""i) '*• (3-225)

where/(a, a* — (d/da.)) is obtained from the original power series expansion off(a, at) and replacing each a by a and each af by a* — (3/ 3a). Note that if/ is inantinormal order, then

(3.2.26)

since (B/da) - 1 = 0 . This is just the analog of Theorem 2 (3.2.14) for antinormalordering. It again replaces commutation relations by differentiations of ordinarycomplex functions.

The proof of the theorem is rather tedious. If we use the completeness relation


(3.2.24), we may write

145

f »2

- T • • • flt!flm • • • a"atW|a><a| — . (3.2.27)J w

We next need the following lemma to move the projection operator |a><a| to the left.

LEMMA

To prove this, we have

*,«><«,_(«•+_),«><«,

e~"'— e"af\0X0\e"a

oa

(3.2.28a)

By repeated application of the lemma we see that

<7tr|a><a| = / a * + J L j | a > < 0 £ | .

Since a* and a are independent variables, we see that

Q.E.D.

(3.2.28b)

so that we may expand the right side of (3.2.28b) by the binomial theorem, whichgives

fltr|a><a| ( 3-2-2 8 c )

Consider now the last two terms in (3.2.27)

C d*«. C d2a.

I = J atV|a><«| - = J aV»<« | - ,since o°|a><a| = oe»|a)<a|. If we use (3.2.28c) this becomes

r ' r! f a , * , ,dxda.*1 = Z "77 r, a*r~'«* T"J la><al .

where we have written d*«. = rfa rfa*. (To be correct, we should change to realvariables, but the reader may verify that our procedure gives the correct result


without the extra labor.) We now integrate this expression for / by parts s-times withrespect to a. Since the integrated parts vanish at infinity due to the presence of thefactor exp (—a*a) in |a)<a| which goes to zero faster than any finite power of a anda* then, / becomes

We may sum the series by the binomial theorem, and we have

/ = J|«><a| - ( a * - _ ) « • • 1.

If we proceed in this way to move |<x)<a| through the other operators we see that

jf(a, ai)|«><«| = J|«X«| V I • • • ( « * - i ) « - -

and the theorem is proved.

THEOREM 5

The "/{-representation" [7] is defined by

f{a, flt) = tt d** rfV|a><a'|J?(a, a'*),

where

(3.2.29)

R(x, a'*) = ^/<">(«, «'*) exp [-J|a|« - £|a? + a'a*]. (3.2.30)

To obtain this theorem we apply the completeness relation (3.2.24) twice:

rr</2a </v/(a, at) = — |a><a|/<»»(a, aâ'Xa'l . (3.2.31)

JJ w w

Since/(n> is in normal order, we have

7,at)|«'> = <«|

where we used (3.2.13) and note that / < n ) ( « \ «*) is obtained from/tn)(a, at) byreplacing a by a' and af by a*. By (2.5.15)

<a|a'> = exp [-*|«|* - | |a'|« + a'a*].

If we use these last two results in (3.1.31), the theorem follows. It turns out [8] thatthe /{-function exists in cases where the P-function does not exist, but we shall notconsider these finer mathematical points.


The associated functions turn out to be very useful since they allow us to trans-form quantum operators into ordinary functions whose manipulations are oftenmore familiar.

A problem which often arises is the evaluation of traces of functions of a and at.We show in the next two theorems how traces may be converted to ordinary inte-grals by means of the associated functions.

THEOREM 6

The trace off (a, a*) is given by

Tr/(a, a t)

Tr/(a,flt)

(3.2.32)

(3.2.33)

That is, the trace may be found by integrating either the antinormal or normalassociated functions over the complex a-plane. The traces exist if the integrals exist.

The proof of (3.2.32) follows directly if we take the trace of both sides of (3.2.23)and note that Tr |a><a| = <a|a> = 1. To prove (3.2.33) we introduce the complete-ness ielation in the trace:

i, af) = Tr I f(a, af)|aXa| —J *

J ' *since Tr |«><o| = <»|H>, and we let \u) =/ |a> and <P| = <a|. The theorem then fol-lows if we use (3.2.12).

THEOREM 7

Iff (a, a*) and p(a, a*) are two functions of a and a*, then

Tr P(a, flt)/(a, **) - JV«>(«, «*)/<»>(«, a*) ~ (3.2.34)

or

f </2aTr P{a, a*)f(a, a*) = j /><»>(«, a*)/<«>(a, a*) — . (3.2.35)

To prove (3.2.34) we expand p = p(a) and / =/(n> in power series and use the


cyclic property of traces: Tr ABC — Tr CAB = Tr BCA. Thus

T r f,(oy(B) 2lm

Since at(»+oa(»»-K) is in normal order we use (3.2.33) and obtain

Tr

, a*) . Q.E.D.

A similar proof holds for (3.2.35).The associated functions/<n)(«, «*) and/ ( 0 )(a, a*) of the complex variable a can

be thought of as functions of two real variables x and y where a = * + ity. As suchwe may define a two-dimensional Fourier transform of them. It will sometimeshappen that a problem is simpler to solve in the transform variables than in theoriginal variables x and y (or a and a*). Accordingly, we define the Fourier trans-forms as

, (*) = *-««***•••»/<•>(«, a*) — .

(3.2.36)

(3.2.37)

(In real variables we have00

Fw(kx, ky) = ^ Me-i<k**+*'*f<'»(x, V) dx dy

where+ I*

However, it is just as simple to work in the complex space directly with I and £*independent variables as well as a and a*.)

We can now prove the following theorem.

THEOREM 8

Tr/(a, at) = ^"'(0,0) = F™(fi, 0) (3.2.38)

That is, the trace off {a, a*) is found by evaluating the Fourier transform of eitherassociated functions at £ = I* = 0.


The theorem may be proved by letting f = $ * = 0 in (3.2.36) and (3.2.37) andcomparing with Theorem 6 above.

We can invert (3.2.36) and (3.2.37). We multiply both sides by exp i[H' + $*<*'*]and integrate over all f-space:

( 3 2 3 9 )

where we interchanged the order of integration. But*

(3.2.40)

When we use this in (3.2.39) we have

f fJ «•<•'+—TO, ^)rf«f-wJ/(a,«*).

= */(«', a'*).We therefore have shown that

fM(a a*\ — I »«l«+4»«»)/?(»

_ a )

d2tf *) —! .

(3.2.41)

We may apply the normal and antinormal ordering operators to the first andsecond, respectively, of these equations and obtain the following interestingtheorem.

THEOREM 9

/(«,flt)=J

, at)

(3.2.42)

which we shall call the Fourier representation of the operators. (Note that we usedthe fact that

(3.2.43)

as may be seen from the power series expansions.)

* We may verify this by changing to real variables.


THEOREM 10

Tr a*)]

(3.2.44)

(3.2.45)

This gives the Fourier transforms in terms of the original function.To prove this theorem which is closely related to the characteristic function to be

discussed later, we note that in (3.2.44) the first two operators are in antinormalorder. If/is in normal order, we may use Theorem 7 which leads directly to (3.2.36)and the theorem is proved. A similar argument may be used to obtain (3.2.45).

The following theorem gives a direct relation between the two associated func-tions.

THEOREM 11

/<»'(«', «'*) = J/<«>(«,/ '

(3.2.46)

Therefore, if/<0) is known, in principle we may obtain/<n).To prove the theorem, we take diagonal matrix elements of both sides of (3.2.23)

in the coherent state representation. This gives

The left side by Theorem 1 is just /(B)(<x', a'*). By (2.5.16)

|<a'|a>|2 - exp - | a - a'|\

and the theorem follows.

(3.2.47)

3.3 ALGEBRAIC PROPERTIES OF BOSON OPERATORS

In this section, we prove a number of theorems which will aid us in puttingfunctions of boson operators into normal and antinormal order.

THEOREM 1

If / is an integer, then

' da'(3.3.1)

& * . « ' ] - - f a " - - 5 - .

3.3 ALGEBRAIC PROPERTIES OF BOSON OPERATORS 151

The theorem may be proved by induction since [a, <jt] = 1. We leave this as anexercise.

THEOREM 2

Let * be a c-number (not an operator) and/ (a , a*) be a function which may beexpanded in a power series in a and a*. Then

e**f{a, x) (3.3.2a)

(3.3.2b)

The theorem is true regardless of the order of/and is therefore true if / is in nor-mal or antinormal order since/ =/<"• = / ( o ) .

To prove the theorem, we have by (3.1.4)

f[a, (3.3.3)

where we have assumed / may be expanded in a power series. The second stepfollows since a and exp xa commute.

Next, if we use (3.1.14) we see that

(3.3.4)

since

and higher order commutators vanish. When we use (3.3.4) in (3.3.3), we have

e**f(a, a V s 0 =/(<*, a* + *), (3-3.5)

and (3.3.2a) follows. A similar proof holds for (3.3.2b).Suppose that we have somehow managed to put/into normal order. However,

e*°/(n> and/^V*^ will not be in normal order. By (3.3.2) we have that

(3.3.6)x> flt).

Since the right sides of both equations are in normal order, we have been able toput c*°/{n> and/We10* into normal order. If we take the diagonal elements of bothsides of (3.3.6) in the coherent state representation, we have by (3.2.12) that

<oc|c*°/(a,

<<x|/(a, a *, **)•(3.3.7a)


From the definition of the normal ordering operator, we also have that

e*°/(a, at) = •/r{e*a/(n)(«, a* + *)}

/ (a , at)e*ot = jr{e*a*fM(u + *, a*)}.(3.3.7b)

It is simple to show that the theorem also holds in the following case

a* - x)r^ = / ( a , at)

- x, a V 0 ' = / ( a , at).(3.3.8a)

If we assume/has somehow been put into antinormal order, we have by (3.2.23),

/<a»(a, a V ° = e*°/<0)(a, at - x) = jf{e**fia>(*, a* - x)}

**fW(fi, at) =/<«)(a - *, tf)e*J - &{<**'/<*(« - x, a*)}, ( 3 3 - 8 b )

since the middle forms are already in antinormal order. By (3.2.23) we may alsowrite these as

ck ' (3'3-8c)e"*f(a, at) - J - - |a><a|{«-y«(a - x, «*)}

THEOREM 3

If/(a, af) is a function which may be expanded in a power series in a and aT,then

|

(3.3.9a)

[at,/(a,at)]= - i . (3.3.9b)

We prove (3.3.9b). Since/ = / ( 0 ) , we have on expanding/(B)

1«) fy.

If A, B, and C are any noncommuting operating operators, it is easy to verify byexpanding the commutators that

[A, BC] = [A, B]C + B\A, C\.

Therefore, we may write the commutator above as

at']}.


The second commutator here on the right obviously vanishes, and the first is foundby (3.3.1) so that

a/««)~1a~'[at,/] - -

Since/ = / ( o ) , (3.3.9b) follows. The proof for (3.3.9a) is similar.I f / i s in normal order, then a/<n) and/ (n)at are not, but we can put them into

normal order by the theorem. Therefore,

at) =

fM

da

(3.3.10)

The right-hand expressions follow since the middle terms are in normal order.Clearly

Alternatively, we have by (3.2.12) that

<«|a/(a,at)|a> = (a + ^ «, «*)(3.3.11)

In case/ is in antinormal order, we easily show that

/(a,

(3.3.12a)

— ( ( a* - —) /<«>(a, a*) j . (3.3.12b)IT \ \ CltJ J

154

THEOREM 4

If m is an integer a n d / =/ (n> =/ ( o > ,

amf(a, at) = ^ [ ( a + —;) /(n)(«, oc*)l

while

/ ( a , at)am

OPERATOR ALGEBRA

(3.3.13)

at»/(a, «+) = ^{(a* - yTfw(*, «*)} •(3.3.14)

For simplicity we omit theP-representation of (3.3.14). This theorem may be provedby induction and using Theorem 3 (3.3.9).

THEOREM 5

Iff (a, at) may be expanded in a power series then

«~V(fl» aV-*01" =/(<»?-*, afe*). (3.3.15)

In particular

= ae"* (3.3.16a)

= a V . (3.3.16b)

We first derive (3.3.16a) by expanding

g(x) = e*° °aer~*° °

in a power series in x. We have that£-(O) = a and

dx '

If we use (3.3.9a), this becomes

so that

'a — _ o


and

Therefore,

155

^ ) = a ^ l - a : + - - - Q.E.D.

To prove the theorem, we expand / in a power series, insert e"»'v** ° = 1d t ri hi i

A similar proof holds for (3.3.16b).To prove the theorem, we expa / p

between each factor of a and at and resume the series. This gives

&"**"/(a,

If we use (3.3.16), the theorem follows. We also prove the following lemma.

LEMMA

If |0> is the vacuum state such that a|Q> = 0 and x and y are parameters, then

ga.t«eTOt|0> = e x p (2^)10) . (3.3.17)

By (3.3.15), we have

Sincee x p

the lemma follows if we apply both sides of the expression above to the vacuumstate.

THEOREM 6

If / i s a function of a*a, its normal form is

} («, «*)}. (3.3.18)n=0

PROOF

The completeness relation of the energy eigenstates is

l»Vffl 1

where cfia\ri) = n|«>. Since

/(ata)|n> =/(n)|n>,

(3.3.19)

(3.3.20)

156

it follows that

OPERATOR ALGEBRA

2 /(n)|n><«|. (3.3.21)

If we take the diagonal matrix element on both sides in the coherent state representa-tion, we have by (3.2.12)

, a*)

By (2.5.13), we havea*aL\n

(3.3.22)

(3.3.23)

from which the theorem follows when we apply the normal ordering operator toboth sides.

LEMMA 1

If a; is a parameter, the normal form of exp —xefia is

[£-xat<>]<n) _ 2 — ~- tflal

= *f{exp [(<r* - l)a*a]}. (3.3.24)

PROOF

By the theorem we have

= exp

When we apply the normal ordering operator, the lemma follows.An alternate proof of this lemma which is more general illustrates the use of the

associated function to manipulate operators.Let

(3.3.25)

(3.3.26)

If we differentiate both sides with respect to x, we have

Note that since/is a function of d*a only, it follows that

Wa,f] - 0.


Assume that / is in normal order so that f—fM. If we use (3.3.10), we mayrewrite (3.3.26) as

« .dx l.

J(3.3.27)

Then since fM is in normal order by assumption, if we take the coherent statediagonal matrix elements of both sides since a|a> = a|a>, <a|at == a*<a|, and

, a*)

it follows from (3.3.27) that

dx

(3.3.28)

(3.3.29)

which is the partial differential equation obeyed by the normal associated function.If we make the change of variable

x — In a* a* = exp | ( | —

by the usual rules of differentiation we have

a a a

da* d£ Bt] '

with this change of variable, (3.3.29) becomes

2-

If we integrate, we have

(3.3.31)

(3.3.32)

where g(v) is an arbitrary function of integration which may be determined asfollows. If x = 0 in (3.3.25),/(0, da) = 1 so that/(n)(O, a, a*) = 1 andln/(n»(0,a, a*) = 0. When * = 0, by (3.3.30) we have n = —$ and (3.3.32) reduces to

0 = -2w~* +g(rj), (3.3.33)

which givesg(v)- If we put this in (3.3.32) and change variables back to x and a*,we find that the normal associated function is

/<»>(*, a, a*) = exp [{e~* - l)a*a],

as in the previous case.

158

LEMMA 2

The vacuum state projection

|0X0| =limj«—1

operator |0X0|

= lime-»l

is given

CO / _

1=0

OPI

by

£)W/!

OPERATOR ALGEBRA

(3.3.34)

PROOF

By (3.2.12) and (2.5.13) we have, when we take the diagonal coherent state matrixelement of |0><0|,

/<»>(<x, «*) = <<x|0X0|a> = |<a|0>|s

= e_«»« (3.3.35)

We must write this asCO ( — A1

<a|0><0|«> = / ( n ) ( « , «*) = Hm <?-"*• = Hm ]£ ^-rp- (a*a)'. (3.3.36)

Let us apply the normal ordering operator to the above. This gives

|0><P| = l im f ^ - «t«a». (3.3.37)t—11=0 * !

When we take matrix element in the number representation, we obtain

<n|0X0|»i> = lim f ^f- <n|at'al|ffi>. (3.3.38)

By repeated application of (2.2.10) we have for / ^ m

and for I <,n

Therefore, (3.2.38) becomes

{m-t)\\m - />, (3.3.39a)

(3.3.39b).

: lim (1 - <)m6nm,<—I

(3.3.40)

since(n -l\m -f> =


and we have summed the series. Since

lim (1 - e)™ = dm0,«—i

both sides of (3.3.40) are equal.

LEMMA 3

The normal form of the operator |n><m| is

\n)(m\ =li

159

7 a t fl\\Jn\ m\

This is proved by the same procedure as Lemma 2.

LEMMA 4

The normal form of the projection operator |a><a| is

(3.3.41)

(3.3.42)

where the normal ordering operator is understood to be applied to 0 and ft*.

PROOF

By (3.2.12), (3.2.9), and (2.5.16), we have

where we used Lemma 2 above.

THEOREM 7

The antinormal form of exp ( —xcfia) is given by

00 (1 —(3.3.43)


This theorem is most easily proved by using the differential equation technique forthe antinormal associated function as in Lemma 1 above. The proof is left as anexercise.

THEOREM 8

The normal form o

/(ata)at

while

is

\n=0 HI

l)a= A f

(3.3.44a)

*aa]. (3.3.44b)

PROOF

We prove (3.3.44a) and leave (3.3.44b) as an exercise. By (3.3.10) and (3.3.18) wehave

( oo

«* 2a*rt\n(

«* 2 f(n)e-n0

n - 0- «*n + 1

\B]

- n - l

since the first and last sums cancel in the next to last form. If we let n -*• n + 1 wehave

n=0a*)

f

(3.3.44c)

If we use (3.3.18), the theorem follows.The evaluation of traces plays a very important role in radiation theory, since

mean values of observables are to be compared with experimental measurements.For example, if L is an observable, the mean value of f(L) when the system is in astate |y> is

Tr


where the last step follows since Tr \u)(v\ = <t>|u>. We have already given threeways of evaluating traces of functions of a and a1. The first involves taking the di-agonal matrix elements of/(a, a*) in the |« representation and summing:

Tr/(fl, -5>i/(a,.n=0

A second method consists of obtaining either the normal or antinormal associatedfunction for/and integrating [see (3.2.32) and (3.2.33)]:

«>( *)

The third technique follows from the Fourier transforms of the associatedfunctions. From (3.2.38) we have

T r / f o a*) = FM(0,0) = Fw(0, 0).

In the following theorem we give another method which is often quite useful inevaluating the trace of f(a, a*)exp (—ka*a). Physically this corresponds to anensemble average for a system in thermal equilibrium (Boltzmann distribution) aswe show later.

THEOREM 9

If/(a, a?) is a function of a and aT, then

</(*, a% m (1 - e~x) Tr/(a, o V *

b\ y/l +n fk b]\0,0>, (3.3.45)

where A is a parameter, and

V - 1(3.3.46)

where b and tf are boson operators which commute with a and at, and |0,0) is thevacuum state for a and b:

(3.3.47)a|0,0> B ff|0>o|0>t, = 0

b\0, 0> a b\O\\Q\ = 0.

This theorem allows us to convert the evaluation of such thermal averages to theevaluation of the double, vacuum expectation value of the function with a replacedby v 1 + na + <Jiiftt and at by the adjoint of this linear combination [9].


PROOF

The trace in the number representation |/t> where

aâ\n>a = n\n)tt,

in (3.3.45) is given by

n=0(3.3.48)

Since we are evaluating the trace in a representation in which a*a is diagonal, wesee that any term in/(a, a*) which does not have equal powers of a and a* will giveno contribution to a diagonal matrix element (see Problem 3.13). Accordingly, inthe trace above, we are free to replace a by ka and a* by d*\k in the function wherek is a parameter with no change in the value of the diagonal matrix element. Wemay therefore write (3.3.48) as

</>o - (1 - e~>) | > ! /0 \ K (3.3.49)

We determine k later.To proceed, we would like to carry out the sum over n in (3.3.49). This may be

accomplished as follows. We rewrite (3.3.49) as

if) 00 00

This is identical with the previous equation since the sum over m contributes onlywhen n = m. We next introduce a representation for the Kronecker d by means ofa new set of boson operators b and tf where [b, tf] = 1 together with their eigen-states

n\n\. (3.3.51)

The b and tf are independent of a and a* and therefore commute with each other.Since the eigenkets \n\ form a complete orthornormal set, we may write the Kron-ecker 6 as

6<0| (3.3.52)

where |0>i, is the vacuum state for the 6-operator (see Eq. 2.2.11).Since (3.3.52) is zero unless n = m, we see that if n is a real parameter, we may

rewrite the equation above as

(3.3.53)


Also since by (2.2.11) of Chapter 1 we may write

an

M - „«

\nt)a -

163

(3.3.54)

and since the a's and *'s commute, we may use the last two results to rewrite (3.3.50)

at i (3.3.55)

We have carried out the sums over n and m at the expense of introducing the dummyoperators b and 6f and thereby succeeded in transforming a trace to a vacuumexpectation value.

We should note that up to this point we could have omitted the terms (1 - e~*)and exp — h$a so that we have proved the lemma.

LEMMA

Tr f(a, <0, O\e"abf (ka, i |0,0>, (3.3.56)

which is still another technique for evaluating the trace of a function of a and a*.The /t and A: are arbitrary.

We may next use (3.3.2a) to commute exp fiab through/(a, af) since b commuteswith both a and af. This gives for (3.3.55) on inserting exp {-futb) exp (pab) to theright o f /

(1M* = <0,0\f\ka, -.

By (3.3.17), we have (* -*• —X, y since b* commutes with a and

, 0> = exp -

so that

0,0|/f*«, I (a*

We show next that

1 —

(3.3.57)

(3.3.58)

164

To show this we look for a solution of the form

, 0> - e9l"-af- 6

OPERATOR ALGEBRA

(3.3.59)

where 9 is only a function of af and b*. If we differentiate both sides with respectto /i, we have

, 0> = abe9\O, 0> = e9 —10,0>.3ft

(3.3.60)

Note that 3913ft commutes with exp 9 since 9 is a function of af and only. If wemultiply from the left by exp — 9, we have

e~9ab e9\0,0) = —10,0>,3ft' '

or on inserting c^g"* = 1 between a and b we have

99e-9ae9e~9be9\0, 0> = — |0,0>. (3.3.61)

If we use (3.3.9a) we have if we le t / = r*

(3.3.62)as?

since e9 commutes with 39/3d*. Similarly,


e~*be9 = b + - ^ ,

39

However,

6|0,0>=0,

so that the above reduces to

39 39 39\ 39

If we again use (3.3.9a) and le t / = d9/d&t, we have

39 39 3*9

( 3- 3 < 6 3 )

(3.3.64)


and since a|0,0> = 0, (3.3.64) becomes

165

3*9 39 39 39,!

Since 9 contains only a* and b* which commute, we may look for a solution of(3.3.65) of the form

9 = /<0t) + Bifttfbt, (3.3.66)

such that by (3.3.59) when p - 0

or.4(0) - 0; B(0)

(3.3.67a)

(3.3.67b)

When we put (3.3.66) into (3.3.65) and equate equal powers of a V , we see that Aand B satisfy the equations

dA

so that by (3.3.67b)

so thatA(ft) = - log (1

« f lF |0.0>-1 r:«

which is (3.3.58).If we use this with fii = e~K the thermal average (3.3.57) becomes

</>o = <0,0|/ ka, TW + f*b) exp - a V |0,0>, (3.3.68)L K\ i yf* / J

where we have let

1" ~ «* - 1 *

Since it is obvious that since <0,0|ct = 0 and <0,0|6f = 0,


and we may rewrite </>0 as

, \ (at 0>

1 ( at 0 , 0).

(3.3.70)

The last step follows since we imagine/expanded in a power series and we inser*exp {—na*b*lft) exp (na^tf/ft) = 1 between each power of a and (a* + fib). Theseries may be resummed to give (3.3.70).

If we use (3.3.2), we have

, I (3.3.71)

Finally, we may choose k and ft in any convenient way. Let us choose them sothat the two linear combinations are adjoints, that is, so that

c —tea H

This requires that

jfc = ^ 1 +n

Therefore (3.3.71) reduces to

</>o = <0,0\f[y/T+~fia+jn

which completes the proof.

THEOREM 10

If £ is a complex parameter and 17 is real,

To prove this, let

(3.3.72)

(3.3.73)

(3.3.74)


If we differentiate X with respect to lij, we have

dX

167

(3.3.75)

Assume that by some means we have put X(ji, a, at) into normal form whereX = X<-n\ Then by (3.3.10), we have

(3.3.76)

Since both sides are now in normal order, we may take diagonal matrix elements ofboth sides in the coherent state representation. This transforms the operator equa-tion (3.3.76) to the c-number partial differential equation

j , a, a*) + !•da* '

(3.3.77)

We may easily solve this subject to the initial condition (3.3.74) by the substitution

(3.3.78)e9,where

such that

C(7?)a*,

,4(0) = 5(0) = C(0) = 0.

(3.3.79)

(3.3.80)

If we substitute this into (3.3.77) and equate equal powers of a and a*, we find thatA, B, and C satisfy the simple equations

dA

dB

SC

The solutions which satisfy (3.3.80) are easily found so that

7, a) = exp [—1»72|?|2 + /»7fa +

If we apply the normal ordering operator to both sides, we obtain the first form of(3.3.72).

We could repeat the procedure by putting both sides of (3.2.75) into antinormalorder. An alternative simpler proof follows directly from (3.1.20).


3.4 CHARACTERISTIC FUNCTIONS [10]; THE WIGNERDISTRIBUTION FUNCTION

We have shown in Section 1.20 that the expectation value of an observableM may be evaluated in the SP or HP by [see (1.20.5)]

- Tr Ms(t0)p8(t) = Tr MH(t)pH(t0),where

and ps(t) satisfies

(3.4.1a)

(3.4.2a)

(3.4.3)

If we generalize p to be

The averages in (3.4.1) are quantum averages and assume we know that thesystem is in state IY>H('o)> at t = /„. If we only know that the system is instate \y>B(ro)> with probability pv, we must take a further average over theprobability distribution/;,,. Thus we generalize (3.4.1) to

(3.4.1b)

(3.4.2b)

we may rewrite this as

«M(0» = Tr Ps(t)Ms = Tr pH(t0)MH(t), (3.4.1c)where ps(t) still satisfies (3.4.3) since pv is the initial distribution and istime independent.

Often one is interested in the moments of some system operator A. Its/th moment in the Schrodinger picture is given by

((A1)) = Tr p(t)A\ (3.4.4)

It is sometimes easier to evaluate the average

CM, 0 = «c 'M < 0» = Tr p(tyiA = Tr p(to)eiUM, (3.4.5)

where f is a real parameter than to evaluate the moments (3.4.4) directly.Furthermore, all the moments of A can be found from CM, 0 s u i c e i t ' s

clear that

l-o(3.4.6)

3.4 CHARACTERISTIC FUNCTIONS 169

That is, the /th moment is obtained by differentiating C I times and thenletting | = 0. CA($, t) is called a moment generating function or the charac-teristic function for A. Furthermore, we may evaluate the characteristicfunction in either of the two pictures. The choice is a matter of convenience.

If we know the characteristic function, we may obtain the diagonal matrixelements of the density operator in the representation in which A is diagonal.(For the electromagnetic field, we show below thafwe may obtain the entiredensity operator from the characteristic function.)

To prove the statement above we write (3.4.5) as

, t) = Tr (3.4.7)

Let us assume that A is a hermitian operator which satisfied the eigenvalueequation

A\A') = A'\A'). (3.4.8)

To be definite assume that the A' are continuous from — oo to +oo. Theeigenvectors form a complete orthonormal set:

fv—(

dA'\A')(A'\ = 1(3.4.9)

{A'\A") = 6{A' - A").

If we insert the completeness relation twice in (3.4.7) on either side of theexponential, we have

, 0 = 2 P ){A'\e*A\A")M'lj<0> dA' dA". (3.4.10)

Since

{A'\eÂ\A") = SA\A'\A") = e'tA' 6{A' - A"), (3.4.11)

we may carry out the integration over A" in (3.4.10) and obtain

CAS, 0 f" dA'J—CO

(3.4.12)

since

(y>(t)\A')*. (3.4.13)

However, \(y>(t)\A')\* dA' is the probability that a measurement of A whenthe system is in state |y>(f)> will yield the value between A' and A' + dA'.The ensemble average when the system is in state |y(0) with probability pv is

P(A\ t) dA' = 2 Pv ')!* dA' = (A'\p{i)\A') dA', (3.4.14)


where we have used (3.4.2b). This is the ensemble probability distributionfunction for the observable A and is the diagonal matrix element of thedensity operator p(t) in the ^(-representation. When we substitute (3.4.14)into (3.4.12), we have

J(3.4.15)

where | and A' are real. Thus the characteristic function for A is just theFourier transform of the probability distribution function P(A', t). Accord-ingly, we may invert (3.4.15) to obtain

7- {= {A'\p(t)\A'). (3.4.16)

Therefore, the characteristic function for A determines the diagonal matrixelements for the density operator in the ^-representation. This is all theinformation needed to determine the moments of A.

Let us next consider the special case of a harmonic oscillator describedby the boson operators a and a*. The density operator will be a function ofa and a* with unit trace:

Tr/foe*, 0 - 1 . (3. 4.17)

There are three useful ways to define characteristic functions depending onthe moments of interest. These are

" ^ V (3.4.18a)

(3.4.18b)

(3.4.18c)

where f is a complex parameter and r\ is real. We call these the antinormal,normal, and Wigner [11] characteristic functions, respectively. By (3.4.17)we see that

C(fl)(0, 0 = C(o)(0, 0 = C(w)(0, 0 = 1. (3.4.19)

The first of these is most useful if we want the normally ordered moments

C<B)(f, 0 = «e" l l"°V l '5a» = Tr

C(a)(f, 0 = «e" {V**o t» = Tr />(0e"'*Vl'**ot

C<w)(f, 0 = «e*»<*«+l*«t>)) = T r

li-o

uam= Tr pit)allam. (3.4.20)

3.4 CHARACTERISTIC FUNCTIONS

The second form yields directly the antinonnally ordered moments

171

= Tr rfOaV".while the third form yields the symmetric moments

(3.4.21)

- Tr (Mtfa + £ V]1.In these, we have treated £ and I* as independent variables.

We may use (3.3.72)

(3.4.22)

= J*Vf*Wm\ (3.4.23)to show the connection among these characteristic functions. If we use thisand the definitions (3.4.18), we see that

tf-'tf, 0 = *4i»tl*ltCto)(£, 0 = eM'''«Cw(f, 0, (3-4.24)

so that we may easily obtain all three characteristic functions if one is known.We next show that the density operator for a mode of the radiation field

may be determined uniquely from the characteristic function [10] (3.4.18).By (3.2.34) since the exponential operators in (3.4.18a) are in normal order,it follows that

= fp< 0 )(a,J

(3.4.25)

where p<0) is the antinormal associated function for the density operator.(It is a function of a, a*, and t.) Therefore, C(B)(f, t) is the double Fouriertransform of p(aK It follows from (3.2.37) and (3.2.41) that

7T

Similarly, we have

Therefore, the density operator is given by

p(a, a\ t) - ««f{p(a)(a, 0 } =

(3.4.26)

(3.4.27)

(3.4.28)


We may use (3.2.23) and (3.4.26) to obtain the direct relation

— |a><a|/>to)(a,/)IT

^ f V & .4.29)

J

f ^ #•>(£, 0 f e-*<*«+*V) |a><a| & . (3J ft J *

We next use (3.4.18c) to obtain the Wigner distribution function [11].This form is particularly appropriate if we are interested in obtaining momentsof/> and q directly where by (2.1.20)

€•*./£<.' + .)

lh(O' a* - a) a* - - = (to? - ip).(3.4.30)

If we substitute for a and a* in (3.4.18c), define two new real parameters Xand n by the relations

/ — A — i /—f t2co V 2

1

then (3.4.18c) becomes

C(W)(A, A«, 0 = Tr p(p, q,

From (3.1.27), we may write

E = g<(^+"1>) = e"V»»

since [#,/>] = i^. This may be rewritten as

E

By (3.1.4), we see that

""V"V*w> = exp

(3.4.31)

(3.4.32)

(3.4.33)

(3.4.34)

(3.4.35)

where we used (3.1.7). If we use this in (3.4.34), we see that (3.4.32) becomes

CM(X, (i, t) = Tr p (p, q, O ^ ^ e ^ . (3.4.36)

3.4 CHARACTERISTIC FUNCTIONS

Let us evaluate the trace in the ^-representation where

1'W)

173

(3.4.37)

dq'W){q'\ = 1.

Furthermore, it follows from (1.10.17) and its adjoint that

(3.4.38)

Since Tr AB — TiBA, we may rewrite (3.4.36) as

){X,n, f) = Tr e^"Xp, q,

-r-)dq\ (3.4.39)

where we evaluated the trace in the ^-representation and used (3.4.38).As in the case of C<0) and C (B) , we may obtain the Fourier transform of

C(w», which is

J f, 0 (3.4.40)

This is the Wigner distribution function and is the analog of the associatedfunctions p(o) and p(n) for the density operator. Since

( 3 4

(3.4.42)

if we integrate both sides of (3.4.40), we see that

fJ

W(«,a*, t)— = C(w)(0, 0 = 1.n

We shall express the Wigner distribution function in the notation commonin the literature and point out a few of its properties. For this purpose, we

174

introduce a change of variables. Let

1

OPERATOR ALGEBRA

1

' + ip')

' + ip').(3.4.43)

y/2Ha>

We define P(p', q', t) in terms of W(x, a*, /) so that

00

fw(«,a*,0— •J T*

0.4.44)

Because

d Re a d Im a = /— - 4 = dg' dj>' = — dp' rf«', (3.4.45)\yJ2H<O

we have on using (3.4.43), (3.4.31), and (3.4.40) that

1P(P', q\ 0 - Witt, a', 0

Â , 0 (3.4.46)

since 77* </2f = HdX dp\2. If we use (3.4.39), carry out the integral over Afirst and then over q", we find that (3.4.46) reduces to

Pip

where we used

Finally, we let

then P reduces

2ff J-vo

the integral

y

to

P

= 2TT diq' - q").

h

, (3.4.47)

(3.4.48)

(3.4.49)

P(P', q', 0 = "T f" e -^ 'VThe density operator is

while the scalar product,

/>(<) = 2 p

= V<4'. 0 .

' - y> dy. (3.4:50)

(3-4.51)

(3-4-52)

3.4 CHARACTERISTIC FUNCTIONS 175

is the Schrodinger wave function in the ^-representation. If we substitute(3.4.51) into (3.4.50) and use (3.4.52), the Wigner distribution functionbecomes

'> q', Q-lPr^zf e-m»»-»y,*iq' - y, t)V(q' + y, t) dy. (3.4.53)

If we integrate Pip', q', i) over all//, we have that

[1 Pv \V<q', Ol2 (mixed state)Pip',q',t)dp' = \* (3.4.54)

{\viq',t)\2 (pure state).*Here we used the result

£(3.4.55)

Therefore, if we integrate the Wigner distribution over all p ' , we obtain theprobability that a measurement of q will yield the value q' at time t.

If we next integrate (3.4.53) over all q' we find

[2 Pv \®iP', 01" (mixed state)Pip',q',t)dq' = \" (3.4.56)

»'»Ol2 (pure state)£

where

', 0 ', t) dq',

is the Schrodinger wave function in the /^-representation.To prove this, we change variables in (3.4.53) and let

u~q'~y q' == i(» + «),

v = q' + y p' = i(» - «),so that

Therefore,

dy' </y =i -4

dudv = i du dv.

(3.4.57)

(3.4.58a)

(3.4.58b)

(3.4.59)

, «', 0 dq' =y)

". Q.E.D. (3.4.60)

* In a pure state, we assume we know the initial y>(q', 0) (see Section 3.16).


Therefore, if we integrate the Wigner distribution function over all valuesof q', we obtain the probability that a measurement of p on the system instate |y(0) will give a value betweenp' and/?' + dp'.

3.5 THE POISSON DISTRIBUTION

If a harmonic oscillator is initially in a coherent state |a'>, the densityoperator is

Kfl ,a t ,0) = |a')<a'|. (3.5.1)

Such a state represents a coherent signal in the cavity mode, since

<a(0)> = Trp(O)a = <a'|a|a'> = oc' = |a'| e*>

<at(O)> = Tr p(0)at = (a'|at|a'> = a'* B |a'| ei<f, ( 3 t }

and the field has a definite amplitude and phase. We have normalized so that(a'| a') = 1. The mean number of photons is by definition

« <aV«|a'> = |a'|2.h = Tr (3.5.3)

The state is a minimum uncertainty state as we have previously shown.In the number representation, the diagonal matrix elements of (3.5.1) are

(n\p(a, a\n!

(3.5.4)

where we have used (2.5.13). This is just a Poisson distribution. If we measurea* a when the oscillator is in state |a '), this gives us the probability of ob-taining n photons.

We next obtain the antinormal and normal associated functions for thecoherent state |<x')(a'|. We first show that

tr(3.5.5)

|a'><a'| = i

s tf{n d(x* - a'*) <5(a - a')}

= TTT <J(a* - a'*) 5(a - a')|a>(a|

so that

/5<B)(a, a*, 0) = lim ee-^-'-«'*«—'> = » 0(a* - a'*) <J(a - a'). (3.5.6)

To prove the result above, we must first show that

limee-*'*-*'*"*-*' = v(5(a - a'*) d(a - a'). (3.5.7)

3.5 THE POISSON DISTRIBUTION 177

We note first that if a* ^ a'* and a # a', the left side vanishes as e -*• oo inagreement with the right side, while if a* = a'* or a = a', both sides becomeinfinite as c -»• oo. Finally, we note that

while

lim c J e-*<«*-"'*><«-«'> _ ? _ j i m 6 _ _,. if«-»oo J 7T «-»oo t 6

d2aTr <5(a* - a'*) 5(a - a') — = 1,

7T

(3.5.8)

(3.5.9)

which proves (3.5.7). We recall that

p(a, a\ 0) = fp<tt)(a, a*, 0)|a)(a| ^

(a, a*)}. (3.5.10)

If we use (3.5.6) here, (3.5.5) follows. As a special case, we note that fora' = a ' * = 0

n (5(oc) <5(a*)}. (3.5.11)

To obtain /»(n)(a, a*, 0) we have by (3.3.42)

|a')<a'| = ^f{/5{n)(a, a*, 0)} = lim ^{e^'-*'^}, (3.5.12)

so thatpln)(<x., «.*, 0) = lim e^1'-*''2 = ./T-^la'Xa'l}. (3.5.13)

So far we have considered a pure coherent state in which the amplitudeand phase are known. Suppose, however, that the phase of the oscillatorswere random. The density operator then becomes

/>(a, aT,0) = |a'Xa'|, (3.5.14)

where the bar indicates that we average over the phases. That is, if we let

a' = |a'| eiv, (3.5.15)then

Since by (2.5.12b)

p(a, a\ 0) = |a'><a'| = ^— . (3.5.16)2TT

I 2 ^n=0 m=0 71!

(a'*fl)"m!

(3.5.17)

178

the density operator becomes

Since

(3.5.18) reduces to

n,m=0 m! Jo

2TT

OPERATOR ALGEBRA

.-»)* ^£ (3.5.18)

(3.5.19)

i, a*, 0)

n!This is also seen to be a Poisson distribution since

n\

(3.5.20)

(3,5.21)

However, all phase information has been lost. One easily sees that Tr pz(0) ?*1 so that we no longer have a pure state (see Section 3.16). This correspondsto giving up all information on the phase. It means that we make no measure-ments on the phase although the oscillators may still have a definite phase.

The normal associated function for (3.5.14) is given by (3.5.20), (3.2.12),and (2.5.13)

pM{x, «*, 0) = (a\pia, a\ 0)|a) = |<a|a')|* = e^'-?oo f£\n

n!

where(«!)* '

h = Tr d*a p(a, a\ 0) = |a'|2.

(3.5.22)

(3.5.23)

However, it is well-known that the modified zero order Bessel function 70(z)is given by

W«) = f ^ 7 T • (3-5-24)We therefore have that

^"'(oc, a*, 0) = e-*-'«' /0(2n jocj2), (3.5.25)

which differs greatly from (3.5.13) for the pure coherent state. We calla'}(<x'| the random phase coherent state.

3.5 THE POISSON DISTRIBUTION 179

We next obtain the characteristic function C(B)(f, r) for this state. It is by(3.5.20) that

C(n>(f, r) = Tr p (0«* l V V*"

= Tr e *( f — \n)(n\ei^'"feii'<a\

n=o n!(3.5.26)

However,

while

,„ _ 0,

When we use these and the orthogonality relations, we obtain

The Laguerre polynomials are given by the power series

so that

(3.5.27a)

(3.5.27b)

(3.5.28)

(3.5.29)

n=o n!

Since the Laguerre polynomials have the well-known generating function,

e"' f - ^ L^), (3.5.31)0 Ri

where / 0 is the ordinary Bessel function of the first kind of order zero, C(n)

becomesCf">(f, 0 = /0(2Vi JJ If I). (3.5.32a)

The other characteristic functions then are

f, 0 - e-^'u^/h n |f |) (3.5.32b)

C(w)(|, 0 = e-*"''*'\{2jfi v III). (3.5.32c)One easily sees that (a) = (af) = 0 for this state while (a*a) = n.


The coherent state represents a coherent signal with a definite phase whilethe state |a')(cc'| represents a coherent signal with no knowledge of the phase.At optical frequencies phase measurements are difficult, and the lattersituation represents the field of a laser above threshold to a good approxima-tion as we shall see.

3.6 THE EXPONENTIAL DISTRIBUTION

Consider the density operator

p(a, af) = (1 - e-x)e-*°\ (3.6.1)

where X = fihw. In a later section we show that this density operator willmaximize the entropy subject to the constraints that Tr p — 1 and that theaverage energy of the system is known.

If we use the completeness relation in the number representation, p maybe written as

n=0

from which it follows that

Tr/> = (1

since Tr \n){n\ = <n|n) = 1 and

The average energy of the oscillator is

(H) = fta)(a1a) = hoi Tr a1apCO

.—An

(3.6.2)

(3.6.3)

(3.6.4)

n=0(3.6.5)

where we used (3.6.2). If we differentiate both sides of (3.6.4) with respectto x, we obtain

or on multiplying both sides by x,

x (3.6.6)

3.6 THE EXPONENTIAL DISTRIBUTION


<fl> =hoy

As - 0 in the classical limit this becomes

Ho)(n).

181

(3.6.7)

(3.6.8)

That is, by the law of equipartition of energy, the average energy is \kT perdegree of freedom. Therefore,

whereHa> e* - 1

A = .kT

s ii, (3.6.9)

(3.6.10)

This is the Planck distribution law for radiation in a cavity to be in thermalequilibrium at temperature T. It is therefore described by the density operatorin (3.6.1).

In the number representation (3.6.2), we see that

> flt)in> (3.6.11)

gives the probability of finding n photons when we measure the oscillatorenergy. The form suggests we call it an exponential distribution. By (3.6.5),we see that

J« = 2 "K") = — :

o e — 1or

so that we may also write p(ri) as

w " ~ (i + / ^ » *

The reader may show for this distribution that

(a) = Tr pa = 0

<af) = Tr pa1 = 0.

(3.6.12)

(3.6.13)

(3.6.14)

(3.6.15)


3.7 GENERALIZED WICK'S THEOREM FOR BOSONOPERATORS

We have already given several methods for evaluating traces. The methodgiven in Theorem 9, Section 3.3 is especially useful for evaluating thermalaverages, and it may be generalized to include many independent oscillators(see Problem 3.17). In this section we give yet another method of evaluatingthermal averages which is called the generalized Wick's theorem [12].

Let Vu Wit • • • be either boson creation or annihilation operators corre-sponding to various independent oscillators whose hamiltonian is

Ho = (3.7.1)

When the field is in thermal equilibrium, its density operator is the Boltzmanndistribution

where1

(3.7.2)

(3.7.3)

We would like to evaluate thermal averages of the form

T r y>! y>t • • • y 2 tVz ' * * (3.7.4)

when there are an even number of creation and annihilation operators.When there are two operators, we first show that

1 - «**'(3.7.5)

where we use +A;- if yx = a] and — X} if y>± = af.To prove this, we note that (y)x Vz)* vanishes if (a) both yx and y% are

creation operators (as does the commutator [y>u yd on the right), if (b) bothare annihilation operators (as does [y>u y>2]), and if (c) Vi and y>2 are a creationand annihilation operator referring to two different modes of the field. Theonly nonvanishing cases then are when (a) Vi — °J» Va — af> o r (*) Vi — atand y>t = a) for a single mode/. But in case (a), we easily see that

(a] a,)0 = (3.7.6)

3.7 GENERALIZED WICK'S THEOREM FOR BOSON OPERATORS

while in case b, since af a] = a] at + 1,

183

(3.7.7)

which verifies (3.7.5). By introducing the commutator [y>lt ya] which is zero,plus 1, or minus 1, all possibilities are included.

We next show that

4>0 + <Vl V3>0<V2 V4>0

(3.7.8)

which says that the thermal average of any product of four creation andannihilation operators is the sum of the products of all averages taken inpairs.

To prove (3.7.8), we note that if u, v, and w are any noncommutingoperators, then

[u, vw] = [«, v]w + v[u, w], (3.7.9)

which may be verified by expanding all commutators. If we use this, we seethat

(3.7.10)

where we let v — y>2 and w = y>a y4. If we use (3.7.9) again on the last com-mutator, we have (this time u = tplt v = y>3, and w — y j

[y>u y>2 Va V<] = V* + * + ]• (3.7.11)

Since [%, y>f] for two boson operators is a c-number, we may move operatorsthrough the commutators, and write this as

i . Wz V* V*\ = [V Va

- Vz V» V* Vi> (3.7.12)

where the last form is just the original commutator written out.Consider next the thermal average

V* Vi)o =Tr

Tr

Tr €+

y3 yt

Tr(3.7.13)

where we inserted exp — /?//•„ exp + @H0 = 1 between tpt and yx. However,from (3.3.16) we have


so that we may writee/>B' Wl e-'a' = Vl e±Xi, (3.7.15)

where we use +Ay if y>t = a) and — X} if yx = a,. Therefore, (3.7.13) becomes

(n v* v* vi>o =T r

Tr

= e±x'(Vi V* Va V4>o. (3-7.16)

where we have used the cyclic properties of traces in the second step.Let us return to (3.7.12) and take the thermal average of both sides and

use (3.7.16). We then find after minor algebra that

- e±XlVa V>3 Vt)o

(3.7.17)

Since the commutators are not operators they are not affected by the thermalaverage. If we use (3.7.5) here, we see that (3.7.8) follows.

The generalized Wick's theorem we wish to prove is

Vz •' • Vi '' •+ <Vi Vsn>o<V2 Vs * ' ' V8»-i>o- (3.7.18)

By repeated application, this reduces to sums of products of all pairs ofthermal averages.

We-have proved (3.7.18) for n = 2 (four operators). Let us assume it istrue for 2(n — 1) factors and show that it is then true for In, and the proofwill be complete, since it is true for n — 2.

We again evaluate by the trick used to show (3.7.8). To do this we take thethermal average of the commutator

Vi"' V2n>0 +

(Vi Vt * * * V

tVl» Vi]<V3 • • • V«n>0 +

+ [Vl» VinMVi V3'" V«n-l)o-We have of course used the generalization of (3.7.13) and (3.7.9) repeatedly.Again, if y>t = a}, we use +Xt and if ipt = a, we use — A,. When we dividethrough by (1 - e±x*) and use (3.7.5), we see that (3.7.18) follows. Thus if thetheorem is true for 2(« — 1) factors (all thermal averages above now involve

3.8 WICK'S THEOREM FOR BOSON OPERATORS 185

2n — 2 factors), it is then true for In and the theorem follows, since it istrue for 4 factors. This theorem is easily generalized to the case where theoperators are fermions rather than bosons [12].

We leave as an exercise to show that if y»(0 is a creation or annihilationoperator in the interaction picture, then the thermal average of the timeordered product of 2n operators is given by

(3-7.19)

(3.7.20)

(3.7.21)

where

and

+ <P{Vl VinWiVi ' • ' Vin-l}>0,

(t, < h),

, ym(tm)]

l -

3.8 WICK'S THEOREM FOR BOSON OPERATORS

In Chapter 1 (1.16.53), we developed a perturbation theory solution forthe Schrodinger equation in the interaction picture given by

= 1 + 1 i1)"1: f' * * IV -"dtnn=i\i^/ n! Jo Jof ) , (3.8.1)

n=i\i^/ n! Jo Jowhere

Vj(t) = e^HotVse-wmut} ( 3 g 2 )

and P is the Dyson time ordering operator. If again Ho is the free fieldhamiltonian

Ho = I Hcoâ,, (3.8.3)i

then in the interaction picture

atf) == a

so that boson operators obey the same commutation relations in the inter-action picture as they do in the Schrodinger picture.

We now proceed to develop Wick's theorem [13] which allows us to convert


the time-ordered operators appearing in (3.8.1) into normally orderedoperators. We must first develop some notation.

Let y>lt y>2 , . . . represent either creation or annihilation operators forvarious modes of the radiation field in the interaction picture and letfi> ft> • • • be the c-number associated functions in the interaction picture.For example, if y>t = aUtj) and y2 — tf*(fj)

or in this case

We next show that

(3.8.5)

(3.8.6)

(3.8.7)

(3.8.8)

(3.8.9)

The last term in (3.8.7) is the vacuum matrix element for modes j and k.

PROOF.

Assume tt < tx. Then

If Vi Vt is in normal order, then

and

where P is the Dyson time-ordering operator defined by

< , x , x, M ' » ) ••( '•) i f '«

and «/f is the normal ordering operator defined byi f V* »

since the vacuum matrix element of operators in normal order vanish and (3.8.7)follows. If Vi y>t is not in normal order, then we may write

(3.8.10)

and since y2 Vi is in normal order, we have


If we take the vacuum matrix element of both sides and note that such matrix ele-ments vanish when the operator is in normal order, we have

<0, Otad) Vs(tj\O, 0> = [*</,


and (3.8.7) again follows when t2 < tt. A similar argument goes through whenh < h-

If we adopt the short-hand notation for the vacuum matrix element

(0, ,0> (3.8.11)

which is called a contraction of Vi and v"2, we may rewrite (3.8.7) as

(3.8.12)

which converts a time-ordered product of two operators into a normal product plusa contraction which is a c-number.

To develop the notation further, we define contractions inside the normal order-ing operator by

vs • • •} = vi vt n n ^{n • • •} (3.8.13)

Vz V» V* V Vs. (3.8.14)

and the extension involving more contractions is straightforward.We next prove the following lemma.

LEMMA

Let vVt-i be an operator at a time tn+1 which is earlier than the times of any of theoperators yu y 2 , . . . , y>n. Then

Vn) Vn+l = ^"{Vl • • ' V« Vn+l} + -^{Vl Vt' ' ' Vn-! V«Vjfl}

}. (3.8.15)

To prove this we note first that if Vn+\is a n annihilation operator earlier than thetimes of all other operators, all contractions with it vanish, leaving only the firstterm on the right above which, by definition of the Jf operator, is identically equalto the left-hand side. We therefore must only prove that (3.8.15) is true whenis a creation operator.


Since the vf are all c-numbers, we can therefore assume with no loss of generalitythat we have written them in normal order so that

V2 • ' • Vn} = Vl V2 • • • Vn, (3.8.16)

• • • vB} y>n+1 will not be in normal order. We may put it into normal orderas follows. We write the identity

Vl V2 • • ' Vn Vn+l = Vn+l Vl V2 • • • Vn ~ [Vn+l, Vl V2 ' ' ' V«L (3.8.17)

since [A, B] = — [B, A]. Note that the first term on the right is in normal order.We may use (3.7.9) repeatedly and write the commutator as

fVn+1, Vl Vj • ' * Viil = tVn+1, Vl] Vi • • • V» + Vl [Vn+l, V2 ' ' ' Vn]

= [Vn+l, Vl] V2 ' * ' Vn + Vl [V»+l, V2] V3 * • * Vn + Vl V2 [Vn+l, Vs ' ' * Vn]

= [Vn+l, Vl] V2 • • ' Vn + [Vn+l. V2] Vl V8 ' ' ' V»

+ • • • + [Vn+l, Vn] Vl V2 • * * Vn-1- (3.8.18)

We used the fact that the commutator is a c-number and moved all operatorsthrough the commutators in the last step.

Accordingly, when we use (3.8.18) and change [Vn+i» V*] (k =» 1, 2 , . . . , n) to-lVk, Vn+il (3.8.17) becomes

Vl Vi • ' • Vn Vn+l = Vn+l Vl V2 ' ' ' Vn + [Vl, Vn+l] V2 ' - " Vn

+ lVz, Vn+l] Vl Vs • * ' Vn + • • • + [Vn, Vn+J Vl V2 ' ' * Vn-1- (3.8.19a)

Since Vn+i is a creation operator at a time earlier than any other operator and vx • • •Vn is in normal order, all terms on the right are either in normal order or are com-mutators, which are c-numbers, and we may write this as

V2 • * • Vn} Vn+l Vn Vn+l}

Vn}

' ' Vn}

V2 ' • ' V»-l}, (3.8.19b)

when we use the definition of the -operator. Now by (3.8.8)

n+l)} = Vkih)

since tM.x < tk (k = 1 , 2 , . . . , n). Since vn+1 is a creation operator, we may putthis into normal order by

V* Vn+l = Vn+l V* + [V*, Vn+l] = P{Vk(fk) V,

If we take vacuum matrix elements, we therefore see that

<0,0\P{vk Vn+i}|0,0> = [VJ

= Vk Vn+l

(3.8.20)

(3.8^1)


since <0,0|vn+i V&|0,0) = 0. If we use this result in (3.8.19b) and use the notation(3.8.13) and (3.8.14), we see that (3.8.15) follows when Vn+i is either a creation orannihilation operator at a time earlier than all other operators and the lemmafollows.

We may generalize the lemma as follows. Since the contraction Ax A2 is a c-

number, we may insert it inside each of the ordering operators in (3.8.1S). Therefore,we have the generalization

2 Vl • • • Wn) Vn+l = -^{Ai A2 ^ • • • Vn+l} + ^ { ^ A2 Vl • • • Vn Vn+l}

+ JT{AX Vn Vn+i} x A2 Vi (3.8.22)

In fact, we may insert as many contractions as we like in each term above.We next state Wick's theorem which we shall prove by induction. A Dyson time-

ordered product of boson creation and annihilation operators in the interactionpicture can be expressed as a sum of normal ordered operators by means of

P{Vl V2 * • * Vn} = -^{Vl V2 • * • Vn} + - "{Vi V2 Vs • ' ' Vn} + -^{Vl V2 Vs " * ' Vn}

+ ••

+ ••• Vn-1 Vn}

i " ' ' Vn-3 Vn-2

V 2 V 3 V 4 " > V n }

lV»}

i

LV2 V3 V4 • • * Vn-2 Vn-1 Vn}- (3.8.23)

We should understand clearly the notation used in the theorem since it is a greatlabor saver in nth order perturbation theory. The time-ordered operator equals (a)a term in which all operators are placed directly in normal order (b) plus a sum ofall normal products times all possible contractions between every pair of operators;(c) plus a sum of the normal products containing all possible contractions of twopairs of operators . . . (d) and so on until we have a term in which all operatorpairs are contracted.

To prove (3.8.23) by induction, we assume it is true as it stands when n factorsare involved. We then shall show it is also true for n + 1. Since we have shown ittrue for two factors in (3.8.12), it will be true for three and so on.

Since (3.8.23) is true, we multiply each term in the equation from the right by theoperator Vn+i('n+i) where /„+1 is the earliest time of any operator. Then, by defini-tion of the P-operator, we have that

P{Vi V2 • • • Vn} Vn+i = P{Vi V2 • • • Vn Vn+l}, (3.8.24)


PiVl V2 • • • Vn Vn+l} = V2 ' • * V»)Vn+l V2 Vs • • ' Vn) Vn+l

V2 Vs (3.8.25)


We now apply the lemma (3.8.15) and its generalization (3.8.22) and so on to eachterm on the right of (3.8.25). We then have

• • • Vn

V8 • • • ft, VV+l} Vt Vs • * ' Vn Vi

and we see that the theorem is true for n + 1 factors. Q.E.D.There is a minor difficulty that remains to be discussed. There may be cases

when two or more operators are evaluated at the same time so that the P-operator is undefined. Consider, for example, a term like

s(t)} ye},

where y>3, rpt, and y>5 are operators at the same time. In this case we maintainthe normal ordering of these operators. We may define the jP-operator inthese cases if we agree to evaluate the creation operators at a slightly latertime t + At so that the P-operator will be equivalent to the ^"-operator.After we apply the theorem, we then let At-*- 0.

Let us assume that y>3 and y>t are creation operators and that y>s is anannihilation operator. We then see that

V*A(-*0

P{y>i ft ip,(t + At) A/) y6(0

We then apply the theorem and we have contractions such as y>z(t+ At) y>s(t)

among the almost equal time operators. However, these will all vanish since

lim va(f + Arty5(0 = lim <0,0|P{%(* + At) ys(0}|0,0),A4-+0 ' ' A»-»0

because the operators will all be in normal order. Therefore, we may as wellomit all equal time contractions when using the theorem from the start.

PART III. ARBITRARY OPERATORS

3.9 GENERALIZATION OF ORDERING TECHNIQUES TOARBITRARY QUANTUM OPERATORS [14]

In the case of bosons we have developed a one-to-one correspondencebetween operators in various orders and c-number associated functions. In

3.9 GENERALIZATION OF ORDERING TECHNIQUES 191

this section we generalize this correspondence to include any set of quantumoperators.

Let us consider a complete set of noncommuting operators alt a2 at

in the Schrodinger picture which obey some set of commutation or anti-commutation relations. The anticommutator of two operators is defined by[°i> as\+ — aflt + afif Suppose we have some function Q of these operatorswhich may be expanded in a power series. We may use the commutation (oranticommutation) relations to reorder the terms in the function into somepredetermined or chosen order. Let this chosen order be ax, a2, a3,..., at.Therefore, Q in chosen order becomes

I r* (3.9.1)

We put a superscript c to indicate we have put the function in chosen orderwhich of course is equal to the function in the original order. Once we haveput Q in chosen order, we may define an associated c-number function bymeans of

«. «,) = 1 *' • I QrU „«? «? • • ' «?', (3-9.2)where we replace the operator a€ by the c-number af which is real or complexdepending on whether a{ is hennitian or not. We put a bar to remind us weare now dealing with a quasi-classical or c-number function.

We next define a linear chosen ordering operator # by means of

, • • •, a,)}, (3.9.3)

where <& tells us to replace each a,- by the corresponding operator and writeall terms in chosen order. We may also define the inverse operator <8~x

(compare with Section 3.2).We may give a formal representation to the <£ operator by means of

f• «,) n <>(«< -n (3.9.4)

where the ^-functions are operators in the chosen order. In the integration, weformally replace each <xf by the corresponding at and put all terms in thechosen order. But since this is exactly what # tells us to do, the two expres-sions in (3.9.4) are formally equivalent.

If the operator at is hermitian and the a,- real, we may represent the d-function by means of

<5(<x - a) = — f"2n J-<

(3.9.5)


while if it is nonhermitian, the operator and its adjoint will both be presentand we then let

3(« - a) «5(a* - a*) = ± JJe-*s«->e-<s*<**-«*> d% (3.9.6)

if the chosen order is a, a*. Here we integrate over the entire complex f -planeand a**f = d*(Re £) </(Im £). In this formal presentation we use (3.9.5) forsimplicity.

If we use (3.9.5), (3.9.4) may be rewritten as

. (3.9.7)

Q\au ..., a,) = (£)' J- • •

f • • • |Wx • • •If we interchange the order of integration of the | ' s and a's, we may writethis as

where

(3.9.8a)

, . . . , a,).

(3.9.8b)

Therefore, J F ^ , . . . , £ , ) is just the Fourier transform of the associatedc-number function 5'(«i «/)• If we invert (3.9.8b) by taking the inverseFourier transform, we obtain

f,) - ( ^ J- • •

QX*t,..., a,) = J- £f, (3.9.8c)

which gives us the associated function in terms of F. If we apply the <€-operator to both sides of (3.9.8c), we obtain (3.9.8a) exactly.

We have seen this same result already in the case of bosons (see Eqs.3.2.36 and 3.2.42). For if we let ax — aT and a2 = a where a and a* are bosonoperators, the chosen order is normal order and we have (on using 3.9.6)

l»\*, a*) (5(a* - a*) 5(a - a)(x, a*)} =

- 2 (Td2a QM(oc, a*)

(3.9.9a)

3.9 GENERALIZATION OF ORDERING TECHNIQUES

where

F<»>(f, f*) = I f | > a g(B)(a, a*)e-i({a+«*a*),

193

(3.9.9b)

as we have found earlier.As an example of (3.9.8) let Q be the density operator p(au . . . , af, t) in

the SchrSdinger picture. Then by (3.9.8)

= J- • • /f 0

where

, - . . , £ f , t ) =

or

...,«„<)=[•••so that

P%ai,..., af, i) = i , . . . , a,, 0}.

(3.9.10a)

(*i> •••»«/». 0>

(3.9.10b)

f , ,0 , (3.9.10c)

(3.9.10d)

p(t) =

while operators in the two pictures are related by

We are often interested in the density operator in the interaction picture%{f) which is related to the Schrodinger picture by

(3.9.11)

(3.9.12)

If we use (3.9.11), we have on solving for %

ih % = eii/h)B4[p, Ho] + ih |£]<r«'*"r»*. (3.9.13)at I at)

If we use (3.9.10a), we have for the equation of motion for the densityoperator in the interaction picture

ih <& J...If we use the result (3.1.13)

at

1 = exp ABA~\

£„ t)

t. (3.9.14)

(3.9.15)

IV194

and insert

OPERATOR ALGEBRA

1 (3.9.16)

between all exponential factors in (3.9.14), it follows from (3.9.12) that

dt

where all operators are in the interaction picture. We should note that althoughp in (3.9.10a) was in chosen order, (9%/3l) as it stands above is not in chosenorder because of the commutator term in the integrand. In practice, we mustput this into chosen order.

If we use (3.9.10a), we see by (3.9.11) that % becomes

«/, 0 " J'''(3.9.18)

which gives the density operator in the interaction picture directly in chosenorder if &{$, t) is known.

Consider next the expectation value of Q

(Q\alf..., a,, 0) - Tr

If we use (3.9.8), we havei, • • •»«/)!• (3-9.19)

r(a1,...,fl /,o>-

where(3.9.20)

(3.9.21)

is a characteristic function and F(Slt..., f,) is given by (3.9.8b) and is theFourier transform of the associated function £ K a i » . . . , af). In (3.9.20) wehave converted the quantum expectation value to an integral over c-numberfunctions. We may express this in another way as follows. By (3.9.4) we havean alternate expression equivalent to (3.9.20)

<6 e(«i . • • •»a/» 0> J ' • J , . . . . <x,)P(oci a,, I)

(3.9.22)

3.9 GENERALIZATION OF ORDERING TECHNIQUES

where we have let

195

, . . . , a,, t)

Tr

- ax) • • • <5(a, - at))

- ax) • • • d(ocf - a,)

(3.9.23)

and where we have used (3.9.5) and (3.9.21). We call P a distribution functionwhich is the Fourier transform of the characteristic function. If we apply theinverse Fourier transform to (3.9.23), we have

I"'' • J dxt • • • a,, 0

Tr eiimp(ai, . . . , a f , t ) . (3.9.24)

The advantage of using the distribution function stems from the fact thatexpectation values of operators in chosen order are evaluated just likeclassical averages if P is considered as a classical probability. We may seethis as follows. We have by (3.9.24)

= f • • • LfW* • • • «fT'P{*x • • • fl*«y (3.9.25)

There will be corrections to this quantum-classical correspondence foroperator means which are not in chosen order as we show later. For example,

, , t) dxx dcc2 (3.9.26)

if ax, a2 is the chosen order.The advantage of using the £"(04,... ,«.f,t) associated function of

(3.9.10) rather than the P-function is that we may very simply recover thedensity operator itself directly at any stage of the calculation by (3.9.10d).This is somewhat more difficult to do with the P-function. We may obtain adirect connection between the P-function and pe quite easily. By (3.9.23) wehave

,..., *„ t) = Tr p(t) <5(ai - at) • • • <5(a, - a,), (3.9.27)

196 OPERATOR ALGEBRA 3.10 OPERATOR DESCRIPTION OF INDEPENDENT ATOMS 197

while by (3.9.4), we have

, ~ a,)

(3.9.28)

If we substitute (3.9.28) into (3.9.27) we have

P(*, 0 = f • • • fdg p<(/?, t)K(§, a),

where the kernel is

, a) = Tr

(3.9.29a)

(3.9.29b)

and where a = ax, a af,d@ = d$x • • • dfi,, and so on. From the kerneland pe we may obtain P in principle. The kernel may be evaluated by usinga representation for the <5-functions such as (3.9.5).

From (3.9.10) and (3.9.22), we see that

=J • • •= J - • • J«/«! • • •

£„ 0 Tr

• • • a/)P(ax a,, 0, (3-9.30)

which offers two independent ways of calculating the mean value of Q. Ingeneral, for practical reasons, the last form is generally preferred.

3.10 OPERATOR DESCRIPTION OF INDEPENDENT ATOMS

In this section we present an operator description of a single atom andthen generalize it to cover the case of N atoms which do not interact witheach other directly. Such a description will be adequate to discuss the inter-action of an atom with a radiation field, an atom with a reservoir of oscillators(elastic vibrations in a solid) and so on.

For simplicity let us consider the hamiltonian for a single atom given by

A 2m

This hamiltonian satisfies the energy eigenvalue problem

HJl) = C,|/>,

(3.10.1)

(3.10.2)

where e, are the energy eigenvalues and |/) are the eigenvectors. The readershould not confuse the vectors |/> with those of the harmonic oscillator whichare just a special case when H = haxfla.

The vectors |/) form a complete orthonormal set so that

2 l O < f | - l (3.10.3a)i

</|m> = dlm. (3.10.3b)

We may use the completeness relation to express an arbitrary atomicoperator in terms of the operators \k)(l\. To illustrate, let us apply HA toboth sides of (3.10.3a) and use (3.10.2). This gives

HA - J,HA\l)(l\ = 2«il0<fl- (3.10.4)i i

We have therefore expressed the hamiltonian in terms of the projectionoperators |/></|. The expansion coefficients et are just the diagonal matrixelements of HA, namely,

fj = (J\HA\l). (3.10.5)

Consider next any function Q(p, r). If we use the completeness relationtwice, we have

(3.10.6)l.m

where the expansion coefficients are just the matrix elements of Q in theenergy representation

fi.m = </|fil«>- (3.10.7)

If Q involves operators for another system such as the a and a* of the radia-tion field, then Qkl will still contain these operators obviously. In particular,if Q is the interaction energy between the atom and some other system, wemay expand it as in (3.10.6).

The state vector \y>) may also be expanded as

where

and

(3.10.8a)

(3.10.8b)

(3.10.8c)

is the probability the atom is in state |/>. It is just the expectation value of theprojection operator, |/)</|.


Due to the orthogonality of the energy eigenvectors, we always reduceproducts to a bilinear operator, since

|*></|m><B| - dlm\k)(n\. (3.10.9)

This property makes the algebra of these operators relatively simple.Consider next the Heisenberg equation of motion for an arbitrary operator

Q which is a function of atomic operators. If we use a superscript H or S torepresent Heisenberg or Schrodinger pictures, we have

(3.10.10)at

Let us express both HAa and QB by (3.10.4) and (3.10.6), respectively. Then,

(3.10.10) becomes

(it l.m 1k.l.m (3.10.11)

Since the orthogonality relations must be the same in both pictures, theequation above reduces to

i,m k.l.m(3.10.12)

J.m

where we carried out the sum over k. Since the function Q was arbitrary, letus choose it so that it has only nonvanishing matrix elements are Qrt. Thesums then reduce to the single term

ift-j(\r)(s\)H « ( c r - O(|r)<s|)H.at

The solution is clearly

(|r><5|)H = eim"X\r)(s\f,

where at t = 0 the two pictures coincide and we have defined

(3.10.13)

(3.10.14)

(3.10.15)

The reader may quickly verify that (3.10.13) may be written directly as

ih — (|r> <s|)H = [(|r> {s\f, HAB], (3.10.16)

3.10 OPERATOR DESCRIPTION OF INDEPENDENT ATOMS 199

so that the operators \r)(s\ obey the familiar Heisenberg equation of motion.When the atom has only two levels or is interacting with a system that

causes it to make transitions between only two levels, say |1) and |2), wheree2 > ex to be definite, the operators |1><1|, |2>(2|, |1)<2|, and |2)<1| can beput in a one-to-one correspondence with the Pauli spin-£ operators ax, ay, az,and the identity operator.

To carry out this correspondence let us assume the following corre-spondence, namely,

*+ = Kff. + '""„) = 12) <1| (3.10.17a)= o\ (3.10.17b)

(3.10.17c)

(3.10.17d)

To prove the equivalence, we must merely show that the operators satisfythe same algebra as the spin operators, namely,

[a,-, as] = 2iak, (3.10.18)

where i, j , k form an even permutation of x, y, and z, and

[aiy <r,]+ ss <rfa, + atat = 2«5O i,j = x, y, or z. (3.10.19)

If we use the completeness and orthogonality relations (3.10.3), we see thatthe operators in (3.10.17) satisfy (3.10.18) and (3.10.19).

Thus we may visualize the operator a+ = |2>(1| as raising an atom fromstate |1) to state |2) and a_ as the inverse lowering operator. It effectivelyannihilates a particle in the upper level |2) and simultaneously creates theparticle in the lower level |1).

Let us derive a few theorems for the operators of a two level atom whichwill prove useful in our later work.

THEOREM 1

If n is a positive integer, then

(<T + O B = °+°- "= i(*« + 1) (3.10.20)

PROOF.

By (3.10.17) and the orthogonality relations

|2X2|,

so that

Q.E.D.


THEOREM 2

If /(<*+<*-) i s a n v function of <r+<r_ that can be expanded in a power series, then

/(ff+O =/ (0) + [/(D -/(0)K<r_. (3.10.21)

PROOF.

We expand / a sTO

/°(z) = 5 * /* Zn = /*(0) +71=0

But if z = <r+ff_, by Theorem 1 above zn = z. So

and

and the theorem follows directly, provided the sum exists.As an example, we see that

THEOREM 3

If f is a parameter, then

(3.10.22)

(3.10.23a)

(3.10.23b)

From these and (3.10.17), it immediately follows that

= a+eW* + o_e-M = a., cosh J f + «"» sinh | |

l»s _ -i<,+el/iS + iae-H$ = -itr^, sinh ^ f + ay cosh | f.

(3.10.24)

PROOF.

Let

so that

/ ( I ) = eM°'

/(0) = «r+

(3.10.25)

(3.10.26)

3.10 OPERATOR DESCRIPTION OF INDEPENDENT ATOMS

Then

201

where we used (2.8.21). When we integrate, we obtain

A similar proof holds for (3.10.23b).

THEOREM 4

If £ and n are parameters, then

i COS V f2 + »?2 +

We leave the proof of this and the following theorems as exercises.

THEOREM 5

If I and t} are parameters, then

e«S*++ncJ> = cos

THEOREM 6

= cos £ + tat sin £ i = 1, 2,3.

THEOREM 7

If/,/ = 1,2, or 3, thenTr

Tr Oi = 0.

(3.10.27)

(3.10.28)

($az + ijffj. (3.10.29)

(3.10.30)

(3.10.31)

(3.10.32)

(3.10.33)

The density operator describing a single atom in the Schrodinger picturemay obviously also be expanded in the energy representation as

(3.10.34a)

202

where

The expectation value of Q is given by

= Tr/K0Q = Tr

Since

we have

«2(0) =

ALGEBRA

(3.10.34b)

(3.10.35)

(3.10.36)

(3.10.37)

which is simply the trace of the product of the two matrices in the Schrodingerpicture.

All of the algebra above may be repeated if we identify

a\

and require that these operators obey the relations

(3.10.38)

Tr dkl.

(3.10.39a)

(3.10.39b)

(3.10.39c)

We may call a, an annihilation operator for a particle in state |/> and al acreation operator for a particle in state \k). However, to ask whether theseoperators obey commutation or anticommutation relations is a meaninglessquestion. All that is implied is simply expressed by (3.10.38). It is usuallysimply more convenient to write al than it is to write \k) so that we use thenotation above. The Heisenberg equation of motion (3.10.16)

j - aXt)a,(t) - ^ [dXfiaJLO. HAB]

at in

looks more familiar in this notation where we write

(3.10.40)

(3.10.41)

If we have N independent atoms, we take our hamiltonian to be

N(3.10.42)2

m-1 1

311 SOEUTION«©E«THE SCHRODINGER EQUATION

and if all atoms are identical, we have

203

6,<m) = e,. (3.10.43)

That is, the energy levels of all atoms are the same. Therefore, the generaliza-tion to N independent atoms is straightforward.

PART IV. ELEMENTARY APPLICATIONS

3.11 SOLUTION OF THE SCHRODINGER EQUATION BYNORMAL ORDERING; DRIVEN HARMONIC OSCILLATOR [15]

The SchrSdinger equation

has a solution of the form

where U satisfies= U(t, to)\fito)),

'"ft -subject to the initial condition

Let us assume the hamiltonian is of the form

l.m

(3.11.1)

(3.11.2)

(3.11.3)

(3.11.4)

(3.11.5)

where we have put it into normal order. The hlm(i) are c-number expansioncoefficients. Then (3.11.3) becomes

l.mimU. (3.11.6)

If we use (3.3.13), we may rewrite this as

ih — = ^ hlm(t)anjV[ (a + — ) Oln\a., a*, t)\. (3.11.7)dt i,m \\ da.*/ j

where0M(x, a*, 0 = <a|t/(a, a*, *)|a>. (3.11.8)

If we take diagonal coherent state matrix elements of both sides of (3.11.7),

204

we obtain the c-number equation

2J . m

OPERATOR ALGEBRA

(3.11.9)

since the right side is in normal order. When we solve (3.11.9), we obtainIv (0> by

|y(0> = ^{C/<»'(a, a*, t))\W(t0)). (3.11.10)

We may also first transform (3.11.1) to the interaction picture and proceedas an alternative.

As a simple example let us consider a driven harmonic oscillator describedby

H = HcoJa + h[f(t)a +f*(t)J). (3.11.11)

In this case, we see that (3.11.9) becomes

« + 5?) S?)

If we let

where

Qi n)

(3.11.12)

(3.11.13)

(3.11.14)G = A{t) + B(t)oL + C(0a* + D(0a*a,

then (3.11.12) becomes

X d A , d B t d C * , d D . m"\i — + — a + — a* + -— a*aLdt rf/ d< df J

= <oa*a + (ua*(C + Da) + /(«)« + /*(0a* + /(0(C + Da). (3.11.15)

Equating coefficients of a*a, a, and a* we have

dt

(3.11.16)

3.12 TWO WEAKLY COUPLED OSCILLATORS 205

Since U(t = 0) = 1, A(0) = B(0) = C(0) = D(0) = 0. The solutions of(3.11.16) are easily seen to be

D(t) = *-*•«- 1

B(t) = -i fr'-'/CO dt'Jo

C(0 = -i f eto<''-"/*(O A' = - r " ^Jo

,4(0 = - [dffit") [ e^-^fit') dt'.Jo Jo

(3.H.17)

Therefore,

(Olv(o)>= eAMeOWatjrieDW'''}em)a\ip(t0)).

(3.11.18)

If we desire the solution to be in normal order, there is no advantage in goingto the IP for this simple example.

If at t = t0, |v(/0)> = |a>» the coherent state, then it follows from (3.11.18)and/(o)|a) = / ( a ) | a ) that

_ e4«)+B«)«gC(0otg D(t)«ot|a^

If we express | a) as

|a> = e-*l«lVat

then we see that (3.11.19) may be written as

so that a coherent state will always remain a coherent state.If tyi'o)) ~ |0>, then we see that (let a = 0 in 3.11.21)

= eAM+CH)ttt\0),

(3.11.19)

(3.11.20)

(3.11.21)

(3.11.22)so that a driven oscillator starting from the vacuum state develops into acoherent state. This, of course, assumes the oscillator is at absolute zero.

It should be noted that antinormal ordering may also be used in the sameway to solve the SchrSdinger equation.

3.12 TWO WEAKLY COUPLED OSCILLATORS

We may apply the techniques of the last section to solve the Schrodingerequation for two weakly coupled oscillators described by the hamiltonian

H = hmâ + hmtfb + hK(a^b + tfa). (3.12.1)

OPERATOR ALGEBRA

We have on assuming U is in normal order

Since both sides are in normal order and since

(a, a*, ft /?*, 0 -

we have when we let

If we assume G of the form

0n

we see the coefficients satisfy the equations

.dA' dt "'

1 dt =

at

f3a*

. (3.12.2)

(3.12.3)

(3.12.4)

(3.12.5)

(3.12.6)

(3.12.7)

(3.12.8)

(3.12.9)

(3.12.10)

where ,4(0) = B(0) = C(0) = Z)(0) = 0. From (3.12.7) and (3.12.10) weobtain

jf (A + 1) + iwM + 1) = -IKD (3.12.11)

dD— + to8D = -iK(A + 1), (3.12.12)

3.13 DISTRIBUTION FUNCTION FOR TWO-LEVEL ATOM

or on substituting D from (3.12.11) into (3.12.12) we have

207

so that

where

1),

A(t) + 1 - e-i{ai+mtW2 cos Tt

sin Tf,

r =Similarly, one finds that

Thus

If at t = 0, |y(0)> = |a, j8), then

A(t) = B(t)

C(t) = D(t).

(3.12.13)

(3.12.14)

(3.12.15)

(3.12.16)

^ Q^ (3.12.17)

so again we find that a coherent state remains a coherent state. If a = /? = 0,the vacuum state remains a vacuum state since these oscillators are notdriven.


To illustrate the techniques developed in Sections 3.9 and 3.10 let us obtainthe distribution function for a two-level atom. Let

/ = 1 , 2 (3.13.1)

be the projection operator for the atom in levels 1 and 2, respectively, and let

A/=| l><2| A/* = |2><1| (3.13.2)

be the dipole moment operators. The hamiltonian by (3.10.4) is

^12X21, (3.13.3)

where we assume e2 > <rx.Since

(J\m) = dlm, (3.13.4)


it follows directly from (3.13.1) and (3.13.2) that

N,2 = N{ M2 = A/t2 = 0. (3.13.5)

The reader may therefore verify the following commutation relations (seeproblem 3.20) which we shall need

[eiiM, N J = -igM = -i$MeiiM

[eiiM, N2] = i£M = i$Me<*M (3.13.6)

\c , iV2J = —15 M = i? M e

[Nu Nt] = 0.

We must first decide on a chosen ordering for these operators. We pickthe ordering M\ Nl3 Nz, M. According to (3.9.23), the distribution functionis defined by

where we let

4ni=i

= Mf = a\

(3.13.7)

(3.13.8)

at = M = a\.

The density operator satisfies the equation

so that when we use this in (3.13.7) we obtain

M Jt («. 0 = Tr [[Ho, P] TJ 5(«, - «*)}•

(3.13.9)

(3.13.10)

If we use the cyclic property of traces, namely, Tr AB = Tr BA, this may berewritten as

ih (a, 0 = Tr rf0[n 5(af - a,), tfo], (3.13.11)

we represent the <$ functions (see Eqs. 3.9.5 and 3.9.6) as

x e-ii*(J(*-M^)e-i^-r^-Ni)e-iu(^ri-ifi)e-mjt-M)t (3 13 12)


where we let<xx = ~M a3 = ^r 2

0t2 = = *^r j <X ^ »^^.

Thus when we use (3.13.12) and (3.13.13), (3.13.11) becomes

209

Aa, 0 = - ^ f • • • |at \A'lT)n J J

(3.13.14)

where/ = Tr {p(t)[ei**JfV*liVSlJV**f, ( e ^ + e2N2)]}. (3.13.15)

We note that the ^-function is in chosen order but the commutator termsare not. Our first task is to use the commutation relations (3.13.6) to put allterms in chosen order. Since

[AB, C] ss A [B, C] + [A, C]B,

we have on repeated application of (3.13.16),

(3.13.16)

(3.13.17)

which is now in chosen order. Similarly, we see that

fe«'*VWVWVl* JV2] = _[^l*Jte 'fc^Hs«riV»Jf, tf J . (3.13.18)

We have used the fact that iVj and N2 commute. Therefore, (3.13.15) becomes

j = _ ( € 2 _ €l) Tr p(fi{t£*Mtjf "***>"&****

- e^MteihNleiitNteiiMim}. (3.13.19)

Since both operators in the curly brackets are in chosen order, we may use(3.9.4) and write

/=-(e2-c1)J---Jda'Trp(0

W+h^r'+S>^'+^'> J J <5[a; - a j ) , (3.13.20)X

where

where(3.13.21)

(3.13.22)


and the at are given by (3.13.8). From the definition of the distributionfunction (3.13.7), we see that (3.13.20) may be written as

- -(«t ~ «i) [' • • [<*«' P(a', tXi

= -(«• - ex) f • • J da' P(a\ 0

' * —

We next integrate the first term by parts with respect to Jt'* and thesecond with respect to M'. This gives

( ^ O j ( 3 . 1 3 . 2 4 )

where we assume the distribution function vanishes at the limits of integration.We now substitute (3.13.24) into (3.13.14) and carry out the f integration.

This gives

at *> / . . . / *

JT)

where a = (Ji*, Jfx, Jf^ Jf). We have therefore obtained the e-numberequation obeyed by the distribution function which is equivalent to thedensity operator equation.

We may obtain solutions of (3.13.25) very easily by the method of charac-teristics (See Appendix A). We may rewrite (3.13.25) as

The characteristic equations are therefore

—ioi2iJK

0. (3.13.26)

(3.13.27)

3.14 DISTRIBUTION FUNCTION FOR HARMONIC OSCILLATOR

We therefore have that= 0,

or

211

(3.13.28)

(3.13.29)

(3.13.30)

Therefore, any arbitrary function of J?(0), J?*(0), J^10, and ^T20 such as

satisfies (3.13.26). The functional form of P must be determined by the initialconditions. If we know that Jf± = Jf19, JV% = Jf^, St = ^ ( 0 ) , andJ(* = J?*(Q) at / = 0, then

P =

- J>\). (3.13.32)

From (3.9.22), it follows that the averages of M, M\ Nu and N2 are givenby

(M(0)

(M\t)) =

••• f(3.13.33)

when measurements were made at t = 0. Therefore, we see that the averagenumber of atoms in levels 1 and 2 does not change with time while the dipolemoment oscillates at frequency co.n.

3.14 DISTRIBUTION FUNCTION FOR HARMONIC OSCILLATOR

The hamiltonian for an oscillator is

Let us consider the case in which the operators are in normal order and obtainthe equation of motion for the distribution function

P(oc, a*, t) = Tr />(/) <$(a* - a1) d(a - a). (3.14.1)

212

Then

OPERATOR ALGEBRA

dPih — = Tr [Ho, p] 6(OL* - a f) <S(a - a)

ot

= Tr {/>(Ot«5(a* - «f) <J(a - a), W a ] }

(3.14.2)IT -I .1

where

7 = Tr rfOt^'V* Jo]. (3.14.3)

We have used (3.13.9), the cyclic properties of traces and (3.9.6). By (3.13.16)we may rewrite I as

(3.14.4)

(3.14.5)

}. (3.14.6)

(3.14.7)

J = TrAlso since

[A, BC] == [A, B]C + B[A, C],we have

/ = Tr /<0{e*([«'*V , af]a + ^[e'**"1, a])

But

9aTherefore,

/ = Tr /K0{—c'SoatjfVs*°t + eîfae'1*01}.

Since these are in chosen (normal) order, we have by (3.9.4)

P(a',

where we used (3.14.1). This may be rewritten as

(3.14.8)

' ,«", l) , (3.14.9)

3.15 GENERATING FUNCTION FOR OSCILLATOR EIGENFUNCTIONS 213

where we integrated by parts. When we put this into (3.14.2) and integrateover dH, we obtain

B * a * •" h a']p(a'a'*' °)<5(a' ~a) " ~ a*)>

or the distribution function obeys the equation

f£(a, «*. 0 = toff (aP) - ^ ( « ^

By Appendix A, the associated characteristic equations are

dt da da*

so that1 — icoa +icoa*

a = aoe-*0"

a* = «•«*•*,

and the solution of (3.14.10) is

P ( a , a ^ = g[«e<<oU*<rto<],

where g is an arbitrary function.

3.15 GENERATING FUNCTION FOR OSCILLATOREIGENFUNCTIONS

The coherent state is given by (2.5.12) as

|oc> = e-^ | a |Vot |0>

If \q') is a coordinate eigenvector, then

where

(3.14.11)

(3.14.12)

(3.14.13)

«„(?') = <«'|n>

(3.15.1)

(3.15.2)

(3.15.3)

2 1 4 OPERATOR ALGEBRA

are the oscillator energy eigenfunctions in the coordinate representation. Ifwe use (2.5.38), we have

(3.15.4)

Therefore, by expanding the left side in powers of a, the coefficient ofan is un(q')j\fn\. However, a generating function for hermite polynomialsis given by

n!(3.15.5)

This may be shown as follows. We first differentiate both sides of (3.15.5) withrespect to x twice and with respect to z once. This gives

(3.15.6)

(3.15.7)

o dx n\

o dx* n\

. nz.n-l

n\(3.15.8)

If we multiply both sides of (3.15.6) by -2x, (3.15.7) by 1, and (3.15.8) by2z, and add, we have

0 ='x2 dx n)n\

But the hermite polynomials satisfy the equation

d*H. _ 2j_ dHn

dx2 X dx

If we make the identification

a

(3.15.9)

Q.E.D. (3.15.10)

(3.15.11)

and use (3.15.5), (3.15.4) becomes

u " ( € ) - ( 3 - 1 5 1 2 )

3.16 DEFINITION OF ENTROPY

Therefore, the normalized oscillator eigenfunctions are

215

(3.15.13)

Note from (3.15.2) that the minimum uncertainty wave packet stateis a generating function for the {un(q')}.

PART V. PRINCIPLE OF MAXIMUM ENTROPY

3.16 DEFINITION OF ENTROPY

The entropy of an ensemble of systems is defined in classical statisticalmechanics by the relation

npt, (3.16.1)

where k is Boltzmann's constant and px is the probability of finding thesystem in state /. The /?j's satisfy the conditions

2,Pi= > " S Pi S A Pi== Pi • yi.io.Z)i

Entropy may be visualized physically as a measure of the lack of knowledgeof the system. If we know the system is in a definite state /, then pt = 8U,and we see by (3.16.1) that the entropy is zero. In this case we have completeknowledge about the system; it is in a definite state.

On the other hand, if we know nothing about the system, it is equallylikely to find the system in any of its possible states /, subject only to theconstraint that ]£/>, = 1. We show that the entropy is a maximum underthese conditions. We therefore maximize S, subject to the constraint %Pi — 1,by the method of Lagrange multipliers. If we vary the p,'s, the variation inSis

dS = -k 2 (1 + In pt) 8Pl = 0, (3.16.3)i

where we set 6S = 0 to find its maximum. At the same time, the variationin the constraint is

2 8Pl = 0. (3.16.4)i

To apply the method of Lagrange multipliers, we multiply (3.16.4) by anundetermined parameter X and add to (3.16.3). Then we have


Each dpi is now independent, and this equation will be satisfied if and onlyif each term is zero:

\npt = - ( 1 + A).

From this we conclude that each p% must be a constant independent of /,the state of the system; that is, the probability of finding the system in anyof its states is equally likely. We then have no information about the stateof the system. Entropy is therefore a measure of the lack of information aboutthe states of the elements of the ensemble, as we stated. This is the startingpoint of Shannon's theory of communication [16]. Jaynes [17, 18] hasproposed that entropy be used as a fundamental postulate of statisticalmechanics.

In quantum statistical mechanics [19] entropy is defined in terms of thedensity matrix as

S= -kTrplnp, (3.16.5)subject to the constraint

lrp = l. (3.16.6)

It should be noted that both />'s appearing in (3.16.5) are evaluated at thesame time.

For purposes of evaluating the entropy, we must evaluate the trace insome representation. Let the set {|«)} be some complete orthonormal set.Then (3.16.5) means

S = -kZ(n\p\m)(m\ In p\n). (3.16.7)n.m

If we transform from the representation {\n}} to a representation in which pis diagonal by means of a similarity transformation, then (3.16.7) reduces to

S=-kZPcllnpt, (3.16.8)a

wherepa — (<x|/»|oc), and {|a)} is the representation in which p is diagonal andpa are the diagonal matrix elements.

The question now arises as to how to find the density matrix already dis-cussed in detail. We know that the entropy (3.16.5) is a measure of lack ofknowledge about the states of the elements of the ensemble, just as in theclassical case. For if we maximize 5 subject to (3.16.6), just as in the classicalcase we find

2(1 +\np + X)dp = O, (3.16.9)

or p = const. This tells us that the entropy is a maximum when the proba-bility of finding the system in any of its possible states is the same. On theother hand, if we know the system is in a pure state |y>), then 5 = 0 since/ V — <W-

3.16 DEFINITION OF ENTROPY 217

We now suppose that we know something about the system, for example,its average energy. The average energy is

(E) = TrpH, (3.16.10)

where H is the hamiltonian. This knowledge about the system must bereflected in the choice of a density operator to describe the ensemble. We mayregard it as an added constraint and choose p so that it maximizes the entropysubject to the constraints (3.16.6) and (3.16.10). Then, when we vary p,

Tr (1 + In p) dp = 0Tr dp = 0

TrH8p = 0.

If we multiply the second of these by the undetermined multiplier X and thethird by /3 and add to the first, we have

Tr (1 + X + In p + pH) dp = 0.

Since dp is arbitrary and all variations are now independent, this will besatisfied if and only if

ln/D= - 1 - X-pHor (3.16.11)

p = e-(m><r/rH

We may determine X as follows: we take the trace of both sides of (3.16.11),use (3.16.6), and find that


P = (3.16.12)

where Z is called the partition function. To determine /?, we use the con-straint (3.16.10). We have

»H a

- = - £ In Z. (3.16.13)

From this, we can, in principle, solve for /? in terms of the average energy ofthe system.

As a particular example, we consider a cavity filled with electromagneticradiation* in thermal equilibrium with the walls at temperature T. The

* In the following chapter we show that the radiation field in a cavity is equivalent toa set of fictitious harmonic oscillators. Each normal mode of frequency o>, is associatedwith a harmonic oscillator with hamiltonian hwpja^

218OPERATOR ALGEBRA

average energy contained in one mode of the cavity at frequency co may bewritten as nhco, where h is the average number of quanta. Since the hamil-tonian for this mode is

we have, by (3.16.13),

nhco

where

H = had*a

00

hto£(n\afa exp [-xafa]\n)2 = 2 22=2

m=0exp [-xa?a]\m)

(3.16.14)

3.16 DEFINITION OF ENTROPY 219

In the energy representation, where H\En) = En\En), the matrix elementsof/>, from (3.16.18), are

(3.16.19)Em

The probability of finding the system in state Em (or of finding one elementof the ensemble in Em) is therefore

Pm =g~PEm

(3.16.20)

200

<£>

( 3 W 1 5 )

_

- ] - ^ - (3.16.16)

From this we may solve for /? in terms of «. However, from the corre-spondence principle, as h-*-0 the energy E must become the averageclassical energy contained in a cavity mode; that is, E-*- kT as h -*• 0 sincefrom the classical equipartition-of-energy theorem, we get |fcrper degree offreedom. The electric and magnetic fields each correspond to one degree offreedom. Therefore, in the limit, as h -*• 0, we have by (3.16.16)

•o that ] / * r

(3-16-17>

Trexp(-H/kT)' (3.16.18)

(This is the diagonal matrix element of p in the energy representation.) Thiscorresponds to a Maxwell-Boltzmann probability distribution. In thisrepresentation, we may therefore write the density operator as

P = I \Em)pm<EJ,Em

where pm is given by (3.16.20).We also observe that

2Em

(3.16.21)

(3.16.22)

so that the ensemble represents a mixed state; that is, knowledge of theaverage energy of the system is not sufficient to determine the state of thesystem completely.

The maximum entropy for this ensemble is

= -kTrplnp=-kTr (?-— In* ^ + klnZ,\ Z / Z T

where

Z = Tr e-»H = 2 e"^-

is the partition function and

(3.16.24)

(3.16.25)

The partition function determines the thermodynamic properties of thesystem.

We repeat, for emphasis, that when the average energy is known the en-semble may be chosen to be made up of a large number of elements, each ofwhich is in a state, say \Em), and weighted with probability pm given by(3.16.20).


If additional measurements are made on other variables in the problem,we add their averages as additional constraints and again maximize theentropy. In this way we obtain the best estimate of the density operatorpossible, subject to our knowledge of the system.

3.17 DENSITY OPERATOR FOR SPIN-J PARTICLES [20]

We use the principle of entropy maximization presented in the previoussection to obtain a density operator for a beam (ensemble) of spin-£ particles.

Lets = Tr/ja,

where ax, ay, and at be the Pauli spin matrices,

(0 -i\

(3.17.1)

0 l\

in the <r2 representation. That is, s represents the ensemble average of o.We would like to obtain p by maximizing the entropy subject to the con-straints Tr p =s 1 and the knowledge of s. This gives

Trdp[l + l n p + A + a . o ] = 0, (3.17.3)

where a.x, a,, a,, are Lagrange multipliers. Therefore, we have that

» » P - ( « . . , )

Tr exp (—a • o)

we must determine the a.x, av, and a, in terms of the measured ensembleaverages sx, sy, and st.

We leave it as an exercise for the reader to show that

-«•• r u i i s i n h Me = I cosh a — a • a,I I I I *

l«lsince (a • a)8 = a2/ where / is the identity matrix

and

|a| = a,1.

Since Tr o = 0 and Tr / = 2, we see that

Tr e""" = 2cosh |ct|,

(3.17.5)

(3.17.6)

(3.17.7)

(3.17.8)

3.17 DENSITY OPERATOR FOR SPIN-i PARTICLES [20]

so that we may write our optimum density operator as

tanh |a|i f tanh|o| "I2L |a| J

221

(3.17.9)

We next have when we use this in (3.17.1) that

tanh|as = T r - \ a -

loci(a • o)a

1 tanh lociTr o(a • c).

Since Tr

2 H0 if i jtj and Tr of = 2, (3.17.10) reduces to

tanh |oc|s = — a.

Then (3.17.9) becomes

(3.17.10)

(3.17.11)

(3.17.12)

This density operator maximizes the entropy subject to Tr p = 1 and measure-ments of o on the ensemble. In the (^-representation it becomes

. if 1 + sz sx- isJ

2Lsx + isv 1-3.1P B ; | . . \=P'-

In the representation in which p is diagonal, we have

l p + W 0 "12L 0 1- ls l J '

(3.17.13)

(3.17.14)

where |s| = y/sx* + sv* + sz2. (Prove this.) Then we easily see that

Tr ,>* = Tr p'2 = *(1 + s*) < 1, (3.17.15)

where the inequality follows from the general theory.If a beam is unpolarized, then sx = sy = sx = 0 since it is equally likely

that the spins are pointing in any direction. Thus

p = \h (3.17.16)

for an unpolarized beam. In case the beam is completely polarized, then it isin a pure state so that by (3.17.15) the equality holds and

|s| = ±1 , (3.17.17)so that

P = f° °1Lo u

(3.17.18)


When the beam is in a mixed state (partially polarized) |s| < 1 and we maydefine the degree of polarization, P, in the direction s by

P=|s|; (3.17.19)

P = 0 is unpolarized and P = 1 is completely polarized.Let us assume at t = 0, we have measured s so that p(G) is given by

(3.17.13). If we pass our beam into a region in which there is a magneticfield, H(f), the hamiltonian is

H = — a • H(t).

The density operator in the SP satisfies

dt ~ 2

(3.17.20)

(3.17.21)

We would like to find p(t).Since any 2 x 2 matrix can be expanded in terms of a and /, we write

pit) = \[s*{t)I + s(r) • a), (3.17.22)

where by (3.17.12) s(0) = s and JO(0) = 1. If we use this in (3.17.21), weobtain

2

since / commutes with a • H. We may rewrite the commutator as

[o • H, a • s] = (a • H)(o • s) - (a • s)(o • H)3 3

However,

(3.17.24)

(3.17.25)

where / , / , k form an even permutation of 1, 2, 3. Thus

[a - H, c • s] = 2i 2 HiSjO* - 2iH x s • c, (3.17.26)tt.i.W

since in (3.17.24) the i =j terms vanish. Therefore, (3.17.23) becomes

ds0 _ ds- J + -At At

(3.17.27)

3.17 DENSITY OPERATOR FOR SPIN-i PARTICLES [20] 223

If we take the trace of both sides since Tr / = 2, Tr <r< = 0 we see that

dso(t)dt

= 0,

so from the initial conditions we see that

so(t) = 1

(3.17.28)

(3.17.29)

for all time. If we next multiply both sides of (3.17.27) from the left by ait

trace and note that Tr o Oy = 2<5,7, we obtain the three equations for s(/)

ds(t)dt

+yU(t) x 8(0- (3.17.30)

To proceed further we must specify H(/). An interesting case to consider is

H(/) = [hx cos <ot, ht sin <ot, Ho]. (3.17.31)

In this case we have

Asx , .— = — oiosv + ynx sin cot stAt

where w0 = yH0.If we let

At

As,

At

= +co0 sx — yhx cos cot sz

cot sy — sin cot sx),

we may rewrite these as

dt

As_At

dt

s± = sx ± isv

= i<o0s+ -

-ico0s_

If we let

(3.17.32)

(3.17.33)

(3.17.34)

(3.17.35)

(3.17.36)

(3.17.37)

(3.17.38)

(3.17.39)

224

these reduce to equations with constant coefficients:

«f...

dtiAcoS+ = —iyh1st

dS_—— — i Aci>S_

OPERATOR ALGEBRA

(3.17.40)

(3.17.41)

., , v"x^_-SJ . (3.17.42)at

where Aeo = co — <o0. If we add the first two of these we have that

•7 (S+ + S_) = i Aco(S_ - S+)dt

_2Awds1

yhx dt '(3.17.43)

where we used (3.17.42). From this it follows that one integral of the equationsof motion is

S+(t) + S_(0 ~ " ^ ^ 0 ) 1 •» cv (3.17.44)

We may. use this to eliminate st in (3.17.40) and (3.17.41):

From these we easily see that S± satisfy

where

fl* = (Aco)8

so that the solutions may be written as

S+(0 = L(0) - V^Tcil cos at + A sin ,

i (^J C l ] cos Qi + A* sin

(3.17.45)

(3.17.46)

(3.17.47)

(3.17.48)

:)" (3.17.49)

!9

(3.17.50)

3.17 DENSTTY OPERATOR FOR SPIN-i PARTICLES [20] 225

and A is a constant of integration yet to be determined. The amplitudes of thecos Qt term were chosen so that 5^(0) = 5±(0). If we use (3.17.44) and theabove we see that

s.(o - f x t {[2s«(°) - c » ( ^ J ] c o s Q r + ( x + A * > s i n " ' - Ci(ir)}-(3.17.51)

From (3.17.34) it follows that

dt

so if we use (3.17.51) we see that

(3.17.52)

(3.17.53)

When we use this and (3.17.44) we have that

Sx(t) = 2 jS [[A<osx(0) + y/»iS,(O)] cos at + a sv(0) sin at

From (3.17.40) we see that

dS+(0) _dt

yhtsz(0) - A<u s,(0)i (3.17.54)

-i[yhlSt(0) + Aai s+(0)]. (3.17.55)

(3.17.56)

If we use this and (3.17.49), we see that

A = -ia-l[yhlSx(0) + Aa)s+(0)].

After minor algebra we therefore obtain

sx(t) — a~* cos <w*{Atu[y/i1s.8(0) + Aetwx(0)] cos at

+ Acoii s,(0) sin at - AeoyAÔ) + ^

+ 5177L' {[^5,(0) + Aa>sx(0)] sin at - a sy(0) cos at} (3.17.57)

s/0 = - ^ ^LZ

+ Aft>5x(0)] sin at - a sjp) cos at}

"' {&<o[yhisz(0) + Act) sx(0)] cos at

sv(0) sin Qf - y^! Aco sz(0) + (3.17.58)


(3.17.59)

(3.17.60)

The magnetic moment isft = -\yha.

Its expectation value in the SP is

<ft) = - \ y h Tr p(t)a = -\yh Tr[7 + s(t) • a]a.

Since Tr ot = 0 and Tr a^ = 2<5W, we see that

(JJI(O) = -$yhs(.t). (3.17.61)

Therefore, we may give a direct physical interpretation to the "components"of the density operator, s(t). If we restrict ourselves to resonance, Aeo = 0and by (3.17.54), (3.17.57), and (3.17.58), we see that

<j"*(0> = — kyh{sx(0) cos <*V + sin a)ot[st(O) sin yhjt — sv(0) cos yhj]}

(3.17.62)

</*&)) = - | y ^ { - c o s co0t[sz(0) sin yhtt - 5,(0) cos yhxt] + sin (O0tsj0)}

(3.17.63)

W O ) = -hyh{sz(0) cos y / ^ + sv(0) sin yhj}. (3.17.64)

= — Jy^ sin yftrt sin a>ot

</«,(0> = +4y^ sin yhxt cos wof

- * y * cos

If the beam is polarized along the z axis, sx(0) = sv(0) = 0 and st(0) = 1. Inthis case

(3.17.65)

This represents a precession of fix and //„ about the z axis and a "nutation"of {nt) between "up" and "down."

By virtue of (3.17.61) and (3.17.30), we see that (n(0) obeys the equationof motion

T <ft> - -y<|t> x H(t), (3.17.66)dt

which is the equation of motion of a classical dipole in a magnetic field andrepresents an example of Ehrenfest's theorem.

We may also obtain the Heisenberg equation of motion for a:

(3.17.67)

(3.17.68)

We may easily show that this reduces to

da- = - y o x

PROBLEMS 227

PROBLEMS

3.1 If A and B are two noncommuting operators, show that

eê8^ - exp [^Be^].

3.2 If A = xy + 3*1 Bx 3y and I is a real variable that commutes with x and y,show that

dei(Axe~iiA = * cosh f + i sinh $ —

dya

= y cosh f + i sinh f — .ox

3.3 If ? and i\ are parameters independent of y, show that

exp fl-r- — 'L \ ^

3.4 Show that

[at «.-*«*«] «, ( e - _ i ^ V =

3.5 Show that if m is an integer

[aU, atro] = /"«tm

[d*a, am] = —ma"1.

3.6 If |0> is the boson vacuum state, show that

—ca

where a; is a parameter and /(a1^) is any function of a* that may be expandedin a Taylor series.

3.7 Show that

3.8 If a; is a parameter and m is an integer, verify that

eâm = (a - x)me^ '}

3.9 If f and »; are parameters and (a, flf) and (6, fef) are two independent sets ofboson operators, find the normal form of


3.10 Express e-*«ta in (a) its diagonal representation, (b) in the coherent staterepresentation, and (c) in the /{-representation.

3.11 By means of Theorem 6, Section 3.2, show that

Tr e'»**V«* = n c5[Re £] (5[Im f],

where Re and Im mean real and imaginary parts, respectively.3.12 Find the Fourier transform of the antinormal associated function for e-***"

where A is real. Use the result to evaluate Tr erx*\3.13 Evaluate the thermal average

(a) in the number representation, (b) by Theorem 7, Section 3.2, and (c)by Theorem 9, Section 3.3.

3.14 If an ensemble of harmonic oscillators is in a coherent state |a')(a'|, obtainthe three characteristic functions C(o), C<B), and C(w) as well as the threeassociated functions pw, pin\ and P of Section 3.4.

3.15 If the oscillators of Problem 3.14 are described by p = |<x'><a'| where the barindicates we are averaging over random phases, find p<a) and P.

3.16 If p = (1 - «-A)e-*»to, find C ( a \ C<n>, C(w> as well as P.3.17 Generalize Theorem 9 of Section 3.3 for the case

/fa. «l;«i.«!;-.. I l l 0 ~e~

where the (ait a\) are independent boson operators such that

[ait af] = [a], a*] = 0

[a,-, flj] = ««,

3.18 Show that the generalized Wick theorem of Section 3.7 for bosons is un-modified in form if the operators y,- are in the interaction picture

provided we let

3.19 Show that the generalized Wick theorem for time-ordered boson operatorsbecomes

REFERENCES 229

3.20 For a two level atom, show that the operators iVf = |/><i| (i = 1,2), Aft|2><1| and M = |1><2| satisfy the following commutation relations

3.213.22

Show that U(t) in (3.11.13) is unitary.Prove Theorems 4-7. Section 3.10.

REFERENCES

[1] The author is deeply indebted to Dr. L. R. Walker of Bell Telephone Laboratoriesfor teaching him many of the operator techniques given in this chapter. See R. Kubo,/ . Phys. Soc. Japan, 7,1100 (1962); F. Coester and H. Kummel, Nucl. Phys., YJ, 477(1960).

[2] G. Weiss and A. Maradudin, / . Math. Phys., 3, 771 (1962).[3] W. Magnus, Commun. Pure Appl. Math., 7, 649 (1954).[4] E. Wichman, / . Math. Phys., 2, 876 (1961).[5] D. Finkelstein, Commun. Pure Appl. Math., 9,245 (1955).[6] A. Messiah, Quantum Mechanics, Vol. 1, New York: Intersdence, 1961, p. 442.[7] R. J. Glauber, Phys. Rev., 131, 2766 (1963). See also N. Chandra and H. Prakash,

Indian J. Pure Appl. Phys., 9,409 (1971), 9, 677 (1971), and 9,688 (1971); H. Hakenand H. D. Vollmer, Z. Phys., 242, 416 (1971); I. D. Dryugin and V. N. Kurashov,Opt. Spectrosc., 29,183 (1970) and 29, 345 (1970).

[8] J. R. Klauder, J. McKenna, and D. G. Currie, / . Math. Phys., 6, 743 (1965).f9] L. R. Walker, private communication.

[10] A. E. Glassgold and D. Halliday, Phys. Rev., 139, A1717 (1965).[11] E. Wigner, Phys. Rev., 40, 749 (1932); M. T. Raiford, Phys. Rev., 1, A1541 (1970);

I. A. Deryugin, V. N. Kurashov, and A. I. Mashchenko, Opt. Spectrosc., 30, 507(1971).

[12] M. Gaudin, Nucl. Phys., 15, 89 (1960).[13] G. C. Wick, Phys. Rev., 80,268 (1950); F. Dyson, Phys. Rev. 75,486 and 1736 (1949).[14] M. Lax and H. Yuen, Phys. Rev., Ill, 362 (1968).[15] H. Heffner and W. H. Louisell, / . Math. Phys., 6,474 (1965).[16] C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, Urbana,

HI.: University of Illinois Press, 1949.[17] E. T. Jaynes, Phys. Rev., 106,620 (1957).[18] E. T. Jaynes, Phys. Rev., 108,171 (1957).[19] R. C. Tolman, The Principles of Statistical Mechanics, Oxford: Clarendon, 1938.[20] H. A. Tolhoek and S. R. de Groot, Physica, 15, 833 (1951).

Quantization of theElectromagnetic Field

Light has wave-like properties in interference and diffraction experiments andparticle-like properties when it is absorbed or emitted by atoms. A satis-factory theory of radiation must explain in a unified way these two apparentlyparadoxical properties. By quantizing the electromagnetic field, Dirac [1]was able to bring about the first successful synthesis of these two aspects ofradiation.

It will be sufficient for our purposes to give a noncovariant (nonrelativistic)formulation of Dirac's theory of radiation along the lines of Fermi's [2]classic paper and Heitler's book [3]. We shall not quantize charges andcurrents which are the sources of the radiation field.

Since we are interested in a phenomenological use of quantized radiationtheory, we begin with the study of a classical lossless LC circuit with avoltage generator and show, by analogy with a harmonic oscillator, how totreat such a circuit quantum-mechanically. In the next section we show howto quantize a classical lossless transmission line. After these introductoryexamples, we begin a more general systematic study of the problem ofquantizing an electromagnetic field in a cavity. In Section 4.3 we show that aclassical radiation field in vacuum is equivalent to an infinite set of uncoupledharmonic oscillators. This equivalence suggests that we quantize the radia-tion field in the same way we quantize a harmonic oscillator; the quantizationis carried out in Section 4.4. The density of modes in a cavity is obtained inSection 4.5.

Section 4.6 gives the commutation relations for fields in a vacuum sincethese relations are closely connected with the theory of measurement and theuncertainty principle. The zero-point field fluctuations are discussed inSection 4.7 and are the source of the natural line width of atoms, the Lambshift, and quantum noise.

In the next two sections we present a simplified treatment of a radiationfield interacting with charges and currents. This should give the reader some

230

4.1 QUANTIZATION OF AN LC CIRCUIT WITH A SOURCE 231

idea of how to treat quantum-mechanically empirical models in quantumelectronics. A very general and thorough treatment of phenomenologicalquantum electrodynamics has been given by Jauch and Watson [4].

References 5 and 6 give a more thorough treatment of some of the topicsof this chapter.

4.1 QUANTIZATION OF AN LC CIRCUIT WITH A SOURCE

We utilize the classical analogy between a lossless LC circuit in series witha voltage generator and a driven harmonic oscillator as a simple example ofthe method to be presented in Section 4.3 for quantizing the radiation fieldin a multimode lossless cavity. We do not quantize the generator since thereaction of the circuit back on the generator may be neglected in both theclassical and quantum theories.

The classical "equation of motion" of a lossless LC circuit in series with avoltage generator, e(t), is

^ * ± (4.1.1)

where q(i) is the charge and coo2 = (LC)'1 is the circuit resonant frequency.

The conjugate variable in the circuit, p(t), is given by

We may use this to write (4.1.1) as

dp==_dt

e(t).

(4.1.2a)

(4.1.2b)

Equations (4.1.1) or (4.1.2a) and (4.1.2b) are equivalent ways of writingKirchhoff's law for an LC circuit with a generator. If we let the charge q(t) bethe analog of the coordinate and the current p(t) be the analog ofthe momentum, (4.1.1) or (4.1.2) describe equally well a driven harmonicoscillator. We may utilize this analogy and visualize (4.1.2) as the hamil-tonian form of the equations of motion for an LC circuit. If we let thehamiltonian be

1 - ' " ' * ' " (4.1.3)

and use the classical equations of motion (1.15.12), we find

dq_dl}__ i

dt ~ dp ~ LP

dp__ _dH_ i_

dt ~ dq ~ C

(4.1.4)

232 QUANTIZATION OF THE ELECTROMAGNETIC FIELD

which agree with (4.1.2). This result justifies the choice of (4.1.3) for thehamiltonian H(t) since (4.1.4) and (4.1.2) are the same. The hamiltonian andthe energy are not always the same.

We may pursue the analogy between the circuit and material oscillatorfurther to quantize the circuit. We associate hermitian operators with thecharge q and the current p and require that they satisfy the commutationrelation

[q,p] = ih, (4.1.5a)

and we have quantized the LC circuit. We may also introduce the non-hermitian operators a and a1 in terms of the charge and current by means of

a = (2ha>0L)-H[a>0Lq + ip]

a f = (.2Hco0L)-H[co0Lq - ip],(4.1.6a)

or

(4.1.6b)

where a>0L = (CDQC)-1. From (4.1.5a) it follows that

[0,0*] = 1. (4.1.5b)

In terms of a and a*, the hermitian operator associated with the hamiltonian(4.1.3) for this nonconservative system is, by (4.1.6),

H{t) =

If we let

f(t) = -e(t)

\J2htOoL•f*«)

(4.1.7)

(4.1.8)

Equation (4.1.7) is the hamiltonian given in (3.11.11) for a driven harmonicoscillator. As we have already solved the SchrSdinger equation in that case,we may take over completely the solution in Chapter 3 [Eq. 3.11.18]. Theformalism is the same but the physical meaning of the operators is different.The number operator da for quanta of the oscillator becomes the numberoperator for photons in the LC circuit, and a and ot are circuit photonannihilation and creation operators, respectively. Now p and q are currentand charge rather than momentum and position. Therefore, from (3.11.18),

4.1 QUANTIZATION OF AN LC CIRCUIT WITH A SOURCE 233

the state of the circuit at time t is related to the state at time t0 = 0 by

|y(f)> = eAWeOU)'tJ^{em)''a}eBMa\Vi0)), (4.1.9)

where A, B, C, and D are given by (3.11.17).In Section 3.11 we showed that if the circuit is initially in a vacuum state,

then at time / it will be in a coherent (minimum uncertainty wave packet)state given by (3.11.22):

^ < " t (4.1.10)

(4.1.11)

(4.1.12)

It is left to the reader to show that

A{t) + A*(t) = -|C(0l2,so that

1.

This minimum uncertainty state may be considered in another way. Fromthe general theory of Chapter 1, the probability that a measurement of a1 awill yield m photons at time t when it is known with certainty the system is inthe vacuum state at t — 0 is

m\ec'W, (4.1.13)

where we used (4.1.10) and (4.1.11). Since from Section 2.5 we know that

1-0

oo-2-7=10. (4.1.14)

it follows from the orthogonality relation (m\l) — £m< that (4.1.13) reduces to

JVo(0 = e~l°imt I | C (°,1 ] • (4.1.15)m!

This probability distribution function is called a Poisson distribution over thephoton energy eigenstates. It is normalized to unity for all time, since

= 1. (4.1.16a)2iVo(0 =

A Poisson distribution is generated by the voltage generator.The average number of photons in the circuit at time t is

ffi-0(4.1.16b)


Therefore m(t) = (eta) = |C(/)|2; this shows that the average number ofphotons calculated in two ways is the same. The mean-square number ofphotons is

m\t) = 2 m2Pn.0(t) = m(t)[m(t) + 1],m=0

(4.1.16c)

which is characteristic of a Poisson distribution. The variance of the minimumuncertainty state is

(Am)2 = m^) - MOFThe state lv(0) will be referred to either as a minimum uncertainty state or asa Poisson state; we see that it may be produced by the application of agenerator to a circuit initially in a vacuum state. It should be emphasizedthat the wave packet is in terms of charge and current—not physical positionand momentum.

We have discussed the state of the circuit at time t in great detail; it isworthwhile to study the properties of the assumed initial vacuum state withmore care. We discuss the zero-point energy and zero-point fluctuations ofthe charge and current at t = 0.

Before the generator is turned on, the hamiltonian is given by (4.1.3) withe(t) = 0. If the state |y(0)) = |0>, the average values of the charge, current,and energy are easily found to be

<«> = <0|9|0> = 0

0

<0|p|0> - 0

(4.1.17)

<«ffl>

(Aq)

We see that, in the vacuum state, the average value of the charge and currentis zero but the average of their squares is not zero. Therefore the charge hasa zero-point fluctuation given by (A<y)2 = ft(o0C/2 and the current has azero-point fluctuation of (A/?)2 = ha>0Lj2 while the zero-point energy ish<o0}2. If <^2) and </>2) were zero, as they are classically when the current isinitially unexcited, then, the uncertainty principle would clearly be violated.We know that Af A/> = hjl is a direct consequence of the commutationrelation [q, p] = ih so that the zero-point fluctuations of q and p resultbecause the circuit has been quantized.

We consider next the zero-point energy h<o0j2. This also arises because qand p do not commute. However, since energy is not absolute, it may be

4.2 QUANTIZATION OF A LOSSLESS TRANSMISSION LINE 235

measured from the zero-point level. This means that the hamiltonian (4.1.3)can be redefined by [with e =0]

H = - - p2 + — fl2 — — - = titoda.

In this case we still find that for state |0>

(p) = <0|p|0> = 0 (q) = (0\q\0) = 0

(4.1.18)

(4.1.19)

2 2o)0L 2

From this, the average energy is

hcooiaâ) = 0.

By the redefinition, the zero-point energy has disappeared, but the zero-point charge and current fluctuations, < 2> and </>2>, have not disappeared;they are always present to ensure that the uncertainty principle is not violated.Since the zero-point energy may be subtracted out in this trivial way, weneglect it.

4.2 QUANTIZATION OF A LOSSLESS TRANSMISSION LINE

Another simple circuit example easily treated quantum-mechanically is alossless transmission line whose classical "equations of motion" are

dz

dz

dt

dt '

(4.2.1)

where V(z, t) is the voltage, /(z, t) is the current, and L and C are theinductance and capacitance per unit length, respectively. These equationsmay be decoupled in the usual way, and both V and / satisfy the waveequation

where c is the velocity of propagation given by

1—LC

(4.2.3)

It is well known that the wave equation (4.2.2) has forward- and backward-wave solutions. By proper choice of boundary conditions, we may have


standing waves or traveling waves. Since the next section deals with the waveequation in great detail, for simplicity we consider here only one forward-plane-wave solution of (4.2.2). We write a forward wave solution in the form

V(z, t)V C

(4.2.4)

where z0 is the length of the transmission line under study and a and itscomplex conjugate a* are arbitrary constants. The factor y/hco/2Cz0 is putfirst for normalization purposes. Furthermore, the propagation constant k is

2z:, (4.2.5)c A

where sL\C = Zo is the characteristic impedance of the line and X is thewavelength of the wave on the line.

We assume that <a is given and choose the length of line z0 to be a fixedintegral number of wavelengths

zo = mX — ~m, (4.2.6)

where m is a fixed integer. In this way, we restrict ourselves to one forwardmode only.

The energy contained in this length of line is

H = \ ["\cv\z, i) + Ll\z, t)] dz2 Jo

-I CV\z, t) dz, (4.2.7)

where we used (4.2.4) and (4.2.6). If we use (4.2.4), the energy is given by

H = hcoa*a. (4.2.8)

The choice of the normalization factor of V and / is responsible for the hcoin this classical result. We are just expressing the energy in units of hco.

Let us consider the significance of the arbitrary constants a and a*. If wespecify them, by (4.2.4) we specify the mode completely. We may thereforeregard a and a* as the quantities that describe the state of the system. Infact, a and a* may define two new real quantities q and p by means of

1 (<oq + ip)

- {wq - ip);(4.2.9)

4.2 QUANTIZATION OF A LOSSLESS TRANSMISSION LINE 237

so that the energy contained in a length of line of m wavelengths is, by(4.2.8) and (4.2.9),

H = Kp2 + «>V)- (4-2.10)

We have therefore shown classically that a single forward mode on a trans-mission line m wavelengths long is completely equivalent to a single harmonicoscillator. The mode is completely specified if a and a* (ptp and a) are given;they indicate the state of excitation of the mode.

We now see how to quantize the transmission line. We let p and q behermitian operators related to the nonhermitian operators a and o t by meansof (4.2.9). The voltage and current are therefore operators. We impose thequantum conditions

[q,p] = ih [0,^1 = 1, (4.2.11)

and the rest is straightforward. The operators a(t) and e f(0 in the Heisenbergpicture are

a(f) = ^ f * * ' * = ae-*""flte«w (4.2.12)

where the hamiltonian (without the zero-point energy) is

H — (4.2.13)

The voltage and current operators may therefore be written in the Heisenbergpicture by (4.2.4) and (4.2.12) as

Vn(z,i)=. + = , / ^ / H ( Z , 0 . (4.2.14)

It is clear that the voltage and current for this single mode commute.Let us derive the Heisenberg-operator equations of motion for VH(z, t)

and /H(Z, t). We have

(4.2.15)dt

dt H> H'

where the hamiltonian in the Heisenberg picture is

HH — hcoaHaB = hcoaâ.

Since

[a, a+fl] = a [at, afa] = —

(4.2.16)

(4.2.17)


in both the Schrddinger and Heisenberg pictures, from (4.2.14), (4.2.16), and(4.2.17), Eqs. (4.2.15) reduce to

ih—^ =dt

ihdlH

dt

2Cz0

L dz

ihdIH

C dz(4.2.18a)

(4.2.18b)

These are the classical equations (4.2.1). Again we have shown the equivalencein form between the classical "equations of motion" and the Heisenberg-operator equations of motion.

The state of the system (one mode of the transmission line) at time t isgiven as usual by

|V(O> = r^ ' I r tO)) , (4.2.19)where |y(0)> is its initial state of excitation and da is the photon-numberoperator for the number of photons in the length of line.

Since the hermitian operators VH and IH commute for this mode, theycan be measured simultaneously. However, it is easy to show that VB(z, t)and dIHJdz do not commute.

4 3 EQUIVALENCE OF CLASSICAL RADIATION FIELD INCAVITY TO INFINITE SET OF OSCILLATORS

In this section we review briefly the classical theory of radiation in asource-free cavity. We cast the theory in canonical form and show theequivalence of the field to an infinite set of harmonic oscillators. In thefollowing section, we quantize the field in the cavity by quantizing theoscillators as done in the previous two sections.

We use mks units throughout; symbols that are universally used will notbe defined since their meaning is assumed to be well known to the reader.

Maxwell Field Equations

If no sources are present, the electromagnetic field is determined by theequations

div B = 0 (4.3.1)

curlE=-?dt

div D = 0

curl H =dt

(4.3.2)

(4.3.3)

(4-3.4)

4.3 EQUIVALENCE OF CLASSICAL RADIATION FIELD IN CAVITY 239

whereB = jU0H D = «0E (4.3.5)

and fioeo = c~2, where fi0 and e0 are for free space.We may satisfy (4.3.1) identically if we let

B = curl A,

and (4.3.2) will be satisfied identically if we let

E = _ 9 Adt

(4.3.6)

(4.3.7)

where A and V are the scalar and vector potentials.Since Maxwell's equations are gauge invariant, it is easy to show that

when no sources are present we may work in the Coulomb gauge in which

div A = 0

V=0,(4.3.8)

so that both B and E are determined by A alone. In this gauge, the fields aregiven by

B = curl A = «0H div A = 0

£=_3A V = Q <4-3-9>dt

If we substitute these in (4.3.4) and use (4.3.5), we find that A(r, t)satisfies the wave equation

V2A = ^ ^ 4 . (4.3.10)

The fields in vacuum in the Coulomb gauge are determined by this waveequation.

Energy and Momentum of the Field

The energy H contained in the field inside a cavity is

( 4 - 3 - u )

where dr = dxdy dz is a volume element and the integration is carried outover the volume of the cavity. We may also associate a momentum G with


the classical field in the cavity by means of Poynting's theorem given by

G = \ f(E x H) dr = ~«o f ( | ^ x curl A ) dr, (4.3.12)

where we have used (4.3.9).

Expansion of A(r, f) in Cavity Normal Modes

The electric and magnetic fields in the Coulomb gauge in a vacuum aredetermined by giving the values of Ax, Ayt At at each point (x, y, z) at timet. If we think of Ax, Av, and As as the variables used to describe the field, wesee that there will be an uncountably infinite number. By the following pro-cedure we may describe the field by another infinite set of variables that areenumerable or countable.

We assume that the radiation field is contained in a cavity with perfectlyconducting walls. For simplicity, we let the cavity be a cube of total volumeT = I?. The procedure is independent of the shape of the cavity. If we areinterested in a radiation field in free space, we may let T ->• oo after the cal-culations are complete. If we impose boundary conditions on the fields, thesolution of the wave equation will have an infinite discrete set of normal-modesolutions orthogonal to one another and complete in the sense that anyarbitrary field in the cavity can be expressed as a sum of these normal modeswith suitable amplitudes. The amplitudes of each mode can then be usedinstead of Ax, Ay, and At to describe the field in the cavity. Since the modesare discrete and infinite, the amplitudes (new field "variables") are thencountably infinite. The boundary conditions force a discrete set of modes thatallows description of the field by a countable set of variables.

By the familiar procedure of separation of variables we may assume asolution of (4.3.10) of the form

A(r,0 (4.3.13)

where the coefficient eo~J/i is used for normalization purposes. If we sub-

stitute this into the wave equation, then for each / we have

^ + 0,

(4.3.14)

(4.3.15)

where cof is the separation constant for each /.We may obtain standing-wave solutions if we require that at the walls the

tangential component of E and the normal component of B vanish. From


(4.3.9) and (4.3.13), we have

"ilu. =

From (4.3.8),

curl uz|norm = 0 (on walls). (4.3.16)

div u,(r) = 0 (4.3.17)

everywhere in the cavity.The solutions of (4.3.14) that satisfy (4.3.16) give a discrete set of normal

modes that are orthogonal and normalized to unity:

1cftvitydlm. (4.3.18)

The Uj(r) and qt(t) are real in order that A be real.The u{(r) are sin kt • r or cos k, • r for the cubical cavity. For other geom-

etries the u,'s will be other complete sets of functions. The boundaryconditions and geometry determine the different modes which are dis-tinguished by the index /. In general, three or four numbers are needed tospecify a mode; / is an abbreviation for this set of numbers, as we shall see.

The normalization condition (4.3.18) removes all freedom from choosingthe amplitude of u,. The ut's are completely specified known functions. Theamplitude of each normal mode in (4.3.13) needed to specify a particularfield configuration is qt(t). If each qt(t) is given, the field is just as completelydetermined as if the value of Ax, Av, and Ax at each point in space at time twere given. The qt are taken as a new set of variables to describe the field.

The amplitudes qî) satisfy (4.3.15), the equation of motion of a harmonicoscillator. Therefore with each mode of the field there may be associated aradiation oscillator of frequency cu,. We next show that the energy containedin the total field is just the energy of the infinite set of uncoupled radiationoscillators.

If we substitute (4.3.13) into (4.3.11), we have for the energy in the cavity

H = £ Z Mm » i < " « ^ + ; I <lAm curl B, • curl nm dr.l.m ./cavity Z l.m ./cavity

(4.3.19)

If we use the orthonormality condition (4.3.18), the first double sum in(4.3.19) reduces to the single sum \ 2 <7i2- By the vector identity,

i

curl u, • curl um = um • curl curl u, + div (um x curl u,), (4.3.20)

the last integral in (4.3.19) reduces by means of Gauss's theorem to

J um • curl curl u, dr + (um x curl u,) • dS, (4.3.21)cavity Jwalls


where dS is an element of area on the cavity wall. The surface integralvanishes because of (4.3.16). The first integral in (4.3.21) is, by a well-knownidentity,

Jum • (grad div u, - V\) dr = ^ - Ju , . um dr = ^ - dlm. (4.3.22)

We have used (4.3.14), (4.3.17) and (4.3.18). When we put this in (4.3.19),the hamiltonian for the field is

(4.3.23)

But / / , is the energy of a harmonic oscillator of frequency « „ and so thefield energy is equivalent to an infinite set of uncoupled radiation oscillators.

The hamiltonian equations of motion for the /th oscillator are

—-dqt

(4.3.24)

Here/?, and qt are called canonically conjugate variables.We'are, as usual, able to define two complex variables a, and aj for each

mode in the now familiar way by

Pi a\(4.3.25)

The equations for a, and

dat(t)

dt

and have solutions

aft)1 =

are

d<-iw,a,(0 a!(0 t

dt

(t) = ate*"1.

(4.3.26)

(4.3.27)

In terms of Cj and a], (4.3.23) is given by

H = i 2, fio>t(a\ai + wl)- (4.3.28)

In this purely classical theory, a, and a] commute. We have preserved theorder for the quantum treatment in the following section.

4.3 EQUIVALENCE OF CLASSICAL RADIATION FIELD IN CAVITY

To summarize, we may write the fields in standing waves as

243

(4.3.29)

H 7= 2 9^) curing).

Plane-Ware Representation of the Fields

It is often convenient to represent the fields in terms of plane travelingwaves rather than standing waves.

We write the vector potential as a linear superposition of plane waves inthe form

A(r, 0 21 JTt <T=1 V 2ft)j€0T

r -

+ aja exp [- f(k, • r - ofi]}- (4-3.30)

We now explain the meaning of the symbols. The vector SJ(J and the numbersala and a\a are constants.

The vector k, is the propagation constant and if2

k,2 = - 7 , (4.3.31)c

then each term in the series (4.3.30) satisfies the wave equation, which weassume is familiar to the reader.

From the Coulomb-gauge condition div A = 0, we see that

«b,.k,-O. (4.3.32)

This is called the transversality condition. Here k{ is the direction of propaga-tion of the plane wave. Since E = —dA/dt, we see from (4.3.30) and (4.3.32)that £ and A are transverse to the direction of propagation in the absence ofsources. It is an auxiliary condition imposed by working in the Coulombgauge.

The vectors Sn and &l2 are unit vectors used to specify the polarization ofthe plane wave. Since each polarization is independent, the total field mustbe summed over both polarizations in (4.3.30). For convenience, we shallchoose e ; i perpendicular to @I2 so that

« t l • II2 = 0. (4.3.33)


It is often convenient to define a unit vector in the direction k, as

(4.3.34)

Equations (4,3.32) and (4.3.33) may then be written

K ' K~ = <U< °, <*' = 1,2 ft, • «„ = 0. (4.3.35)

It is convenient to require that the vector potential satisfy periodic boundaryconditions on opposite faces of the cavity in order to make the modesdiscrete. Iff, j , and ft are three unit vectors along the cube edges, the positionvector is r = x\ + y\ + zk and the propagation vector is kt = kji +fcjj + fcjisft- The periodic boundary conditions require that

A(r + LI, 0 = A(r + L), t) = A(r + Lk, t) = A(r, 0,

and are satisfied if

k, = — (/J + / J + Z3k) (4.3.36)

where lu l2, and /3 are integers from — oo to + oo. That is, the propagationconstants are restricted to a discrete set of values by virtue of the boundaryconditions.

For each triple of integers (lu l2, IJ there are two traveling modes (onefor each polarization a), according to (4.3.30). If we let (lx, l2, /3) go to(—h, ~U, —la) (which is designated simply by /-»- —/), from (4.3.36),

From (4.3.31),k_, = - k , .

0 , = <»_,.

(4.3.37)

(4.3.38)

Therefore, from (4.3.30), (4.3.37), and (4.3.38), we see that, if we change / to—/, the two corresponding plane waves travel in opposite directions. Wetherefore have included in (4.3.30) forward and backward modes.

The sum 2 is now understood as a shorthand notation for

00 00 CO

1 = I I I•I ll*«—00 Zt=—00 Ij«s—00

(4.3.39)

The set (lu l2, l3, a) gives a mode of a given polarization.It is easy to verify that the vector potential is real.We next show that the vector potential A(r, r) and the electric field

E = —dA/dt may be expressed in terms of canonically conjugate variables.For this purpose, we let

«to(0 - a.y""» (4-3.40)


and

0 (r) = —— ,_ ' — u£ — — -p , (4.3.41)

which satisfy the orthonormality relations

u£(r) • u,v(r) dr = 6lvdaa..Jc

(4.3.42)cavity

The variables ala and a\a may be used to describe the field, and we mayagain introduce real variables pla and qla by

a,, = ipla)

1(4.3.43)

The electric field E = -dA/dt so that, by (4.3.30) and (4.3.40),

exp (ik, • r) - al(i) exp ( - i k , • r)],o

(4.3.44a)while the magnetic field H = curl A//*o is

E(r, 0 = i

H(r, 0 = - — 2Cfi1

» * r ) ~ a^

where we used the fact that

curl [K exp (±ik, • r)]

(4.3.44b)

x kj) exp (±ik, • r)

-*(«„ x kj) exp (±ik, . r).

Here |kj| = cojc and fc,|ki| = kt.The hamiltonian for the field in the cavity is

H = \ \ («0E•/cavity

+1."

(4.3.45)

The derivation of this result is given in Appendix B. We have kept the orderof factors of aa* and a*a so that the analysis will also be valid when we treatthese as noncommuting operators, although they commute classically.


Again, as in the treatment for standing waves, from (4.3.28) a radiationoscillator may be associated with each mode of the cavity andplo a n d ^ arecanonically conjugate variables. The total field energy is just the sum of theenergies of each independent normal mode of the field.

The canonical hamiltonian equations of motion for the field, from (4.3.45),are

dHVu

dH(4.3.46)

which, as expected, are the oscillator equations of motion.The multiplying constant in (4.3.30) ( /2a>,e0T)+1^ was chosen so that H

would be measured in units of /too,.

Momentum of the Field

We evaluate the momentum of the field

in Appendix C. It is

G = -.1 (ExH)dr,C Jcarity

(4.3.47)

(4.3.48)

Again the units have been chosen to measure the momentum in units ofKkt although the present analysis is still classical.

4.4 QUANTIZATION OF THE RADIATION FIELD IN VACUUM

The method of quantizing the radiation field is now straightforward. Weassociate hermitian operators with the classical field variables plg and qlg. Itis experimentally known that photons are bosons so that we postulate thatqla and/?/<T satisfy the boson commutation relations. Quantization is necessaryto show the particle nature of light. In terms of the nonhermitian operatorsala and a]c> these relations are

[a*, «ivl = 0 = [al, a\.g.].

That is, since the radiation oscillators are all independent, alg andcommute for different oscillators.

The hamiltonian for the field (without the zero-point energy) is

(4.4.1)

H = (4.4.2)

We have already shown that it is possible to change, the level from whichthe energy is measured.

4.4 QUANTIZATION OF THE RADIATION FIELD IN VACUUM

The momentum is, by (4.3.48) and (4.4.1),

247

l.a

But since k_, = —kj, the sum = °- Therefore,

G (4.4.3)2l.a

We may now take over all the previously derived results for bosons.There is the slight generalization in that there are now an infinite number ofindependent field oscillators instead of one oscillator. We interpret ala anda]g as an annihilation and a creation operator, respectively, for a photon indirection k?, polarization a, and frequency o f. Here Nlg = a1

laalg is thephoton-number operator for the number of photons in the /, a mode. Theeigenvalues of JV/o are nla = 0,1,2 oo. The energy of the photon instate /, a is ho>t, by (4.4.2), and the photon has a momentum fibu by (4.4.3).The photon is therefore seen to exhibit particle-like properties, and travel withvelocity c.

Since each cavity mode is independent, a complete set of state vectors maybe written as a simple product of the state vectors for each mode; that is, astate vector for the radiation field may be written as

l»i>l«i> •' * I O = l«i» »t, • • • . O (4-4.4)

where each subscript 1 , 2 , . . . stands for the quartet of integers (llt /2, /3, a).The state vector for an assembly of noninteracting bosons must be sym-

metric under the interchange of any two of the bosons. It can be shown[1, 5] that, by specifying the number of bosons in each state, we obtain awave function with the correct symmetry so that (4.4.4) correctly describesan assembly of noninteracting bosons. We do not go into this questionalthough such considerations justify the interpretation of ala and a]a as bosonannihilation and creation operators [1].

The effect of ala and d[a on the state vectors (4.4.4) is given by

a]a\..., nla,...)

au\ .. ., nla,...)a t f f | . . . , 0 , . . . )

Nla\ . . . , nla,...)

(4.4.5)

J ..., «,„,With this choice, these state vectors are normalized to unity.

As in the case of the single oscillator, these operators are in the Schrodingerpicture. They may also easily be generalized to the Heisenberg or inter-action picture. For example, the Heisenberg equations of motion for «,„(*)

248

are

QUANTIZATION OF THE ELECTROMAGNETIC FIELD

= [ f l f a ( 0 > H a ] " - i

The Schrodinger equation is

(4.4.6)

(4.4.7)

If Hs is given by (4.4.2) for an infinite number of noninteracting oscillators,then

IVs(0) = exp ( - i 2 ala^j I*0)>. (4.4.8)

We may express an arbitrary initial state of the system by

lv(0)>= I <*Jh.,n n<a,0)\n1,rt2,...,na>), (4.4.9)m.nj nm

where we have abbreviated /, a by /. Therefore, (4.4.8) is

(4.4.10)

exP ( -

We also have the completeness and orthogonality relations

"*•"* "

n2> • • • » n o » » • • • » " c o l

(4

Since each mode of the radiation field is independent, we may introduce acoherent state as in Section 2.5 for each mode. If we again let / represent /and <r as above, we have

a,|a,> = «,|a,> (4.4.12)

(4.4.13)

All our previous results on the coherent state of an oscillator may be takenover directly. A classical source driving the cavity will excite a mode in acoherent state. We may write a state vector as

(4.4.14)

4.4 QUANTIZATION OF THE RADIATION FIELD IN VACUUM 249

If we take the expectation value of the free electric field (4.3.44a) whenthe cavity is in a state described by (4.4.14), we have

where<a|E(r,0|a> = «?<+)(r,0

,0 =

'"'(r, t) (4.4.15a)

= [<?(->(r, 0]* (4.4.15b)

Therefore, when the field is in a coherent state, its average value looks like aclassical field. In contrast, if the field is in the state |n) = \nt, n2,...,»,),we have that

<n|E(r, r)|n> = 0 (4.4.16)

no matter how highly excited the modes are; that is, no matter how large the/Zi's are. By the correspondence principle when a system is highly excited itsenergy is large compared to Hco and the system should behave classically.Therefore, the eigenstate |n) is very poor to show the classical nature ofthe field whereas the coherent state is ideally suited for this purpose.

We may expand an arbitrary initial state by means of the completenessrelations:

whered*~ _ I T d2<X'= n »

IT I IT

andy<a, 0) = y(al5 <*2 * • • ; 0) = <al

At time t, we have from (4.4.8) and the above that

(4.4.17)

(4.4.18)

(4.4.19)

77

=nf—I J IT

If we write the exponential in normal order, we have

(4.4.20)

(4.4.21)

QUANTIZATION OF THE ELECTROMAGNETIC FIELD

WO) = I I f — «P [(«-*"" - DaIaI]|al><a,|y(0)). (4.4.22)i J ir

250

so that

If at t = 0 the free field is in a coherent state, this shows that it will remainin a coherent state. [Prove this.]

4.5 DENSITY OF MODES

In the ensuing work it will be necessary to know the number of normalmodes in a given frequency range contained in a cavity of volume T. Thisinformation is found in (4.3.36):

(4.5.1)

Each set of integers (lu l2, /s) corresponds to two traveling-wave modes,since there are two polarizations. We may represent each mode of a givenpolarization by a dot in a three-dimensional space, as shown in Figure 4.1.In a small element of volume dlx dla dla, the number of normal modes is

dN = 2dlx dlt dla. (4.5.2)


dN = 2 (—\ dkx dkv dkt. (4.5.3)

In summations of discrete values of / (/1} /2, /3), when we let L -*• co(free-space limit), IJL, IJL, and 13/L become practically continuous variables.We may replace sums by integrals so that

CO

jili )—*-j-A[\dkxdkydkx{ ). (4.5.4)L i £ - » (2TT)8 JJJ

—«

/

/

/

/

J

/

h

///

y

/

/

//

j

/

/-

/

J

///

Figure 4.1 Diagram of normal modes in a cavity.Each triple of integers (/lt lt, 73) corresponds to amode of fixed polarization.

4.6 COMMUTATION RELATIONS FOR FIELDS IN VACUUM 251

We may transform from rectangular coordinates (kx, ky, kt) to polarcoordinates by means of

k = k(sin 0 cos q>, sin fl sin q>, cos 0), (4.5.5)

so that the element of volume in k space is

dkx dkv dkt = k*dk sin 6 dd d<p = k* dk da, (4.5.6)

where dCl is an element of solid angle about the direction of propagation k.We may then write (4.5.3) as

dN = ll—

Obviously, the total number of modes is

00.

(4.5.7)

(4.5.8)

In (4.S.7) dN gives the number of modes in volume L? in a solid angledd about the direction of propagation k in the range between k and k + dk.Since eo2 = c 2 ^ , we may also write the number of modes in I? in the fre-quency range eo to <a + dco. We have

and so

dN « 21~Jw2 d(o dCl = 2 (-Tva dv da,

(4.5.9)

(4.5.10)

where co = 2m>. Therefore, the number of oscillators per unit volume withangular frequency between co and co + dm in solid angle dQ. is

2ft)2

(4.5.11)

where g(a>) is called the mode density.

4.6 COMMUTATION RELATIONS FOR FIELDS IN VACUUMAT EQUAL TIMES

We have seen in Section 1.12 that the commutation relations for observablesare closely associated with the problem of measurement of these observables.The physical observables of electromagnetic fields are D = c0E and B =fiJH.. From (4.3.44), these fields are represented by hermitian operators, sincea\a is the adjoint of aio. It is of interest to use the commutation relations for


these operators to find the commutation relations for the physical observables£ and H to see the quantum restrictions on their measurement imposed bythe uncertainty relations. Such questions are of great importance in quantumelectronics.

By (4.3.30) the vector potential is also a hermitian operator althoughclassically it is not a physical observable. However, it obviously plays a veryuseful role in the theory, and so we begin this section by evaluating thecommutator of D and A.

Commutation Relations for D and A at Equal Times

The vector potential in the SchrSdinger picture is, by (4.3.30),

Tua 2eo{e0T

i, exp («k, • r) + a\. exp (-ik, • r)], (4.6.1a)

where we use the convention that ala(t) is in the Heisenberg picture whilealc = ala(0) is in the Schrodinger picture at / = 0. The fixed time for theSchrodinger picture could be taken as t0 but nothing is gained by this choice.

The electric displacement operator D = e0E in the SchrSdinger picture is,by (4.3.44a),

(4.6.1b)l.a

The commutation relations for ala and a\, are given by (4.4.1) and must bethe same in both the Heisenberg and Schrodinger pictures.

Let ^ ( r ) be the fth component of Ag(r, 0) in (4.6.1a) and D,(r) be theythcomponent of Ds(r, 0) in (4.6.1b). If we use the commutation relations(4.4.1), then we see that

2 T I.P>

where

p = r — r .

- f t , • p)], (4.6.2a)

(4.6.2b)

We may next carry out the sum over the polarization index a. We knowthat £{, ka, and e,2 are three mutually perpendicular unit vectors. Thenumber (€tl),. is the component of the vector Stl on the cartesian axis x{ andis just the direction cosine of the angle between tn and the xi axis. Similarly,(Siz); is the direction cosine between €t2 and the xt axis, and (£,)< is the direc-tion cosine between fe, and xi axis. By the well-known properties of directioncosines, we then have

4.6 COMMUTATION RELATIONS FOR FIELDS IN VACUUM

or

253

Z9=1

If we substitute this in (4.6.2), we have for the commutation relations in thecavity

% , • p). (4.6.4)[Air), Df{v')) = - % I [dtl - &)«T 1

We have been able to combine exp (—ik, • p) and exp (+/k, • p) since£_, = — fc+, and the sum over / is for all positive and negative integers.

The commutator At(T) and D,(r') in a cavity is therefore given by theunusual-looking result in (4.6.4V For free space, we may let the cavity becomeinfinitely large (L3 = T-+ OO). When we do this, we may use (4.5.4) in theright side of (4.6.4) to replace the sums by integrals, and we have in free space

(4.6.5)where we define

and dk = dkx dkv 3kt is a volume element in Ar-space. Here d{iT is called the

transverse 8 function and is not the ordinary Dirac 8 function, which is

oo

=thi f f f(2TT) JJJ

00

We derive some very useful properties of the transverse 6 function inAppendix D. For example, we show there that [Eq. D.6)]

2 d8ii ( P ) = 0, (4.6.8)

where p = r — r' and xly x2, x3 are the components of r. From this, if we takethe divergence of both sides of (4.6.5), we have

(4.6.9,

Since Dtf) jt 0, then


This, however, is the requirement for using the Coulomb gauge, so that thecommutation relation (4.6.5) is valid for this gauge.

We may use the fact that commutation relations are identical in both theHeisenberg and Schrodinger pictures to write from (4.6.5)

where t is the same time in A, and Dt.We may next show that

[A{(r,t),Af(T',t)]

(4.6.10)

(4.6.11)

To do this, we work in the Schrodinger picture. By (4.6.1a) and the com-mutation relations (4.4.1), we have

\Alx\ Alx')\ = J , ~ L - (•„)<(«„),[«? (ft, • p) - exp (-i% • p)],l.a 2(Ot€0T

where p = r — r'. We may use (4.6.3) to carry out the sum over a so thatthe above commutator becomes

[A/f), AM] = ih 2 — [*„ - &)<&),] sin k, • p. (4.6.12)I COf€0T

Since to, = c|k,| and k_, = —k,, we see that tD,-1[dtf — (fcjX&Vl *s e v e n

under an interchange of — / for / while sin k, • p is odd, and so the sumvanishes. Therefore, (4.6.11) is true in the Schrodinger picture and hencein the Heisenberg picture.

Commutation Relations for D and B at Equal Times

The physical observables are D and B (or E and H). It is therefore veryimportant to know their commutation relations at equal times.

By the same procedure used to prove (4.6.11), it is easily shown that

(4.6.13a)

(4.6.13b)[* i(r,0,**(r'.0]«-0,

where Bf in the Schrodinger picture is given by

B*(r,0) - - i Y2 T

X tfta,.exp (ft,. r)

In the proof of (4.6.13a), we need the result

2 (K x t,U«to x tO, - 2 («toM«to),-

exp ( - f t , , r)].

(4.6.14)

(4-6.15)

4.6 COMMUTATION RELATIONS FOR FIELDS IN VACUUM 255

The proof of this follows easily if we let eu, S,2, and t , form a right-handsystem with

*to • ©to-* ^ ^<rir' *> ^* s = ^ » ^

g,, • £, = 0 (4.6.16)

Then, Sn x fc, = — e,2 and el2 x fe, = e(1, and the proof is evident.From the theory of Section 1.12, it may be concluded from (4.6.13) that

any two compounds of D (or E) may be simultaneously measured withoutany mutual interference. The same remark applies for any two com-ponents of B (or H) since all components commute.

In Appendix £ , we derive the commutation relations

and, 0, £,(r', 0] = 0

[D/r, 0, B£r', t)] =

. . 9

ih

(4.6.17)

(4.6.18a)

(4.6.18b)

where we use (4.6.18a) if i,j, and k form a cyclic permutation of 1, 2, and 3and we use (4.6.18b) if i,j, and k form a cyclic permutation of 1, 3, 2.

From (4.6.17) we conclude that parallel components of D and B may bemeasured simultaneously without mutual interference, and from (4.6.18) wesee that perpendicular components cannot be measured simultaneouslywithout interference.

Heisenberg Equations of Motion for D and B

Since D, B, E, and H are operators, they satisfy the Heisenberg equationsof motion. The hamiltonian for the field may be written as

H = \ f p - D2(r', 0 + J- B2(r\2J Lc0 i"o

(4.6.19)

It is a hermitian operator given here in the Heisenberg picture. In AppendixF, we show that

dti/i curl Hi (4.6.20a)

i^-dt

, H] = — lAcurlE^. (4.6.20b)


These are the Heisenberg equations of motion for the operators in theHeisenberg picture, but they are identical in form with the classical Maxwell"equations of motion"; thus we have developed a self-consistent quantumtheory in that it will become classical theory when H-+0. From (4.6.18),as h -*• 0, the operators commute and behave like classical variables.

The remaining two Maxwell equations are also satisfied. Since £,„ • £ t = 0,it is obvious from (4.6.1b) that div D = 0. Also, since (St0. x fcj • fcr = 0,(4.6.14) shows that div B = 0.

4.7 ZERO-POINT FIELD FLUCTUATIONS

In Section 4.1 we discussed the zero-point energy and the zero-pointfluctuations of an LC circuit. In this section we discuss the zero-pointfluctuations of the fields.

We may write the yth component of the electric displacement vectorDj(j, t) in the Heisenberg picture from (4.6.1b) as

Df(r, t) = Z>(+'(r, 0 + Dlr\r, t),where

r, 0 =

(4.7.1)

(4.7.2)

Here D\+) contains only annihilation operators, and its adjoint £>)"' containsonly creation operators.

The average value of D is

<£>,) = <y(0)|Z>,(r, 0lv(0)>> (4-7.3)

where |y(0)> is the state of the field at t => 0. In general, |y(0)> is given by(4.4.9) as an expansion in the complete set of eigenkets of the energy (orphoton-number operator). For simplicity, we consider the case in which thefield has been prepared in a pure energy eigenstate at f = 0 and there arenu quanta in the la mode, that is,

|y(0)> = |«,, n2, . . . , « „ . . . , O , (4.7.4)

where we have abbreviated la by /; that is, the modes are labeled 1 ,2 , . . . , oo.If we use (4.4.5) and the orthogonality relations (4.4.11), the average of D}

for the energy eigenstate (4.7.4) is

(D,) = 0, (4.7.5)

that is, the average value of the electric displacement vector is zero when thefield is in an energy eigenstate. It is also easily shown that th? average of B is

(Bt) = 0 (4.7.6)

4.7 ZERO-POINT FIELD FLUCTUATIONS 257

for (4.7.4). This is the analog of the classical result that the time average ofD(r, t) and B(r, t) is zero for each harmonic component.

Next we consider the expectation value of Df for the state |y(0)> in(4.7.4). From (4.7.1), we have

D? = D{tn + Di"'2 + Di-)Di+) + Di+)Dlr\ (4.7.7)9 3 * 9 * 9 3 * 7 7 \ *

If we calculate (Df), we have by (4.4.5), (4.4.11), (4.4.1), and (4.7.2), afterminor algebra,

/(r,0> = I f s ^ ( ^ W + i). (4-7.8)l.v T

Since

D2 = 2 ^ / (4.7.9)

we have

l,2. (4.7.10)Tl.a

But (e^), is just the direction cosine between ela and the xi axis, and so}2 — 1- Therefore, the average value of the total field is

T l.a

If the nla = 0 (the vacuum state), then

2T l,<r 2

(4.7.11)

(4.7.12)

This is called the zero-point field fluctuation. Each mode of the field con-tributes an amount hoajl; since there are an infinite number of modes inthe cavity, the zero-point field fluctuation is infinite.

This infinity is not fundamental since we have not specified the means bywhich we shall measure the field. In electrical engineering terminology, thebandwidth of the detecting instrument would be finite and so it would notdetect an infinite value for these zero-point field fluctuations. We mightenvision an electric charge interacting with the field as a means of detection.In this case the charge would occupy some small volume of space, AF, andit would take some small amount of time, At, for a measurable effect to occur.This would imply that, to obtain physically meaningful results, we wouldhave to compute a space-time average:

ffAV At

f dr f dtDsB(r, t). (4.7.13)JAV Jit


The expectation value of Df in the vacuum state is then

JT dr dt dt

From (4.7.2) we see that

, t)Dtf, t')\0).

(4.7.14)

(OlD/r, f)/>,(r\ OlO> = I ^ ! ( € „ ) / exp {/[k,. (r - r') - o>,(* - t')]}.I.a 2 T

(4.7.15)3

If we sum overy to get the total field, since £ (KV = 1, we have for (4.7.14)£

<0| 2 [D/AK. A,)f |0> = dt dt'

"X I " exp {/[k, • p - aft - t')]}, (4.7.16)r

where p = r — r'. We may carry out the integral over dt and dt' and changethe sum over / to an integral over k by means of (4.5.4), and we have

x fff d t <^d T > exp (ik • p)4 sin2 \ck At, (4.7.17)

where w, ->• c|k| = ck. As long as At and AK are finite, the average value ofthe field in the region measured will be finite, from the above integral,whereas if A V and At -*• 0, it becomes infinite. Restricting knowledge of thefield to a limited region of space time has effectively placed a bandwidthlimitation on the detecting instrument.

The zero-point field fluctuations give rise to measurable effects in quantummechanics. In particular, they account for the Lamb shift of the 2PiA-2St/i

energy levels of atomic hydrogen, where the zero-point fluctuations interactwith the electron. One might say picturesquely that they "induce" spon-taneous emission of the electron in the 2P,A state. The zero-point fluctuationsalso give rise to spontaneous emission in lasers, parametric amplifiers,attenuators, and such, and are the source of quantum noise. They may alsobe said to be the "source" of the natural line width of atoms. We discuss thesequestions in more detail later, after we discuss the interaction of electro-magnetic fields with matter.

4.8 CLASSICAL RADIATION HELD WITH SOURCES [7] 259

4.8 CLASSICAL RADIATION FIELD WITH SOURCES [7]

We next turn our attention to the problem of an electromagnetic fieldinteracting with a given charge and current distribution. This is the fieldgeneralization of an LC circuit with a voltage generator. This and thenext section should give some insight into the way to proceed in suchproblems.

The Maxwell equations with sources in inks units are

div B = 0

curl E = — —dt

div D = p

curl H = — + J,

whereB = fioH D = €0E.

The charge and current must satisfy the continuity equation

dpdiv J + -f = 0,

dt

(4.8.1)

(4.8.2)

(4.8.3)

(4.8.4)

(4.8.5)

(4.8.6)

where J and p are functions of r and t. This follows directly if we take thedivergence of both sides of (4.8.4) and use (4.8.3).

As in the source-free case, (4.8.1) and (4.8.2) are identically satisfied ifwe let

B = curl A (4.8.7a)

dt(4.8.7b)

Since it is well known that the Maxwell equations are gauge invariant, weshall work in the Coulomb gauge as we did in the source-free field case, inwhich

div A = 0. (4.8.8)

However, when p and J are not zero, we may no longer let V = 0.If we now substitute (4.8.7a) and (4.8.7b) into (4.8.4) and use div A

and (4.8.5), we obtain, instead of the wave equation (4.3.10),0,

(4.8.9)


while if we put (4.8.7b) into (4.8.3) and use (4.8.5) and (4.8.8), then Vsatisfies the Poisson equation

- - P. (4.8.10)

We are assuming that p and J are given functions. Therefore, the potentialV is determined by the well-known result that

(4.8.11)

where dr — dx' dy' dz'.It is known from vector analysis that any vector A may always be written

as the sum of its transverse components A r and its longitudinal componentsL

A = AT + AL, (4.8.12)where, by definition,

div kT = 0 curl AL = 0. (4.8.13)

In the Coulomb gauge, since div A = 0, we have, by (4.8.12) and (4.8.13),

divA£ = 0 (4.8.14)

If both div C — curl C — 0, then we may let C = 0. Therefore in theCoulomb gauge

A i = 0,

and the vector potential is purely transverse.We may also decompose the current density as

TT I T£— J T «

wherediv 3T = 0

curl 3L = 0.

Then, from the continuity equation (4.8.6),

(4.8.15)

(4.8.16)

(4.8.17a)

(4.8.17b)

--t> < 4 8 1 8 >ot

that is, dp/dt gives rise only to a longitudinal current.We may satisfy (4.8.17b) identically if we let

3L = Vy, (4.8.19)

where \p is any scalar function since curl grad y> = 0. If we substitute this

4.9 QUANTIZATION OF FIELD WITH CLASSICAL SOURCES 261

into (4.8.18), we see that

2 (4.8.20)V2y = - — .dt

At each instant this is Poisson's equation whose solution is

By (4.8.19) and (4.8.11),

1 d

l r -

* - « • * £ • (4.8.22)

If we substitute (4.8.12) and (4.8.16) into (4.8.9) and use (4.8.22), we have

(4.8.23)

which is the equation that A r must satisfy while the potential is given by(4.8.11). Classical fields with sources must satisfy (4.8.10) and (4.8.23) in theCoulomb gauge.

4.9 QUANTIZATION OF FIELD WITH CLASSICAL SOURCES

To quantize the field, we must find a hamiltonian such that the Heisenbergequations of motion for the field quantities, considered as operators reduceto the Maxwell equations. The hamiltonian is not necessarily the same thingas the energy in general. In treating empirical models, trial-and-error methodsmust be used to find a hamiltonian such that the correct Heisenberg equationsof motion have the same form as the classical equations of motion for systemshaving a classical analog.

Since the potential V is determined by the given charge distribution(4.8.11), it cannot be treated as an independent field variable. Therefore Vis not a field operator in the quantum theory. The vector potential as well asB, H, D, and E will be treated as operators in the quantum case.

We begin by taking as the hamiltonian

As noted above, the justification for this hamiltonian follows when we obtainfrom it the correct equations of motion. We may, however, identify the first


integral as the energy contained in the field in the absence of sources. Thesecond integral is the interaction energy between the field and current, andthe last term gives the Coulomb energy between the charges present.

We now postulate the following commutation relations based on the resultsin the source-free case (Section 4.6). In the Schrodinger picture we have

[Ak(r), = -MtlT(r - r') (4.9.2)

where dT is the transverse 8 function. We must show that these commutationrelations together with the hamiltonian (4.9.1) give the correct equations ofmotion.

We know that D and A are related by

Since V is a c-number, grad V is also, and so

[Ak(r), D,(r')] = -eSAk(r), A,(t') + | ^ (r\

(4.9.3)

, ^ r ' ) ] = - ihdklT(x - r'). (4.9.4)

Since grad V is a c-number, the commutation relations for the source-freefield and the field with sources are identical. We may therefore use the source-free expansions for A and A in the Schrodinger picture when sources arepresent. Therefore, from (4.6.1a) we have

As(r, 0) = i j - ^ - K\?to exp (ft,. r) + a), exp ( - f t , • r)], (4.9.5)

and we take for Ag(r, 0) the value —D8(r, 0)/€0 in (4.6.1b):

r. 0) = - i J J ^ L K\*u, exp (ft, • r) - a], exp ( - f t , . r)]. (4.9.6)

When sources are present, D is given by (4.9.3) and k8 is still given by(4.9.6).

Since the commutation relations (4.9.2) must be the same in the Heisenbergand Schrodinger pictures, we may now derive the Heisenberg equations ofmotion for A(r, i). We have

ih^=[Ak,H]. (4.9.7)

4.9 QUANTIZATION OF FIELD WITH CLASSICAL SOURCES 263

In (4.9.1), since p is a c-number, all components of A commute with theCoulomb term in H. Also, J is a c-number, and by (4.9.2) all components ofA commute with the interaction term in (4.9.1). Furthermore, Ak commuteswith all spatial derivatives of A and therefore commutes with the term(curl A)2 in (4.9.1). The only term with which Ak fails to commute is theterm involving (A)2 in (4.9.1). In Appendix G we show that

itxAk | ° (A)2,

(4.9.8)

from which we conclude that the longitudinal component of AL — 0 whilethe transverse part gives an identity. This is in agreement with the classicalresult (4.8.15).

The next Heisenberg equation of motion to be considered is

dt(4.9.9)

We see immediately that Ak commutes with all terms in the hamiltonian(4.9.1) except the curl A term and the J • A term. We leave as an exercise toshow that

and

), f — [curl' A(r')]2l = ihc*\J 2^ J

;(r),fj(r').A(r')dr'l = - -

(4.9.10)

(4.9.11)

All quantities are tacitly assumed to apply at the same time.When we put (4.9.10) and (4.9.11) into (4.9.9), we obtain (4.8.23), the

correct field equation. We have therefore justified the use of the hamiltonian(4.9.1).

If we substitute the field expansions (4.9.5) and (4.9.6) into the hamiltonian(4.9.1), we find that H becomes

2 2o),€0L

lu, • J(r) exp ( - ik, • r) 1 dr. (4.9.12)

The first term is the familiar source-free field energy, the next is the Coulombenergy, and the last is the interaction energy. The time dependence in H in

2 6 4 QUANTIZATION OF THE ELECTROMAGNETIC FIELD

(4.9.12) is contained in the explicit time dependence of p and / whereas theoperators atv and a\o are time-independent in the Schrodinger picture.

4.10 DENSITY OPERATOR FOR RADIATION FIELD

We use the principle of entropy maximization introduced in Chapter 3 todetermine the density operator which describes a mode of the radiation fieldwhen we have made various measurements. Let us assume we measure theaverage energy and the average electric and magnetic field of a mode. Wetherefore have

S= -kTrpXnp (4.10.1)

Tr p = 1 (4.10.2)

<H> = Tr pH = hco T r pa1 a (4.10.3)

*f - a) (4.10.4)

if + a). (4.10.5)

If we maximize the entropy subject to the measured values, we have on usingLagrange multipliers

Tr (1 + In p + Xt + 0H — X*p - X#) dp = 0. (4.10.6)

This will be satisfied by„ -—(l+Ai)-—pH+Xt»+X3t (A 1ft 7\

When we use (4.10.2), we determine the normalizing constant exp — (1 + XJso that

e-fiH+Xit>+Xt<i(4.10.8)

Next we express H, p, and q in terms of a and a t and let

X = fihco

(4.10.9)

Then we may rewrite p as

j f

(4.10.10)

4.10 DENSITY OPERATOR FOR RADIATION FIELD 265

We must next determine the Lagrange multipliers X, w, and w* in terms ofthe measured ensemble averages (E), (p), and {q). For this purpose, let usintroduce the new operators

c = a — w

ct = flt-w*, (4.10.11)

where w and w* are c-numbers. We see that [c, c*] = 1 so we may establish aset of basis vectors

c*c\n)e = n\n)e

c\n)e = 4n\n- l ) e (4.10.12)

and use these to evaluate the trace in (4.10.10). Thus

P = e Vt = C1 - «~V

(see Section 3.6). Also, we have

(fa = (c* -f- w*)(c + w),so that

while

(H) = Hco(afa) = hco Tr {[c*c + cw* + c*w + \wf]}

(p) = / f™ Tr fl(cf - c + w* - w) = i !™ (w* - w)V_2 V 2

(q) = LA Tr p^ + c + w* + w) = UL (w* + w),V2«) V2a)

(4.10.13)

(4.10.14)

(4.10.15)

(4.10.16)

2w

which follow since

Tr e~XeUc = 0 = Tr e-^W (4.10.17)

Thus w and w* are directly expressible in terms of the (ensemble) average ofthe electric and magnetic field and X is expressible in terms of the averageenergy.

Let us consider first the case in which the cavity is filled with thermalradiation only in equilibrium with the walls at temperature T. Then theaverage electric and magnetic fields are zero:

and

<£) =

= 0 ->• w = w* = 0,

hco hco

(4.10.18a)

(4.10.18b)

2gg QUANTIZATION OF THE ELECTROMAGNETIC FIELD

so that A = Ha>/kT. Then

p = (1 - e-x)e-**Xtt - (1 - e"A)9l{-(l - e~A)a*a} (4.10.19)

corresponds to pure noise. We used (3.3.24). As T-*- 0, (1 — er*) -> 1 andby (3.3.34) we see that

p—>|0>(0|. (4.10.20)

That is, when the cavity walls go to zero temperature, p approaches the purevacuum state with zero entropy.

If we again use (3.3.24) for c*c, we see that

p = (1 - e"A)5R{exp - (1 - e"A)(a* - w*)(a - w)}, (4.10.21)

for the case in which (p) and (q) are not zero. If we have signal plus noise:

fico . „ ho>JKo/kT _ ^signal |w|2. (4.10.22)

and (4.10.21) describes signal plus noise. If we again let T-+ 0 and use(3.3.42), we see that (4.10.21) becomes

X—-to(4.10.23)

which is a pure noiseless coherent state with zero entropy.For the general signal plus noise density operator the entropy is in the

cVrepresentation

S = -k Tr p In /> = - fe ( l - e~l) % e-Xn[-Xn + In (1 - e~%o

or

(4.10.24)

where

Hco= <£) -

(4-10.25)

Let us next consider a case in which the signal w = \w\ei<p has a randomphase. We may use (3.3.24) to write the density operator as

(4.10.26)

PROBLEMS

where e = 1 — g-*. Then, we have

p = €e-*

/! m-o m!

If we average over phases, since

then

±lit Jo 'im*

P =

If we again use (3.3.24) this becomes

267

(4.10.27)

(4.10.28)

i=o /! JTl y/nwhich cannot be summed. In case X -*• oo and e -+ 1, this becomes

(4.10.29)

i1=0 / !

(4.10.30)

which describes a system of signals with random phase and zero thermalnoise. In this case the entropy is not zero since it does not correspond toa pure state. This follows, since

Note that p represents a Poisson distribution over the eigenstates \n) where*\) = n\n).

PROBLEMS

4.1 Show that, when no sources are present, in the Coulomb gauge we may deter-mine the fields entirely from the vector potential.

4.2 Using (4.3.14) and the boundary conditions (4.3.16), show that the normalmodes in a cavity are orthogonal. Discuss the case in which modes are de-generate.


4.3 Verify (4.6.20b).4 4 Find the commutator [/^(r, /) , Bv(i, t )], where t * t.4.5 Calculate [VB(z,t\ 9IH(z',0/&'], where VH aaAIB are given by (4.2.14).4.6 Using the notation of (4.7.1) and (4.7.2), find S^D2}, where 91 is the normal-

ordering operator. Evaluate <0|3t{D*}|0>, where |0> is the field vacuum state.Compare this result with (4.7.12).

4.7 Carry out the commutation relations indicated to verify (4.9.10) and (4.9.11).

REFERENCES

[1] P. A. M. Dirac, Proc. Roy. Soc. (London), Ser. A, 114, 243 (1927).[2] E. Fermi, Rev. Mod. Phys., 4, 87 (1932).[3] W. Heiller, The Quantum Theory of Radiation, 2nd ed., Fair Lawn, N.J.: Oxford

University Press, 1944, Chaps. 1 and 2.[4] J. M. Jauch and K. M. Watson, Phys. Rev., 74, 950 (1948); 74,1485 (1948); and 75,

1249 (1949).[51 S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, New York:

Harper & Row, 1961.I6J A. I. Akhiezer and V. B. Berestetsky, "Quantum Electrodynamics," U.S. At. Energy

Comm. Transl. 2876.[7] The treatment given here was motivated by lecture notes of D. Walecka of Stanford

University.

Interaction of Radiationwith Matter

In Dirac's theory of radiation, he considers an atom and the radiation fieldwith which it interacts as a single system whose energy is represented by (a)the energy of the atom alone, (b) the energy of the radiation field alone, and(c) a small term equal to the coupling energy between the atom and the field.

The interaction term is obviously necessary if the atom and field are toaffect one another. A very simple "model" due to Fermi [1] will illustrate theinteraction. We consider a pendulum of resonant frequency <o0, whichcorresponds to the atom, and a vibrating string of resonant frequency (olt

which corresponds to the radiation field. When they are uncoupled, theyvibrate independently, and the energy is the sum of the energy of the pendu-lum and the energy of the string. If we connect the two by a small masslesselastic thread, we may transfer energy between the two systems. If, at t = 0,the string is vibrating and the pendulum is at rest, after a time t some of theenergy will be transferred to the pendulum. If the two frequencies are thesame, complete transfer of energy can take place. This corresponds toabsorption of radiation by an atom.

In the reverse process, in which the pendulum is initially excited and thestring at rest, if the frequencies of the string and the pendulum are the same,the transfer of energy to the string corresponds to emission of radiation by anatom.

In Section 5.1 we present the nonrelativistic hamiltonian for a one-electronatom in the presence of a radiation field, in line with the mechanical modeldiscussed above. The solution of the Schrodinger equation in this case isimpossible and we must resort to time-dependent perturbation theory toobtain approximate solutions. In Section 5.2 we use perturbation theory toexplain absorption and emission of radiation by an atom. In Section 5.3 wepresent the Wigner-Weisskopff theory of natural line width and obtain theLamb shift. In the following four sections we obtain the Kramers-Heisenberg

269

270 INTERACTION OF RADIATION WITH MATTER

scattering cross section and study the special cases of Rayleigh, Thomson,and Raman scattering. In Section 5.8 we discuss resonance fluorescence.

The Doppler effect, which is the change in frequency of light emitted froma moving source, is very simply explained by the wave theory of light. InSection 5.9 we show that in the quantum theory the Doppler effect is explainedfrom the conservation of energy and momentum of the emitting atom andemitted photon. In Section 5.10 we give the quantum explanation of a typicalwave-like phenomenon such as the propagation of light in a vacuum.

The semiclassical theory of a two-level spin-resonance experiment presentedin Section 5.11, followed by a short account of the effect of collision broaden-ing on the line width of the resonance in Section 5.12. The spin-resonanceexperiment is again discussed in the final section where we quantize the radia-tion field.

5.1 HAMILTONIAN OF AN ATOM IN A RADIATION FIELD

According to Dirac, the energy of an atom interacting with a radiation fieldis considered a single system. For simplicity, the atom is assumed to have asingle electron of charge e and mass m in a potential V(r), where r is theposition of the electron. The electron momentum is p. The electron spin isneglected, and in this book we are not concerned with energies sufficientlylarge that relativistic effects need be considered.

The radiation field may be described by the vector potential A(r, /) in theCoulomb gauge, div A = 0. For simplicity, the source of the radiation field(charges and currents) is not considered.

The nonrelativistic hamiltonian for the atom and radiation field in mksunits is [compare with (2.10.4)]

H = — (p - eA)2 + eV(r) + Hr2m

(5.1.1)

where HT is the energy of the radiation field in the absence of the atom. Sincediv A = 0, we may write (5.1.1) as

2m-m

The first two terms give the energy of the free atom, namely,

tf. Pi2m

+ eV(r).

(5.1.2)

(5.1.3)

The term Hr is the energy of the quantized source-free radiation field in theabsence of the atom,

H - 2 Hmâla,. + *), (5.1.4)

5.2 ABSORPTION AND EMISSION OF RADIATION BY AN ATOM 271

which is familiar from Chapter 4. We combine them into Ho, defined by

H0 = Ha + Hr, (5.1.5)

as the unperturbed hamiltonian.The next term in (5.1.2), which is of first order in the coupling constant, e,

we call

e_m

(5.1.6)

This term is small compared with Ha and Hr and represents the interactionbetween the electron momentum p and radiation field A. It is large comparedwith the last term of order e2 in (5.1.2)

= £- A2,2m

(5.1.7)

which represents the energy of mutual interaction between different radiationoscillators of the radiation field through the coupling of the electron to thefield.

The hamiltonian is given in the Schrodinger picture so that we must use theexpansion for the vector potential in Hx and Hz in the Schrodinger picturegiven by (4.6.1a), namely,

, exp (ik,. r) + a\a exp (—ik, • r)], (5.1.8)

where aXa and a\a obey the boson commutator relations and we assume thatthe atom and field are contained in a cubic cavity of volume r.

The hamiltonian (5.1.2), together with the Schrddinger equation of motion

(5.1.9)

and the vector potential (5.1.8) constitute the entire formulation of the Diractheory of radiation for the interaction of a one-electron atom with an electro-magnetic field that has been quantized in the nonrelativistic limit. There isleft only the problem of solving (5.1.9); this cannot be done exactly, and soperturbation calculations must be used.

5.2 ABSORPTION AND EMISSION OF RADIATION BY ANATOM

We use the perturbation theory results of Section 1.21 to show how thetheory explains the absorption and emission of a quantum of energy from theradiation field.


We identify system A in Section 1.21 with the atom and system B with theradiation field. The unperturbed hamiltonian is given by (5.1.5). The freeatom satisfies the eigenvalue problem

Ha\s) - e,|5), (5.2.1)

where the atomic state vectors are orthogonal and complete for the descriptionof the atom. The number operator of a radiation oscillator of polarizationita, momentum /2k, and energy hwl satisfies the eigenvalue equation

t i i I 1 ( 5 2 2 )

These are orthogonal and complete for describing the field. An energy eigen-state for the free atom and field is

|s; nlVl, nlta > = |s> TT I » O = \s; {«,„}>, (5.2.3)

with energy e, + ]£ hco^^.

The interaction energy between the atom and field in the SP by (5.1.6) and(5.1.8) is

IT 8 e

Var = - —e (5.2.4)" r <k l" r

ZJ 73iaue

If we take matrix elements between atomic states \s) and \s') of both sides, weshall need

*" - (5.2.5)

where we have expressed this in the coordinate representation and

Wto - Ms')- (5.2.6)These atomic wave functions vanish when r is greater than about 10~8 cm,the approximate atomic diameter. The wave vector kx = 2TT/A ~ 105 cm"1 atoptical wavelengths. Therefore, in the range of integration over which th^wave functions are nonzero exp ± iktr ~ exp ±1 x 10~s pa 1. Therefore, inthe dipole approximation, we let

(sle^'pls') ^ <s|p|s'), (5.2.7)

so that from (5.2.4) we obtain

mWe next show that

— <s|p|s') = i(m

(5.2.8)

(5.2.9a)

5.2 ABSORPTION AND EMISSION OF RADIATION BY AN ATOM

where

273

|i*, = ex,,, B e(s\x\s') = e(s'\x\s)* (5.2.9b)

is the atomic dipole matrix element between states \s) and \s') and

Hco,,. — €, — €,. = —H<ot.,. (5.2.9c)To show this, we write the Heisenberg equation of motion for xt{i = 1,

2, 3) for the free atom. It is

dt ^ ifi ihl 2m

Since xt commutes with V(x) and

we have

(5-2.10)

(5.2.11)

~ = = ~ lxi> "a\- (5.Z.12)dt m in

If we take the s, 5' matrix element of both sides of this equation and use(5.2.1) and its adjoint, (5.2.9) follows.

Therefore, if we use (5.2.9), (5.2.8) becomes

If we let/^V2€0«lL

3

= exp - ;n

(5.2.13)

(5.2.14)

then it easily follows thatI.a

(5.2.15)The probability of finding the atom in state s at time t given that it was in

state s' at time 0 by (1.21.27) and the above is

|ciu(a, t\S; 0)|2

( « y w x ^ rt) r (f 0)i_ 2 («»• ywx^ - rt.) rdh (fdt2 « « Tr M

0 L 3 I'.*' J(o,(ov J° *'0I."

(5.2.16)

where we have used the commutation relation [au,


As yet we have not specified the state of the field at t = 0 when the inter-action was turned on. We consider several cases.

CASE1 MONOCHROMATIC PLANE POLARIZED FIELD

We assume that we have measured the energy in each mode of the radiation fieldand found that only one mode was excited so that all others are in the vacuum state.We recall that a mode is specified completely by giving its wave vectors k, and itspolarization a. Its frequency w, = c|k,| is then determined. The density operator isthen symbolically written as

Pr(°) = l*V.><"l*r.l X

In this case the traces are easily evaluated since

(5.2.17)

(5.2.18)

and the a2 and an terms all vanish. Then (5.2.16) reduces to

icli)(s tw o)i* - m"' y '*'*' t*M'1*

, 4 sin2 | (a>M , + co,)f 4 sin2 \{<o,,, ~* * ) + » V « *X ((1

(„,„, _(5.2.19)

where the sum over /, a, I', a' has reduced to the sum over /, a due to the Kronecker<5's.

AbsorptionAssume at / = 0 the atom is in its ground state \g) with energy e, and we

ask for the probability that at time t it is found in an excited state \e) withenergy ee > eg. Then

ha>s, = " % „ = €e - et > 0 (5.2.2Q)

Since all <o, > 0, then ifo)eg g£ tt),0, (5.2.21)

the last (sine)2 term in (5.2.19) will be much larger than the first and we have

( 5 2 2 2 )

The transition probability per second is

d (11 » oi.n 2 sin (cut0 - (oet)t. (5.2.23)


We note that

275

since (a) if x = 0:<-»to irx

,. ,. sin xt tcosxthm hm = hm hm = oot—mx-*0 1TX *-»« «-»0 Tf

and (b) if x yi 0:.. sin a;hm*-»» itX

0,

because of the oscillations of the sine, and (c)

5HL£f da. = ! =

—oo WX •/—o

Therefore, as t -*• oo, (5.2.23) becomes

- o>ea).

(5.2.24)

(5.2.25)

(5.2.26)

(5.2.27)

(5.2.28)

This gives the probability per second for the absorption of a photon by theatom when the field is perfectly monochromatic and plane polarized. We haveassumed that the atomic energy levels have no linewidth (no damping) so thatonly if eo,o = o>et (energy conservation) will the atom be able to absorb aphoton and be excited. The process is shown schematically in Figure 5.1.If nltgo = 0, the atom will not be excited. The probability is linearly pro-portional to (oet = to,o. Obviously this is an unrealistic case since it is im-possible to have a completely monochromatic wave. In addition, the readershould note that for the case we have considered in which we measured theenergy precisely of the mode, the average electric fields will be zero.

Emission

Consider next the case in which at t = 0, the atom is in the excited state\e) and let us determine the probability of finding it in its ground state \g) at

tal

Figure S.I Schematic representation of a one-photon absorption by an atom, (a) Initialstate; (b) final state.


time t. Thenno,,, = - /»«„ , (5.2.29)

and the possibility now exists that the first (sine)2 term in (5.2.19) can belarge. There are two different terms. Suppose that «,o<To = 0, that is, that theatom is excited and no radiation field is present. Then (5.2.19) becomes

4 sin8

to. (eo, -. (5.2.30)

This gives the probability of finding the atom in the ground state at time *given that it was in its excited state at t — 0 even when no radiation wasinitially present. That is, the atom may decay spontaneously with the emissionof radiation. This probability is very small unless the emitted radiationsatisfies energy conservation

c , ^ «>„. (5.2.31)

It may be emitted in all directions and in all polarizations, however, so thatwe must sum over all modes and both polarizations. By (4.5.4) and (4.5.10),we replace the sum over / for a single polarization by

so that

- x 2 <otd<o:)8 <r=i Jo2heo(2ncf

Now if we carry out the sum on polarizations, we have

I Vu ' {*J* = \V-S (cos8 a + cos8 iff),

(5.2.33)

(5.2.34)

where a is the angle between y.eir and eh and /? the angle between p.,, andih. If 8 is the angle between |*w and k,, then by the law of cosines,

2 l*i. * P * I* = ll* J * 0 ~ c o s * 0 ) (5.2.35)

(see Figure 5.2). If we take our z-axis for the angular integration in (5.2.33) tobe along jiM, then

dO. = sin 6 dd d<p, (5.2.36)and

F'd<(, ^ 0, we mayextend the lower limit to — oo, and since

£00 sin2 ixt , _ irt

X~~2~'

the probability per second for spontaneous emission is

^ ... 3 I . . |2spon ~ T lca

„ AM2

«» 0)1 =

(5.2.39)

(5.2.40)

Note that the cavity volume, L*, has canceled and that the probability isproportional to coet

3.The remaining term proportional to nlo<,o in (5.2.19) corresponds to induced

emission. It vanishes unless radiation is initially present. In this case we have

K%, t\e, 0)|2 • « •, (5.2.41)2eohL* <olt

which is identical to the absorption term (5.2.22) so the probability persecond for the atom to decay with the induced emission of a photon by(5.2.28) is

,,ind IT ((5-2.42)

It is induced to emit into the identical mode from which it was able to absorband with the same polarization. This case is shown schematically in Figure5.3.


leX

E, s< • n / t w / E,= <t*{n/-\)k<*J

la) ' <*>' '

Figure 5.3 Schematic representation of a one-photon emission by an atom in an excitedstate, (a) Initial state; (b) final state.

As we have noted it is impossible to obtain a perfectly monochromaticplane polarized wave. Let us consider the following slightly more realisticcase.

CASE 2 PLANE POLARIZED WAVES WITH FREQUENCYSPREAD

Let us assume that the electromagnetic field has a range of frequencies so thatmany modes are excited at t — 0. If we measure the energy of each mode, the den-sity operator is given by

I T K><"i»l- <5-2-43>1

The traces in (5.2.16) may be easily evaluated and we may do the time integrals.This yields

\ c < » ( s t \ s ' 0 ) 1 * ^ 1 ^ 1 '

4sin*|(«>,-«w)

Again we consider absorption and emission separately.

Absorption

At / — 0 the atom is in the ground state \s') = \g) and at time r we ask forthe probability it is in an excited state \s) = \e). Since co,,. = coee, only thelast term in (5.2.44) may be large when

«), as (oet, (5.2.45)and we haye

| c i M f , 0)| „, 2 -2he0n ..• ft),

In contrast to the single mode case in which

(co, —. (5.2.46)

(5.2.47)


many modes are excited. We assume that they are closely spaced in frequencyso that we may again use (4.5.4) and (4.5.10) to convert the sum over / to anintegral. For a single polarization

}. (5.2.48)


2 rcf ff=i Jo

f(eo, - )w)

(5.2.49)

where we have let nu -*• na(colt Q). That is, a given mode is completely speci-fied by giving three integers that fix k, and a polarization a. The frequencya>, — c |k,| is then determined for these plane wave modes. In polar coordi-nates then, nlo is a function of co, and the two polar angles are (0, <p) and a.Our original measurements on the field must determine this dependencecompletely.

We are assuming here that the atomic energy levels are infinitely sharp sothat as t increases, the sine2 term is very highly peaked at co, = coeg. We maythen let the lower limit on the <Oi integral go to — oo and remove the slowlyvarying factors. This gives the absorption probability per second as

, .- v-, .o, -,. - . ._ -.3 ^» i "»•** l iff *{*•«»! nôigg, ii).at 2«€ 0 (2TTC) a=i J

(5.2.50)

If the radiation field has a Lorentzian spectrum with line width y centeredat co0, we have

(y/2), Q) = -°

T (coe, - ft)0)8 (5.2.51)

This will be largest when coe, ^ (o0. If the radiation is polarized and we acceptonly a certain small solid angle AQ, then (5.2.50) reduces to

w:Pol

c I %v• ne0

(5.2.52)

where a is the angle between ftei7 and the polarization direction, en. If inaddition our atoms are in a gaseous state, (j.e, will be randomly oriented and


we should average over all directions of \ttg with respect to en. This gives-

cos2 acos2 a sin a da

A-nso that

p o l _

(5.2.53)

(5.2.54)

When the light is unpolarized, na(a>,, Q) is independent of a; that is,we have equal intensities for both polarizations. If we again only accept asmall solid angle AQ, we find

l2 (cos2« + cos2 /?) AO, (5.2.55)

where a and § are the angles between JAW and e u and £w, respectively. For agas, we again average over all directions for \ttt. Since

cos* a

we have

+ cos* p = — f *'d<p f'sin3 6 d6 = -, (5.2.56)Ait Jo Jo 3

— , (5.2.57)

which is twice as large as the unpolarized case as expected.

Emission

In case the atom is in state \e) at t — 0 and state g at time t, the spontaneousemission term in (5.2.44) is exactly the same as it was in Case 1. The inducedemission term becomes identical to the absorption and we have

wet "It(5.2.58)

That is, the probability per second for induced emission and absorption is thesame. Comparison of (5.2.57) with (5.2.40) shows that

-unpolveg (5-2.59)

CASE 3 BLACKBODY (THERMAL) RADIATION

Let us next consider the case in which the radiation in the cavity is in thermalequilibrium with the walls at temperature T. The density operator for the field at


t = 0 is then given by the Boltzmann distribution

where

IT 0 ~ e-Î.a

_hml

281

(5.2.60)

(5.2.61)

We may use the results of Section 3.6 to evaluate the traces in (5.2.16). From (3.6.9)and (3.6.15), we easily see that

wherent = [exp (A,) -

(5.2.62)

(5.2.63)

The others all vanish. This frequency dependence should be contrasted with theLorentzian shape discussed earlier. The reader might also note that the averageelectric field is again zero in this case as it was in the two previous cases we haveconsidered. In contrast to the two prior cases, we have not determined the energyof each mode of the radiation field. Rather the energy in each mode is distributedover the energy eigenstates with a Boltzmann distribution as seen by (3.6.11).

When we use these traces and carry out the time integrals, (5.2.16) becomes

This differs from (5.2.44) in that nl(r is replaced by «„ independent of a. The black-body radiation filling the cavity is unpolarized and isotropic.

Absorption

In this case (5.2.64) becomes as in the former cases

t ( 5 . 2 . 6 5 )

where we must integrate over all solid angles since the radiation is isotropic.Again we see the (sine)2 term is highly peaked so that on integrating we obtainthe absorption probability per second

(5.2.66)


We next calculate the energy density of the radiation in the cavity. By(4.S.10) the number of modes per unit volume of both polarizations withfrequency lying between to, and a>( + d<ol in all directions is

The average energy per mode is

1( 5 - 2 - 6 8 )

If we let MKJ dvt be the average energy per unit volume lying between v, andv, + dv,, then

«r, dv, ^hv*n(v,)dv,, (5.2.69)

which is the Planck radiation formula. Since 2irv, s= «>«,» we may rewrite(5.2.66) as

where Bee is called the Einstein coefficient for absorption. Einstein postulatedthat the transition probability per second for absorption was proportionalto u, where Btt was independent of frequency and T. We now have an explicitexpression which depends only on the dipole moment matrix elements for theatom.

Emission

The spontaneous emission term is identical with the prior cases so that

w s p o

3irheoc3

which is Einstein's coefficient for spontaneous emission.A similar analysis to that above shows that for induced emission

so thatW l n d a s M?al>8 — B U = B U"at — "eg — Deguv — Dgeuv>

(5.2.71)

(5:2.72)

(5.2.73,

Bgt- « • " c' • ( 5 ' 2 - 7 4 )

Einstein was able to derive the Planck radiation formula without the useof quantum mechanics. It may be of interest to give his argument.

From (5.2.70) and (5.2.71) we easily see that

2h<ot Ztrhv,et


Suppose we have many identical atoms in a cavity filled with electromag-netic radiation in thermal equilibrium with the walls at temperature T. Forsimplicity, let us consider only the two lowest energy levels et > ce. If anatom is in level e, it will spontaneously decay to level g and emit a quantum ofenergy with a probability A9e independent of whether there is radiation in thecavity. In addition, if the atom is in state e there is a probability Bttu, that theatom will be induced to decay and emit a quantum due to the presence of theelectromagnetic field. This probability is obviously proportional to the den-sity of radiation uv present in the cavity. Also, if an atom is in state g, thelower state, it will have a probability Btjuv to absorb a quantum of energy andbe excited to state e.

The atoms are in thermal equilbrium at temperature T and the probabilitythat an atom is in state g is proportional to the Boltzmann factor exp — (c,/kT) while the probability it is in state e is proportional to exp —{zJkT). Inthermal equilibrium, we must have

(A,e + Berur)e-'/kT = BW*'**. (5.2.75)

That is, the spontaneous plus induced emission probability per second timesthe probability the atom is in the upper state e equals the probability ofabsorption per second times the probability the atom is in the lower state g.The A and B coefficients are determined by the properties of the atom and areindependent of the frequency and temperature. As the temperature of thecavity walls gets very large, the radiation density will get large so that

Betut» Aeg,

and it then follows from (5.2.75) as 7*-*- oo that

(5.2.76)

(5.2.77)

Since these probabilities are independent of the radiation frequency and thetemperature, we see that the probability for induced emission of radiationper second equals the probability per second for absorption. If we use thisresult and solve (5.2.75) for u, we obtain

A\B

where we have letexp {hvjkT) - 1

hv = ee - et > 0,

(5.2.78)

and have omitted the subscripts on the Einstein A and B coefficients. Equa-tion (5.2.78) is the Planck radiation formula. We obtain J4 and B from a correctquantum treatment as we have shown. Note that if there were no spontaneous


emission, A -*• 0 and Planck's formula would not follow in violation of ex-perimental results. Therefore, spontaneous emission is necessary to explainblackbody radiation and this necessitates a quantum treatment of the radia-tion field.

CASE 4 COHERENT STATE

We consider next the case in which a single mode of the radiation field is excited ina coherent state. This will, of course, be another case of a purely monochromaticwave with zero line width, but a laser output in single mode operation can approxi-mate this situation due to its high degree of monochromaticity (small linewidth)and high degree of collimation. It may also be plane polarized. Thus

where(5.2.79)

(5.2.80)

In this case the traces of all terms exist in (5.2.16). When we evaluate them and carryout the time integrals we obtain

i t\s' 0)1* -*, t\s , 0)| -

_

Re«?*0 +«-"»«*_<!-»•<'>.< 2 cos <o,,t)

If we let

«l<r = (5.2.82)

and differentiate the above with respect to t, we obtain

<- y? \*u, • V-.A* (sin (a>,w

sin (w; -

5| <5 f s i n [(">««' sin

2o>jsin2(«,/ —7 i-—£-* |) 1\

- «),)J/ (5.2.83)

The reader will recognize the spontaneous emission term which is just as it hasappeared in all prior cases. When the atom is in state \g> at / = 0 and |e> at /, wehave the following result for absorption.

5.3 WIGNER-WEISSKOPF THEORY OF NATURAL LINEWIDTH [2]; 285

Absorption

X

•

z \e • u. I2

51? to,

Jsin (ait + coeil)

\ «I + <»«,

sin [(o>et + co,)

.2la\

sin (cot

Oii-

21]

- (oeg)t sin [(cow

2co, sin 2(co,r — f)

- cot)t + 2g]

,t + Oil

-1 a o iO'eff — <°l Vêff + «>lA«>«» — Oil)'

As f -»• oo, the first term becomes w d(co, 4-co „) . But coeg > 0 and co, > 0 soto, + tuw £ 0 and we neglect it. The second becomes IT J(a>, — (oet) which islarge when cot = o>et. In this case the third term reduces to sin 2f/2cow, inde-pendent of r and is negligible compared with w d(w, — coeg). The fourth andfifth terms exactly cancel when cot -*• ateg, for all t. Thus

•nco.(5.2.85)

which is identical with (5.2.28) except the "exact" number of photons in theexcited mode nla has been replaced by the mean number (fl.la\a\<fillt\xla) =|a J 2 . In the former case there were no interference terms like the last three in(5.2.84). Why?

Emission

A similar argument shows that for induced emission,

...emiss _ (5.2.86)

5.3 WIGNER-WEISSKOPF THEORY OF NATURALLINEWIDTH DO; LAMB SHIFT

In the theory of emission and absorption of radiation by an atom presentedin the previous section, we assumed that the atomic energy levels were in-finitely sharp whereas we know from experiment that the observed emissionand absorption lines have a finite width. There are many interactions whichmay broaden an atomic line, but the most fundamental one is the reaction ofthe radiation field on the atom. That is, when an atom decays spontaneouslyfrom an excited state radiatively, it emits a quantum of energy into theradiation field. This radiation may be reabsorbed by the atom. The reaction ofthe field on the atom gives the atom a linewidth and causes the original levelto be shifted as we show. This is the source of the natural linewidth and theLamb shift. Spontaneous emission is also the source of quantum noise due tothe random times at which emission occurs.


If we believe the uncertainty relation

AEAt~h, (5.3.1)

it states that to measure the energy of the atom to an accuracy A£, a time oforder At is needed. An atom in an excited state decays spontaneously to a lowerstate with the emission of radiation. To make the measurement, we must waitsufficiently long to be sure the atom has decayed, namely, a time of the orderof the lifetime r of the excited state. Therefore, the energy is known only to anaccuracy

A E ~ A (5.3.2)T

which gives a linewidth. We shall show that r -1 is equal to the transitionprobability per second for spontaneous emission.

The time-dependent perturbation theory which led to Fermi's golden rule isvalid only for times sufficiently short that the atom does not change its statesignificantly. It is clearly not adequate to show the decay of the initial state.Accordingly, we must obtain more accurate solutions of the Schrddingerequation for an atom coupled to a radiation field.

We begin by assuming an atom is in some excited state |£> at / = 0 and thatthere is no radiation field present. The initial state is |£; 0) of energy eB.We let c(£, 0; t) be the probability amplitude for finding the atom in state Ewith no photons at time t. The interaction energy is in the SP

V = - — A-pm

Ifc(£,0;0)=l, (5.3.4a)

the atom may decay to some state |7) with the emission of a photon of mo-mentum Kkt and polarization £to. Att = 0

c(I, l to;0) = 0. (5.3.4b)

The nonzero matrix elements which connect the initial states to any other aregiven by

• nr ^.m V 2co(eo/J

!„>*. (5.3.5)


If we use the exact equations (1.16.59), we see that the probability amplitudessatisfy

ih (E, 0; r) = 2 2 V*.*i.iwe-«m>r~»\t!, lla; t)at l i.9

.dcdt

where we have let

ihjt(I, \la, t) = VI.ha.E*e-«'°*'-*>*tc{E, 0; 0,

a>BI =

(5.3.6)

(5.3.7)

(5.3.8)

and we must sum over all states |/> for which the matrix elements are nonzeroand over photons of all momenta and polarizations. It should be noted thatthe equations above are still exact. In first order perturbation theory wereplace c(7, lto, 0 and c(E, 0, 0 on the right side by their initial values, butwe must obtain better solutions now subject to the initial conditions. Un-fortunately, they cannot be solved exactly. If we integrate both sides of (5.3.7)and use (5.3.4), we obtain

c(I, l to; 0 = ^ </, h.W\E, 0) f *-«•«—>«c(JE, 0; t') dt'. (5.3.9)in Jo

We may substitute this into (5.3.6) to obtain the integro-differentialequation

^ (£, 0; t) = - fa 2 1 \VE.o;i.i. I2 fV(^-»""-nc(£, 0; t') dt'. (5.3.10)dt h i ,.<, l" Jo

From our original expansion for the state vector (1.16.29b), we have

, 0, 0) £* c(E, 0; ; 0). (5.3.11)

We would expect an approximate solution of the exact equation (5.3.10) tobehave as

c(E, 0; 0 s* e-HI1i)l"*\ (5.3.12a)where

(5.3.12b)

which would show that the atom decays from its excited state with a lifetimer £

- 1 and has a shift in its energy level due to its interaction with the manydegrees of the radiation field. To obtain an approximate solution of this form,let us take the Laplace transform [3] of both sides. If we let

c(s)Jo

'c(£, 0; /) dt, (5.3.13)


then with the initial condition (5.3.4a), it follows on integration by parts that

Jo dt ' ' ~Also we note that

(5.3.14)

\e-"dt I Vn<'-<(c(£,0; t')dt'Jo Jo

Jo

•fJo

,-u-aix= c(£, 0; *>"""" <**' I eJo Jv

'c(E,0.fyt><ft> c(s)s-iQ ~ s-iCi'

dt

(5.3.15)

where we interchanged the order of integration over t and /'. Thus if wemultiply both sides of (5.3.10) by er" dt and integrate from 0 to oo and usethe results above, we obtain after minor algebra

(5.3.16)+ i21hi i.<, (oEI —

The exact formal solution is given by [3]

is A(s)

< rt+iaoc(E, 0; 0 = - M e"c(s) ds, (5.3.17)

2iTl Jt-im

where the path of integration is a contour parallel to the imaginary s axis. Toevaluate, we must know the poles in the half plane Re (s) < 0 of c(s). If c(s)has a simple pole at

s = - - A€jB = - \ T E - i Aft>E,n

then (5.3.17) becomes

c(£, 0; 0 =

(5.3.18)

(5.3.19)

which is the desired form.We therefore make the Wigner-Weisskopf approximation to solve for the

zeros of A(s) in (5.3.16). We note that if the atom-field interaction is small, asa zeroth approximation A($) = 0 if s = 0. As a next approximation we lets -* 0 in the denominator of the sum in A(s). In other words, we let the polebe shifted approximately to

A(0) -s^ + -ft

Mm 7 2 2 I»-»0+ fr 1 l.a COEI — CO, + IS

= i r

(5.3.20)

5.3 WIGNER-WEISSKOPF THEORY OF NATURAL LINEWEDTH [2]; 289

Thus the W-W approximation consists in calculating the first order shift inthe simple pole of c(s) due to the atom-field interaction. Now [41

hm = hm — ; — — z, - o + x + is 5-o+ La;2 + s2 x2 + s2j

1x

since

lim+ ** + s2)

hm I

0 x/0oo x = 0

• dx = 1,

(5.3.21)

(5.3.22)

(5.3.23)ir(a;s + s')

which are the required properties of a ^-function. If we use (5.3.21), (5.3.20)becomes

- 2 2I l2 2

fi I l.a (OEI — CO,

(5.3.24)

~ *>d. (5.3.25)

We see that the atom is continually emitting and reabsorbing quanta ofradiation. The energy level shift does not require energy to be conserved whilethe damping requires energy conservation. Thus damping is brought about bythe emission and absorption of real photons while the photons emitted andabsorbed which contribute to the energy shift are called virtual photons.

We may calculate TE explicitly. For an atom emitting or absorbing radia-tion at optical frequencies, the atomic wave functions decay in a distance ofthe order of the atomic size, namely, ~ 1 0 - 8 cm. So we are justified in makingthe dipole approximation in (5.3.5) so that

a),). (5.3.26)r,-2=2 I -hi*si.c m2

If we use the results of Section 4.5 to convert the sum over / to an integral,we have

knCeÛ a i Jo Xlircl

By (5.2.35)-(5.2.37), we have

2 f dCa J

(5.3.28)

290

so that

INTERACTION OF RADIATION WITH MATTER

r v1 A' = 2,/ Aireohc

But by (5.2.9) this may be written as

73 Iftr/f _1 E = 2, 7ZsI*E 3nfieoc9

(5.3.29)

(5.3.30)

If we compare this with (5.2.40) we see that TE which is the inverse lifetimeof state \E) is just the sum of the probability per second for the atom to decayspontaneously to all lower states / which have nonzero dipole moment matrixelements. This is a very important result.

Before discussing the energy shift, let us obtain c(I, l,o; t). By (5.3.9) and(5.3.19) we have

c(/, lw; t) = (I, lla\V\E,f

n[(oEI — co, + Acu — ijl E

while the probability of finding the atom in state / with one photon is

. |K,,w £ ,0 |2 {1 + e~™ - 2e-*™cos (coEI - a),|c(J, ll(f; f)| =

(5.3.31)

(5.3.32)

which has a Lorentzian shape of half-width TE. If we sum this over all statesof the field, use (5.3.5), and convert the sum to an integral, we have

el \PEif C"

where we have let

c V

x >, — coEI — A t o ,

> ( 5 - 3 ' 3 3 )

(5.3.34)

and have used (5.3.28). Since the integral is strongly peaked at x = 0, we mayreplace the lower limit by — oo. Then since

(5.3.35)f • xdx _ _ I*00 xcosxtJ-» x2 + a2 J-c,

I00 cos xt dx _ IT _

:* + a2 ~ a6

5.3 WIGNER-WEISSKOPF THEORY OF NATURAL LINEWIDTH [2];

(5.3.33) becomes

291

, 01' - + Ac) [1 -1

(5.3.36)

Finally, if we sum over all states / and neglect Aa>, we have on using (5.3.30)

lfc,;OI2 — l ~e-T*\ (5.3.37)/ 1.9

That is, as t -*• oo, the atom has decayed from its original state to one of thepossible lower states.

We are now ready to return to the troublesome question of calculating theenergy level shift given in (5.3.24). When we use (5.3.5), convert the sum toan integral and use (5.3.28) we obtain

h Aft).6cse07r2m i:coEI —

(5.3.38)

which diverges linearly and we have assumed the pole was only shifted asmall amount. This is a standard problem in quantum electrodynamics andthe removal of this divergence is based on the concept of mass renormalization.To remove the divergence completely, we must use the Dirac relativistictheory of the electron. We do not do this here and only are able to make thedivergence logarithmic instead of linear which is an improvement.

We begin by arguing that contributions to the energy shift due to extremelyhigh energy photons cannot be important since the nonrelativistic approxi-mation for the electron must not be valid for photons of energy /ko, ~ me*.Therefore, we only integrate to some cofax so that

nax , 4co, dco, e z

ft), — a)EI 2m 3vmci

(5.3.39)

But this is very sensitive to the choice of wf**. In an effort to improve on this,let us try to understand the physical origin of the divergence. Let us consider afree electron with momentum p interacting with the radiation field. Initiallywe assume no radiation present. The interaction energy is

7 =m

A -p

(5-3-4°)


Then nonzero matrix elements are

(5.3.41)2eoa>,L3

which connect the initial state of momentum p and no photons to all otherstates. We may not make the dipole approximation in the present casebecause the electron eigenstates are plane waves and do not cut off like boundatomic states. Therefore, we have

(P'k"*''r|p) = "1 L^^'-'e-^'e1^''dx = <5[p'; p - Me,], (5.3.42)

since

(5.3.43)

But (5.3.42) is just a statement of momentum conservation for the photonand electron:

p' + Hkt = p (5.3.44)

Thus the nonzero matrix elements are

<P\m

)%' ;P - *kj- (5-3.45)

The equations for the probability amplitudes corresponding to (5.3.6) and(5.3.7) become

ih & (p, 0; 0 = 2 2 V9.WAjTl*''W, !,„; 0 (5.3.46)dt p' l.a

dtv', Ki't) - V...lta;,,oe

+i°f<tc(V,0; r)

where we have let

hoi.2m

(5.3.47)

(5.3.48)

We proceed to solve in the same way. The electron emits and reabsorbs


photons, and the energy shift is

h Acu = - - T T —:h v !.«• m2 2e0o),I?

- 1

~ >&i)2 ~ (l/2m)p2

In the nonrelativistic approximation

so that

— (p - Mct)2 - — p* + h(ot on Hco,,

2m 2m

. (5.3.49)

(5.3.50)

mcf Jo—I

\I 1\4ireahcJ 3ir(mc)

Y (5.3.51)

-Kj>\

where K still diverges linearly. The problem is that we can never have a "bare"electron in real life. It always carries along with it an electromagnetic fieldwhich contributes to its kinetic energy since it is proportional to p2. The fieldshows up as a continual emission and reabsorption of virtual photons. Thisis just the self energy of the electron. When we measure the kinetic energy of afree electron, we can never separate the field contribution so we actuallyobserve the kinetic energy

(5.3.52)

where JMbare is the mass of a fictitious bare electron without its associatedelectromagnetic field which we have so far been using in the theory. Wetherefore define a new observable mass by the relation

P2 = (1 - 2mbareX)p2

2 m o b g JLTI

i r t _ ^ 42mbareL 4ne0hc 3n mbuec*.

(5.3.53)

where we used (5.3.51). For h<oY**~mc*, since ea/47re0/ie~ 1/137, we seethat this is a small correction.

We now argue that we have solved the energy eigenvalue problem for anatom using the bare mass when in reality we should usemob8. The hamiltonian

294

for the free atom should be


zmobg

+ F(r) - Kp* = tf°Id - Kp*. (5.3.54)

Then we have for our new "correct" unperturbed energy levels

ld y l d 2 > . (5.3.55)When we recalculate the energy level shift, we must add back the term+K(E\pt\E) since it has already been accounted for. Thus by (5.3.39) and(5.3.51), the observed energy shift is

- o ,(5.3.56)

where to this degree of approximation the matrix elements are to be computedusing the unperturbed or "bare" energy eigenvectors and we use the "bare"mass. This follows because although K actually diverges if we take anhatf**- ~ mb&, K<£,\ and corrections to |pjj/|2 and the <oBI in the denomi-nator above would be of order K, and we are not keeping K* terms.

We simplify the integrand above as follows:

oiBl - toEl

If we use the completeness relation, we have

(5.3.57)

(5.3.58)

Thus, by these two, (5.3.56) becomes

—I 2<aBl\VElt L

,C) Jo I <»EI ~

2/i

But

so that

4rre0Hc 7>-tr{mbcfZ2*BI\PEI\*

<oBI,

(5.3.59)

(5.3.60)

T TT^T. 2 «» \9BI\* log (—) • (5.3.61),nc 3ir(jnbc) i \ 0)jB I

5.3 WIGNER-WEISSKOPF THEORY OF NATURAL L1NEWIDTH [2]; 295

If we replace <oIB in the log term by some average value, we may remove itfrom the sum and obtain

2hlog—-

4ire0fic 3ir(m6c) (O>IE) I

We may now evaluate the sum as follows. The atomic hamiltonian is

(5-3-62)

so that

Also

2m,+ V(r), (5.3.63)

(5.3.64)

[Po [Pi, Ha)} = -h* V2K, (5.3.65)

where we sum on i from 1 to 3. If we take the diagonal matrix elements of bothsides for state £ , we have on expanding the double commutator

{EftfriptH, - HaPi) - (PiHtt - HaPi)Pi]]E) = - ^ < £ | V ^ | £ ) , (5.3.66)or

2€B(EjPiPi\E) - 2{E\PiHaPi\E) = -H*(E\VW\E). (5.3.67)

If we insert the completeness relation after the first Pi in both terms, we have

2(€E — CT)(E\Pi\l)(I\Pi\E) — — — <£|V2K|£), (5.3.68)/ 2

or(5.3.69)

i 2As a specific example, let us consider the hydrogen atom. In this case,

where <5(r) = 8{x) 8{y) 8(z). Thus

<£|V2K|£) = - \dt \WE{

(5.3.70)

(5.3.71)

If one looks at the hydrogen wave functions [5], they all vanish at the originunless the quantum number / = 0, that is, for j-states. Any state with ahigher orbital angular momentum vanishes, so there is a Lamb shift only for5-states in this nonrelativistic theory. In this case

for s-states, (5.3.72)

296

where

Thus


ft/mc

me*(5.3.73)

2ire0n3

(5.3.74)

Now e*l%n€4a0 is just the ionization energy for an electron in the ls-state. Ifwe take hayf** = m<* and estimate h(toEl) = 17.Zi^fine^) for the 2J-state, Aft^ turns out to be 1040 Mc/sec in very good agreement with themeasured value of 1057 Mc/sec of Lamb and Retherford. Relativistic calcu-lations of Bethe give exact agreement when the Dirac relativistic wavefunctions are used.

5.4 KRAMERS-HEISENBERG SCATTERING CROSS-SECTION

We next consider the scattering of light by one-electron atoms. Initially weassume the atom is in some bound state \a) and the radiation field is in state\nit n,) with Rj photons of momentum Hkt and polarization St and nf photonsof momentum hkf and polarization ef. The initial atom-field eigenvector is\a; nt, nf) of energy ea + n^Wf + nfi(ot. After the scattering is over, theatom is left in state \b) and the field is in state \nf — 1; nf + 1) so that thefinal energy is «„ + nco^ — 1) + htof(nf + 1). We would like to calculatethe probability for such a scattering process to occur and obtain the differ-ential scattering cross-section.

The atom and field are described by the hamiltonian (5.1.2)

H2,

where

m

"-£*•

(5.4.1)

(5.4.2)

(5.4.3)

If we make the dipole approximation which is valid at optical frequencies and

5.4 KRAMERS-HEISENBERG SCATTERING CROSS-SECTION

let exp ±ik , • r s* 1, and use (5.1.8), we have that

e ^ I h

32m2e0Z? i.

297

(5.4.4)

(5-4.5)

We wish to see if Hx and Hz can cause the atom-field system to make transi-tions between the initial state \a; »*, nf) and the final state \b; nt — \,nf+ 1).In first-order perturbation theory, by Problem 1.15, such transitions will notbe induced to occur, since

.ay 2ft),e0L

(5.4.6)

That is, the first-order matrix elements vanish unless the net number ofphotons changes by unity. Here we lose one photon and gain another with nonet change. [Note that we are interested only in photons (ko £,•) and (kr, ef).]We must therefore see if such a scattering process can occur in second order.According to Problem 1.15, we therefore need the nonzero matrix elements

(f\Hz\i) = 7 ^4meolj i.

n, - 1, nf

>- (5-4-7)

(n,- - 1, nf <, nf) =

The last two terms are clearly zero since they either annihilate or create twophotons, a process we are not considering. However, for the first term we have

( IZT7Z. i V\ ;e v~> *• ' •• -

y/n^fif + 1 / H i a = j , i a = i

0 (otherwise).(5.4.8a)

That is, we may first create the final photon and then annihilate the initialone. For the second term, we have

<nt- - 1, nf + l\alaa\.a.\nt, nf) -

If we use these, (5.4.7) becomes

2m€0L3

.0 (otherwise).(5.4.8b)

(5.4.9)


which vanishes if the atom does not return to its original state after thescattering. Also if no photons of frequency o>{ are present, then nt = 0, andthis term vanishes. By Problem 1.15, the contribution to the probabilityamplitude of Ht is

whereofi

hcOf (5.4.11)

We next consider the contribution of # i to the transition probabilityamplitude in second order. According to Problem 1.15 we need the nonzeromatrix elements (f\Hi\k)(k\Hj\i) where \k) is any intermediate atom-fieldstate | / , H ; . , H £ ) for which the individual matrix elements do not vanish.Under the dipole approximation, we have

2m*€0L8 i.* vc

where

X = (n{- 1, n , + l | (a t o + al)\n'{, n',)(n't, n't\{aVe. + aÔI", . «/)• (5.4.12b)

These are zero except for two cases \nt — 1, «r> and !«<, n, + 1>. Then for thefirst case, the only nonzero term is

- Vn<(«/ + 1) i f - / . a s / and /', a' = i. (5.4.13a)

In the second case, the only nonzero term is

- y/n^n, + 1) if Uasi and T . a ' s / . (5.4.13b)

Therefore, we have

e*h lntnf+1)1(9*.V l if ft n,

We have used the notation

(5.4.14)

(5.4.15)

5.4 KRAMERS-HEISENBERG SCATTERING CROSS-SECTION 299

By Problem 1.15, the contribution of Hx to the transition probability ampli-tude is

f, t/i, 0) = - + 1)

2 ((Pw • */XP/«' «<) I rf«i ««*-«-**W»« f ftA,7 I Jo Jo

(Pw ' Jo Jo\J

(5.4.16)When we summed over the two intermediate photon states when \k) =\I> ni — 15 if), then H(ofk = c4 — €z + /ico/ and ^«ow = ez — ea — Aw, andwhen \k) = |/, M,., nf + 1), then /ia)/jfc = «„ — €; — ^cof and ho>ki = € / —

We next carry out the integration over t2. This gives

^ f(g« • Vbl)(ef • Via) ['ji^

' ^ Cj — ea + h(Of Jo

g/ • Ptf)(g» * P/a) |^ r i | - c t» /€j — eo — Att), Jo

Jjj i.j.'i.i/j

where <ofi is given by (5.4.11). The second term in each integral arises fromh = 0 and is a transient effect due to turning on the interaction suddenly att = 0. It will give a small contribution to the transition probability, since, forexample,

4 sin2 \<at,.3If

is only large as t -»- oo if co = 0. Since energy is conserved cofi = €„ — ea +h(cof — « o , ) ^ 0; thus eb — ex + hoit or eb — ez + hcaf will never be small.We therefore omit the transient contributions and (5.4.17) becomes

/ ' ' 0 ) -

— ea H(of (Of

(5.4.18)


If we add this to the Ht contribution (5.4.10), we obtain

|c<2)(/,</'\0)|2

= / e* \* ",<"/ + 1 ) 4 sin2 \o>fit\2me0L

8/ (o^ w«*

. „ , . 1 v [X4 • P>/)(g/' P/o) , (*/ * P»*X4 • P/J1 I2

(5.4.19)

Next we note that photons of any frequency may occur so we must sumover all final frequencies. Again by Section 4.5 we have in a solid angle dilabout k,

I \c™(f, tli, 0)|2 = f " - £ - = o>) do>, d€l \cw(f, tllt 0)|*/ Jo \2irc)

. (_jLJte£»±*>\\doifWeomc7 2irL8o>< Jo

da,

where

|M|2 =

(5.4.20)

4- (g/

(5.4.21)

This gives the probability of finding the atom and field in state \b, nt—\,nt + 1) at time t when initially it was in state \a, nf, nt). If we differentiate(5.4.20) with respect to time the probability per second is [see (5.2.24)]

4ire0mc

,2sinca/If' ~~T;I t w (of doif \M\

rf 2wLft)< Jo Wfi

'•— I da)f (Of \M\Z 5 ( ta / ( )Jjoit Jo

Ml(n/ + 1) (5.4.22)

The differential scattering cross-section is defined as the transition proba-bility per second per incident number of photons per second per unit area(flux). But the flux for nf photon in L9 is n^lL*. Thus .

da (5.4.23)

5.5 RAYLEIGH SCATTERING 301

or

do_(_(dQ. Urr^mc

\ 2 (Of

• + ! ) •

_ 1 2 \V*'PtfX*/-P/«) + &• PuXe,-pZa)-jI2

TO / L c7 — €B + hoif *i — ea — h<°i J I"

(5.4.24)

This is the Kramers-Heisenberg scattering cross-section for hght scattered byatomic electrons. The quantity

r0 = 47re0mc2 4ire0fic me 137 me— - — — *» 2.8 x 10- l s m (5.4.25)

so that r o8 ~ 9 x 10"30 m2. We shall apply this formula to several particular

cases. The nf term is stimulated and the 1 is spontaneous scattering.

5.5 RAYLEIGH SCATTERING

The first case we consider is elastic scattering in which the atom returns toits original state. Then <of = a = co and the cross-section becomes

da _ if , ] \ « . - _ _L y r(î * PblX^f ' P/a) , (£/ ' VblXCj ' Pjq)"| |2

d£l f f m i L €Z — €„ + h(o ez — ea — hoi J j '

(5.5.1)

We would like to simplify this as follows. Consider the diagonal matrixelement of the commutator [x • et, p • ef] for state \a). We have

2 [**«**. Pieifl = e*Wlxic> Pi]k.l k.l

k.l(5.5.2)

Therefore,

ti'ef-Tl K«l(x ' 3J)(P * e,)\a) - <a|(pifi

If we insert a completeness relation, we have

- * P/a) -

Via)

m m iP/a)


where we used (5.2.9). If we use this in (5.5.1), we obtain

da_ _ (r0Ha\?

dCl~\m

(»,+l).

(5.5.4)

We next consider scattering for which hot = hlitc\% <£ ez — ea and expandthe denominators to first order:

(€z-«a±&0)-XSS(Cz-e«)-1:

so that

da\ m / | z (€Z -

p Io) - («, • poz)(*< • Pz,)]

We first show that the first square bracket term vanishes as follows.

2 fti ft

x) -

- er • <a|x|J> </|

x)(^ • x)|a) = 0. (5.5.6)

n

We have used (5.2.9) and the completeness relation. Therefore, (5.5.5)

becomes

jall m

(5.5.7)= (r0m)2a)4

where we used (5.2.9). Therefore, for long wavelengths for which ,ez — ea, the elastic scattering cross-section varies as X~* which is calledRayleigh's law.

We may visualize the scattering process as an absorption of a photon by theatom in which the atom goes to some virtual state and then decays and emits

5.6 THOMSON SCATTERING 303

another photon of the same energy but with a different polarization and in adifferent direction. The intermediate atomic state is called virtual since theincident photon does not have enough energy to excite the electron to anotherreal state which would conserve energy (ftio «C \ez — eo|).

5.6 THOMSON SCATTERING

Consider again the case of elastic scattering in which (oi = cofyf> h~1\ez — ea\.In order for the dipole approximation to be valid, a>t <£ c/a0 where o0 =An-e^P/me* 5.3 x 10~um. In this case, we may neglect the two sums in(5.4.24) and we obtain

^ = r o2 | e / - e i |

2 ( n / + l ) , (5.6.1)ail

which is the Thomson scattering cross-section. The electron in the atom actsas if we were scattering from a free electron. The cross-section is independentof the frequency provided |«j — ea| « Hco, « eA/a,,.

Assume the incident photon is incident along the z axis and polarized alongthe x axis. Then (see Figure 5.4)

— = (0, 0,1) (5.6.2)

it = (1,0, 0). (5.6.3)

The scattered photon has momentum hkf where «,• = ck{ = cof = ckf and

k, = A:,[cos cos 6, sin 9? cos 8, —sin (5.6.5)

k.

figure 5.4 Polarizations and wave vectors for incident andscattered waves for Thomson scattering. x


These satisfy the requirementsThen

if> = 0, Hf> • £ , = <), and 4 2 ) • kt = 0

for

dQ. lr tf cos1 ?> cos19(11,+ 1) for ef\

where dCl = sin 6 dd dq>. If we do not measure the polarization of the scat-tered photon our detector picks up both polarizations and in this case we

measure

— = ro2[sin2 cos2 0](n, + 1).

dQ,(5.6.7)

when the incident radiation is polarized. If we look in the direction 0 = it\2,all emitted photons will be polarized along &p. If we put in a polarizer whichadmits only photons polarized along if\ we measure only the first case in(5.6.6) while if our polarizer admits light to our detector polarized along lj2 ) ,we measure the second case in (5.6.6).

Consider next the case in which the incident radiation is unpolarized. Wemay treat this by averaging (5.6.6) over all angles q>:

da_dQ. unpolarized at 277 Jo2 J dCl W V ' '

for

2 [cos2 0 for ef

Then, if we admit both polarizations to our detector we measure

and

da_dQ. unpolarized ox

+ cos2 0),

(5.6.8)

(5.6.9)

("/ +

= — ros(nf + 1),

K2 \2'd<p f'siJo Jo

sin 0 d0(l + cos2 0)

(5.6.10)

which is the total cross-section when the incident light is unpolarized and weaccept both polarizations in the scattered light.

Note again that at 6 = w/2, the scattered light is completely polarizedagain along e)w even when the incident light is unpolarized as may be seenfrom (5.6.8).

5.7 RAMAN SCATTERING

So far we have considered elastic scattering in the two limits /icu, <<\ex — ea\ and Ha)i» |er — ea\. Let us next consider inelastic Raman scattering.

5.7 RAMAN SCATTERING 305

hoi-hw.

hw.

• |b>

(a)

Figure 5.5 Raman scattering.

(b)

In this case, the first term in (5.4.24) makes no contribution. We considerfirst the case in which eb > eo. If the incident frequency is coL the scatteredfrequency cos = wL — (cb - ea)/h < (oL and this is called the Stokes line(see Figure 5.5). Then, the Stokes cross-section is

fe) - *'(••)<»,+ 1) IV *-«d\ (5-7.1)a£2/stokes \ft)£/ ~

where we define the Raman tensor as

mand - *« +

1osS

Hcos = feo^ — (eb —

(5.7.2)

(5.7.3)

In the second case, ea > €s and the scattered frequency o>A is higher thanthe incident frequency. This is called the anti-Stokes line. The cross-section is

whereQ/antiatokes

R iL\\R • iL\\

hoiL

(5.7.4)

(5.7.5)

It is of interest to study the angular dependence of the Raman cross-section. In an arbitrary coordinate system (x\ y'', z'), the tensor R'v will havenine components. We may associate an ellipsoid with this tensor defined by

<p 'ijX] = constant, (5.7.6)

where we use the summation convention and sum over repeated indices from1 to 3. If atf is a rotation matrix then we may rotate to a new set of axes xt

given by [6]

*t = ««** (5.7.7)

306

where the aw satisfy

Then, (5.7.6) becomes


(5.7.8)

(5.7.9)

aMaM =

<p = <xikxkR'ijxnxl = constant.

We choose the ocy elements so that in the xt coordinate system the tensor isdiagonal. That is, we require

.- (5-7-10)

The parenthesis means that we are not considering the k on the right side arepeated index. The R{M are the eigenvalues of the original tensor. If we use(5.7.8) we have

"•mlP-uR'iP-jl = R'mi*jl = «m*K(*A«. (5.7.11)or

* ! > „ = Ku,aml = RM <5mJaiI. (5.7.12)This gives

[/C-Ku^KÔ (5.7.13)

which is a set of homogeneous equations to determine the rotation matrixa,, which will diagonalize R\j. It is necessary and sufficient for these equationsto have a nontrivial solution that the determinant of the coefficients vanish:

This gives the three eigenvalues Ru R2, and Rs. We then have

(5.7.14)

(5.7.15)

which allows us to solve for a.u and <x2, in terms of a^ for / = 1, 2, 3.Let us assume the incident radiation has a wave vector along the z' axis so

that1^ = ^ ( 0 , 0 , 1 )

polarized along the x axis so that

^ = ( 1 , 0 , 0 ) .

If kR has components along the (x', y', z') axes given by

kjj = ^^[cos 0 sin 0 , sin * sin 0 , cos 0 ] ,

(5.7.16)

(5.7.17)

(5.7.18)

5.7 RAMAN SCATTERING 307

we may take the two orthogonal polarization vectors for the Rai .an scatteredlight as

1r v Ir

" IT5—T7I T T

it5—?£\KR x eR |

[ s i n *» ~ c o s *» (5.7.19)

c o s * > c o s 0 s i n ®> ~ s i n ®]- (5.7.20)

We then have in the primed coordinate system for

R'n - cos (5.7.21)

(5.7.22)

(5.7.23)

If we use (5.7.8), we may invert (5.7.10):

or


£R1 -§'-eL = sin (Dfan^, + a122/?2 +

- cos afauOuJ?! + a2S!a12/?2 + a23a13/?3]. (5.7.24)

We may express the af}. in terms of Euler angles (0, 93, y>) (see Figure 5.6). Theelements are [6]

<xu =

a12 =a13 =a21 =

a31 =

OC33

cos y> cos sin 

sin y> sin 6

—sin y> cos 9? — cos 0 sin 95 cos y>

— sin %p sin 95 + cos 6 cos 93 cos y>

cos y> sin 0

sin 0 sin 97

—sin 8 cos 93

COS 0.

(5.7.25)

Figure 5.6 Euler angles.


As a simple example let us consider a molecule for which Ux = i?2. Then wehave

I**1 ' § • ^ | 2 - {sin

= {sin O ^ c o s 2 v + cos2 0 sin2 ip) + R3 sin2 y> sin2 0]

+ cos O(/?! — R3) sin2 0 sin y> cos y>}2.(5.7.26)

In this case, we note that the angle <p has disappeared. When Rt = Rz thecross-section and the Raman tensor in this case are rotationally invariantabout the z axis of the molecule (see Figure 5.6). The z axis is an axis ofsymmetry.

If we scatter light from a gas, then molecules are oriented at random so thatwe must average the cross-section over all angles (0, <p, ip). Thus for a gaswe have

'sin e dd1 f'.= —- sir

8TT2 JO

r 2/to=\ r* r*»

= — I—)(«« + 1) sin 0 d0 <fy{cos 4>(/?x — /?3) sin2 0 sin y cos v4n\cojJ Jo Jo+ sin OIR^cos* y> + cos2 0 sin2 tp) + R3 sin2 y> sin2 0]}

+ 3R1R3 + 3K32) sin2 O + (i?x - /?3)2 cos2 <D}

(5.7.27)A similar calculation for eR* gives

]«ie + 1){(SR^ + 3/?t/?3 + 3J?32) cos2 0 cos2 O

+ (/*! - /?s)2 (sin2 0 + cos2 0 sin2 <J>)}. (5.7.28)

If we do not measure the polarization, our detector collects the sum of theexpressions above. Note that in contrast to Thomson scattering, there is noscattered direction in which the scattered radiation will be totally polarized.

5.8 RESONANCE FLUORESCENCE

We have considered elastic scattering in the limiting cases in which theincident frequency was small compared with \ez — ejjh (Rayleigh scattering)

)/gas pol.

5.9 THE DOPPLER EFFECT [1] 309

and in the case when it was large compared with these atomic energy differ-ences (Thomson scattering). In the event the incident frequency co4 =(ej — ett)lh for any two atomic levels the cross-section (5.4.24) becomesinfinite. This arises because in the derivation we neglected the small but finitelinewidth of the various atomic levels as well as their frequency shift. In thiscase the resonance term will still be extremely large, and we have

da ^ , , 2(5.8.1)

where hco{ «a e7 — €„. This, of course, assumes the sum of the nonresonantterms are negligible.

5.9 THE DOPPLER EFFECT [1]

It is well known that a moving source that emits radiation will have achange in frequency. This fact is explained by the wave theory of light.It may also be explained quite simply in the quantum theory of light by theconservation of energy and momentum.

When an atom initially in an excited state \a) is at rest and decays to state\b), it emits a quantum of light of frequency

with momentum hk = ho&fc.We now suppose that the atom is set in motion in excited state \a) with a

nonrelativistic velocity v. The initial energy is

Ea + £Mv2. (5.9.2)

At some instant the atom decays to state \b) and emits a quantum of frequencyG/. The atom recoils when the quantum is emitted; this changes the velocityof the atom to v' so that the energy of the atom is then Eb + |Afv'2. By thelaw of conservation of energy,

Hco' = (Ea + |Mv2) - (£6 + iMv'8) = hco + £M(v2 - v'2). (5.9.3)

The emitted quantum will have momentum /Ik' = hoi'^.'jc. If the momen-tum of atom and emitted quantum are conserved, we have

(5.9.4)

(5.9.5)

If we square this and neglect the term of order 1/c2, then

MY2 Mv2 v t , 4c~ hoi cos v,2 — 2 c


where # is the angle between v and the direction of the emitted quantum.If we substitute this into (5.9.3) and solve for co', we find

CO ft) (5.9.6)

This is the same as the nonrelativistic result obtained for the Doppler fre-quency shift, using the wave theory of light. That is, when the electromagneticfield is considered to be composed of quanta of light of energy h<x> andmomentum hca/c, the conservation of energy and momentum for the atomand emitted photon gives the same frequency shift obtained when the light isconsidered as a wave phenomenon.

It was shown in Chapter 4 that light quanta in free space act like "par-ticles" with the characteristics above. However, when the radiation field inter-acts with the atom, is there conservation of both energy and momentum?To answer this, we consider a hydrogen atom consisting of a proton ofcharge +e, mass mp, and coordinates xp and an electron of charge — e, massme, and coordinates xe. The nonrelativistic hamiltonian for the atom andradiation field corresponding to (5.1.2) is

2me

- -*- A(x,) ^-A(x e) .p2 , (5.9.7)m.

where we evaluate the vector potentials (5.1.8) at the position of the protonand electron, respectively. We must consider the atom nucleus, the proton inthis case, since the recoil is taken up mainly by the nucleus; this recoil allowsus to explain the Doppler effect.

We introduce new coordinates. Let

mpxp

be the coordinates of the center of gravity of the atom and

p = xP - xe

be the relative coordinates for the two particles.From the above we see that

+ mexe =

x, - xe= p

(5.9.8)

(5.9.9)

(5.9.10)

5.9 THE DOPPLER EFFECT [1]

where M = me + mv. If we solve for *„ and xe, we obtain

Mxv M% + mep

Mxe = M%- mpp

Then the kinetic energy is

rp + W / = ?i

311

(5.9.11)

(5.9.12)

where p = m/njM is the reduced mass. The momenta conjugate to % andp are given by

(Pik = rr

dT

(5.9.13)

(5.9.14)

If we make the familiar assumption that the wavelength of the radiation islong compared with the dimensions of the atom, we may replace xv and xt inthe interaction terms by %, the value of the field at the center of gravity of theatom. With these changes of variables, the hamiltonian (5.9.7) may be written

H = H0 + Hu (5.9.15)where

and

(5.9.16)

(5.9.17)

The first term in Ho represents a free particle of mass M and momentumP|. It gives the motion of the center of gravity. If we use box normalization,[5] its eigenfunctions are

I7*A exp (- P| • ?) , (5.9.18)

where p$ is the momentum eigenvalue. These eigenvalues are discrete sincethis wave function must satisfy periodic boundary conditions on oppositewalls of the box. The energy eigenvalue is p'|2/2M for this state.

The next two terms in (5.9.16) represent the internal motion of the electronrelative to the protons. The energy eigenfunctions for these terms are thehydrogen wave functions [5], designated by \n), with associated energy En.The last term in (5.9.16) gives the energy of the free radiation field. A state


vector will be designated by \nu w 2 , . . . , n , , . . . ) , with the associated energyeigenvalue n^h<ax + • • • + fta>,«, + • • •. Therefore, an eigenket of Ho willbe designated by

IPj, «, Bi, n ), (5.9.19)with energy eigenvalue

(5.9.20)2M

The equations of motion for the probability amplitudes corresponding to(1.16.59) are

dci& <(P n n n )

dt

x c(p£, m, tn-y,..., m , , . . ,)c , (5.9.21)

2D5*.m,mi mi,.

where

2M

- P ^ + Em + ha>1m1 + --- + Ha>lml+---y (5.9.22)

and the interaction is

Hi = ~ ~ I / ? " \«i« « P (ft, • S) + a,1, exp ( - ik, • ©](gIff • pp).

(5.9.23)We may show the conservation of momentum directly by evaluating the

matrix elementx

This is obviously zero unless

Pj" - p£ = ±/fc t, (5.9.24)

which is the law of conservation of momentum for the center of mass of theatom and the photon. If the matrix elements of Ht are zero between any twostates, no transitions are possible so that momentum must be conserved forallowed transitions. Therefore, momentum conservation is contained in thequantized radiation theory.

5.9 THE DOPPLER EFFECT [1J

We may therefore write the nonzero matrix elements of Hx as

|, «, nlf..., i i , , . . .|-Hi|p| — fikj, m, H i , . . . , n, + 1 , . . . )

313

j , n,..., n |Hi|p| + hkt, m,..., n, - 1, . . .)

(5.9.25a)

where by (5.2.9), with a slight change in notation,

P,»m = <n|p,|m> = "7(£» - Em)(n\p\m).n

, (5.9.25b)

(5.9.26)

The conservation of energy follows directly either from a perturbation-theory approximation or from the Wigner-Weisskopf approximation used inthe theory of the natural line width. Therefore, the explanation of the Dopplereffect is contained in the quantized radiation theory.

The Doppler shift in frequency causes the spectral line emitted by a movingatom to be broadened, in addition to the natural broadening discussed in theprevious section. This additional broadening may be seen as follows: thecenter of mass of the molecules has a Maxwell probability distribution ofmomenta. If the gas is at temperature T, then

exp/I —\

pi2 \—-— IIMkTJ

(5.9.27)

will be the probability of finding the molecule with momentum between pând P| + dp'^. When this molecule emits a photon of frequency co't, energyand momentum must be conserved; this puts a constraint on the range ofallowed values of p^. From (5.9.3), we have

2M(Pi2 ~ P ? ) - *(<>>' ~ *) (5.9.28)

with an obvious change of notation, while from (5.9.4) we obtain, on squaringand neglecting the 1/c2 term,

^ ^ ~ — pt, (5.9.29)2M ~cM

where p* is the component of P| in the direction of emission. When we com-bine (5.9.28) and (5.9.29), we see that the component of p^ in the direction ofemission must remain constant if we are to obtain the intensity of the emitted


light at a given frequency co'. The components of pg normal to fc, the directionof emission, may have any value from — a> to +00. The intensity of theemitted light at frequency to' will therefore be proportional to

exp -

where Ha>0 = En — Em and coj is the frequency of the emitted quantum. TheDoppler line shape is therefore gaussian in character.

Another factor of proportionality is the transition probability per secondfrom an initial state, in which the atom has momentum pg, is in excited state\n), and with no quanta present, to a final state pg, ground state \m), and onequantum present. Since the matrix elements (5.9.25) do not involve p{) theintegration of (5.9.27) over the components of pg transverse to the directionof emission is not affected.

The breadth of the line (5.9.30) at half maximum is

d = 2o02kTMe1

log. 2. (5.9.31)

Although kT « Me1, the effect of Doppler broadening is much greater thanthe broadening of the natural line width. However, the Lorentzian naturalline shape has a value for « ' far from a>0 whereas the Doppler effect is small.Far from the line center, the natural broadening is therefore more pronouncedthan the Doppler broadening.

5.10 PROPAGATION OF LIGHT IN VACUUM [1]

In this section we show how the quantum theory of radiation may be usedto calculate the intensity of light propagated in free space and how the phaserelations between the components of the radiation field give a finite velocityof propagation. This problem is related to that of a transmitter and receiverin communications and serves as a background for studying quantum-noiseeffects in such problems.

The transmitter will be represented by a single atom, which we shall call A,located at the origin of a coordinate system. At t = 0, atom A is assumed tobe in an excited state; after a certain time, A will decay to its ground statewith the emission of a photon. The receiver is another atom, B, located adistance r away on the 2 axis of the coordinate system; B is initially in itsground state (Figure 5.7). If the photon emitted by A interacts with B, thephoton will be absorbed, leaving B in an excited state. Since the emittedphoton travels with velocity c, the absorption by B can take place only after

5.10 PROPAGATION OF LIGHT IN VACUUM [1] 315

Figure 5.7 Atom A is excited and decays whileatom B absorbs the emitted photon after time t >rlc.

a time rjc from emission by A. We show how these results are obtained fromthe radiation theory.

To simplify the analysis, we assume that the mean life of atom A is veryshort so that we may say that A emits at some precise time. If ljyA is the meanlife of atom A, yA is large, and from the theory of natural linewidth given inSection 5.3, we see that atom A has a very broad, practically continuousspectrum. On the other hand, we assume that the lifetime of atom B is verylong (yB very small) so that atom B absorbs a very sharp spectral line. Forour purposes, it essentially stays excited once it has absorbed a photon.

We also assume that the wavelength of the radiation emitted by A is longcompared with the dimensions of both A and B. This allows us to use thedipole approximation for both atoms.

For simplicity, we assume that the dipole moments of both A and B be-tween the ground state and excited state have a component only in the xdirection.

The unperturbed hamiltonian for A, B, and the radiation field is given by

= HA

An eigenstate of Ho is written

Hr. (5.10.1)

(5.10.2)

in which atom A is in state \n), B is in state \ri), and there are nu quanta in thefield with momentum Rkt, energy h(ot, and polarization a. Primes distinguishB from A, which is unprimed. The energy associated with state |£0) is

(5.10.3)I.a

If \n) and \s) are two states of A and \ri) and \s') are two states of B, then

ha>n, = En-E, h<o'n., = En.-E,,. (5.10.4)

Also, if m is the electron mass, (5.2.9) applied to A and B in our new notation

316

becomes


= ima>n.,xn.,., (5.10.5)

where pA and pB are the momenta of A and B, respectively.The interaction hamiltonian for the interaction of A and B with the

radiation field is

m m(5.10.6)

since the field A is to be evaluated at the origin for atom A and at a distance ralong the z axis where atom B is located. Then A(r) is given by (5.1.8).Evaluation of the nonzero matrix elements of (5.10.6) between two eigen-states of Ho, which are needed for (1.16.59), is straightforward.

We may write the coupled equations (1.16.59), using (5.10.6), as follows:To begin, the nonzero matrix elements of Ht are

(n; n'; nJHJs; s'; nla + 1) = - —m

h

K' ««IPJS> W) + (n'|cttiroos<"pB|s'><«|s», (5.10.7)

and

n'; s; s'; nla -

CO,

—m

(5.10.8)

Here 0, is the angle between k, and the z axis where B is located. If we assumethat the atoms are small compared with the wavelength, we may remove theexp (±iktr cos 0,) from the integrals above and use (5.10.5) to write thesematrix elements as

llhs (5.10.9)

The first term in (5.10.9) corresponds to the emission or absorption of aphoton by atom A while atom B does not change its state; the second termcorresponds to the absorption or emission of a photon by B while A stays inthe same state.

5.10 PROPAGATION OF LIGHT IN VACUUM [1] 317

We may now use (5.10.9) to write the coupled equations (1.16.59). We have

^-in; n'; nla, t) = -ij, e —L-(elv • xn,)conat t.i.T M 2€oL

8co,

„, s', nlo+\, t)

(5.10.10)

For simplicity, the ground states of A and B are designated by 1, and theonly excited state of interest is designated by 2. We make the followingchanges in notation:

(5-10.11)—eo12 = co2i = -co — o)12 = + c o 2 l = — o

X2 i = X12 = X X12 = X2i = X

and, as noted earlier, we make the simplifying assumption that x and x' havea component in the x direction only.

At t = 0, the initial state of atoms A and B is given by |2; 1; 0; * = 0) sothat

c(2; l ;0;f = 0 ) = 1, (5.10.12)

and all other c's are zero. The problem is to find the probability that, at time t,the final state in which atom A has decayed and the photon has been absorbedby atom B is given by |1; 2; 0; t). This probability is

|c(l;2;0;r)|*. (5.10.13)

Atom B is far enough away from A so that its effect on the spontaneousdecay of A is negligible. Accordingly, we may use the analysis of Section 5.3for natural linewidth. We assume that the initial state of the system (5.10.12)decays as

c ( 2 ; l ; 0 ; 0 = e"yê/2, (5.10.14)

where \\yA is the half life of atom A. If we neglect the presence of atom B,then, as in Section 5.3, the solution of (5.10.10) for c(l; 1; lI<r; t) in whichatom A decays and emits a photon is

CO

V2^0L3V^ " ' -K«i ~ <») + YAV, (5.10.15)

318 INTERACTION OF RADIATION WTTH MATTER

after a time sufficiently long that atom A has certainly decayed to its groundstate by (5.3.31).

In the transition from the state | 1 ; 1; lto> to the final state | 1 ; 2; 0), thematrix elements x u are zero, and so that (5.10.10) reduces to

dc(U2;0;t)dt

since all other terms vanish. (There are only two atomic levels involved.)We may use the approximate result of (5.10.15) in (5.10.16) and integratewith respect to time to obtain

(glff.x)(€to.x')e'il"rcOB*[l -

,a co,[i(eo — a)j) + yJi/2]i(fi), — e>')

(5.10.17)

where, at / = 0, c(l; 2; 0; 0) = 0. We must evaluate the sum in (5.10.17) bytransforming it to an integral. This is slightly tedious; it is done in AppendixH [1], where it is shown that

c(l;2;0;0 = e*(5.10.18)

47rhi[i(oo yA/2]

Therefore, the probability of finding atom B excited is zero if t < r\c, thetime needed for the emitted photon to reach atom B. For / > rfc, we have

W. % 0; 4T =a> — to') * + (yjlf

This probability is inversely proportional to the square of the distance r.We conclude that the theory predicts correctly the velocity of propagation oflight and also gives the correct decrease in intensity of the light from thesource. Again we see the characteristic Lorentzian line shape.

5.11 SEMICLASSICAL THEORY OF ELECTRON-SPINRESONANCE

The theory of electron- and nuclear-spin resonance and the theory of thetwo-level laser have in common the interaction of radiation with a two-levelquantum system. Because of its fundamental nature, we present a rathercomplete analysis of the interaction of a magnetic dipole moment (spin-£) in a

5.11 SEMICLASSICAL THEORY OF ELECTRON-SPIN RESONANCE 319

d-c magnetic field with an rf magnetic field. The case of an electric dipolemoment (e.g., the ammonia molecular-beam maser) interacting with an rfelectric field is formally identical with the ensuing analysis.

The rf magnetic field is treated classically in this section; we do not takeinto account the reaction of the spin back on the field [7]. In Section 5.13 wequantize the radiation field and thereby consider the reaction of the spin ondie field. Fortunately, this simple problem may be solved completely.

The hamiltonian for a magnetic dipole (i in a magnetic field H is

H - - |i • H. (5.11.1)

For an electron of spin angular momentum \h, the magnetic moment is relatedto the spin by

H=-f«, (5-11.2)

where y is the gyromagnetic ratio and ax, ay, at are the Pauli spin operatorsintroduced in Section 2.8. If we combine (5.11.2) and (5.11.1) and assume anapplied magnetic field

H = {Hx cos ott, # ! sin tot, (5.11.3)

where Ho is a d-c field in the z direction and the rf field is circularly polarized,the hamiltonian becomes

H = — [H0o, + z cos <ot + av sin a>f)). (5.11.4)

If we write ax and av in terms of a+ and <r_ [see (2.8.18)], H may be written as

H = r- [H0o,

The Schrodinger equation

(5.11.5)

(5.11.6)

may be solved exactly for H given by (5.11.5). When Hx = 0, from Section2.8 we know that the eigenvalues of a, are ± 1 corresponding to the stationarystates |±1) . The unperturbed energy eigenvalues are therefore

(5.11.7)

where

to0 = yH0. (5.11.8)


The general solution of (5.11.6) when Ht = 0 is (Section 2.9)

IV<0> = c1e~to0'/8|+l> + c2e<<M/2|-l>, (5.11.9)

where tal2 is the probability that az is in state |+1) and |c2|* is the probabilitythat it is in state |—1). Normalization to unity requires that

Icil2 + |c2|2 = 1. (5.11.10)

It is easily shown, by means of (2.8.32) and (2.8.33), and the orthogonalityrelations, that for |y(0> given by (5.11.9)

(5.11.11)

<a,> - -2 - |c2 |2

<OTherefore, the expectation value of <st is a constant while <rx and tf, precessabout the z axis with <<rx>* + (cr,)2 = 4 tal* |c2|

s.When # x ?& 0, we may think of Hx as causing transitions between the two

states | + 1) and | — 1 >. By a simple transformation, we may obtain a solutionof (5.11.6) directly. We let

IK0> - e*»»'>*\x{t)). (5.11.12)

This is not a transformation to the interaction picture. (Why?) When wesubstitute this into (5.11.6), where H is given by (5.11.5), we see that ( )must satisfy

wherea±(t) ss etof'teaê-*0*"1*. (5.11.14)

This may be simplified by differentiating with respect to r. This gives

dt

where we used (2.8.21). The solution is

(5.11.15)

(5.11.16)

5.11 SEMICLASSICAL THEORY OF ELECTRON-SPIN RESONANCE 321

If we substitute this into (5.11.13), the transformation (5.11.12) leads to anequation with no explicit time dependence, namely,

ata_)]\X). (5.11.17)

In this equation a± and at are in the Schrddinger picture. If we had used theinteraction picture, the hamiltonian would still have had an explicit timedependence. We may therefore write the solution of (5.11.17) directly. If welet

Q cos 6 = (o0 — cor» • a I£1 sm 6 = yHu

where Q? = (a> —(5.11.17) is

\x(t)) =

It is easy to show that

(5.11.18)

+ (yHJ* and use (2.8.18), the formal solution of

(5.11.19)

exp [- I = cos r,ax),

which we leave as an exercise. If we let I = (Qt/2) cos 0 and t\ — (Qt/2) sin 6,then we have that

- i sin |Qf(cos 6cr, + sin 0ox)]\y,(0))

(5.11.20)This is the complete solution for the Schrodinger wave function for thehamiltonian (5.11.5).

Since the states | + 1) and | —1) represent a complete orthonormal set, wemay expand \y>(t)) as in (5.11.9), where cx and c2 are now functions of /. Itfollows that

ki(OI2 = K+i|v(O)l2 kt(OI' = K-i |y(0)l2 (5.11.21)

give the probabilities of finding the spin in the state | + 1) and | — 1), respec-tively, at time /.

As an example, we suppose that the system initially is in state |y(0)> = I +1 >.If we use (2.8.32), (2.8.33), and (5.11.20), it follows directly that

|c2(0l2 = sin8 6 sin8 \£lt. (5.11.22)

This is the probability that the spin in state | +1) at t = 0 will be in state | — 1)at time t. If we use (5.11.18) to eliminate 6 and Cl, this probability is

s i n ' ' ( 5 - 1 L 2 3 )


where, in general,|cj(r)|2 = 1 — \ci(t)\*, (5.11.24)

from the normalization condition.If the system is initially in state | — 1), the probability of finding it in state

|+1) at time / is also given by (5.11.23). Therefore, the probabilities ofemission and of absorption are equal.

The transition probability (5.11.23) reaches a value of unity only at reso-nance, that is, when the applied frequency m = eo0 = yH0. Farfrom resonancethe probability of an rf field inducing a transition is negligible.

Before ending this section, we should consider the temporal behavior ofo±{t) and at(i). We may write the Heisenberg equations for these operators,or we may note that

*.*(«) = U-\t, 0)ot8U(t, 0), ( 5-1 1 > 2 5 )

where U(t, 0) is unitary and given by (5.11.20). By using the results of Section2.9, we may obtain these operators in the Heisenberg picture. In particular,it is found that

<*x(0 — (cos* 0 + sin* 0 cos Q.i)<st + sin 0 sin Qtoy + sin 20 sin* \&t<ts.(5.11.26)

This is left as an exercise.We may show that if, at / = 0, |y(0)) = | + 1), then by (5.11.20), (2.8.32),

and (2.8.33)

; \Q.t — i sin JQf cos '

- ieiat/tsin \Clt sin 0| - 1 >, (5.11.27)

so that the expectation values of a,(t), a±(t) for this state are

<y(0k±lv(0> = T/e***'sin \£lt sin 0(cos \Qt ± i sin $Clt cos 0)<V(0kJ V<0> = cos2 IClt + sin* ±Clt cos 20. (5#1 i "28)

At resonance, to = w0, 0 — w/2, and Q — yHlt and it follows from (5.11.28)and (2.8.18) that

(ax) = sin (o0t sin yHxt

(av) = - cos co0t sin yHxt (5.11.29)

(az) = cos yHxt

Therefore, (aa) and (av) precess about Ho with frequency ta0 while (or,) nutatesat frequency y//,; this is the classical behavior of a magnetic moment in ad-c circularly polarized if field, in agreement with Ehrenfest's principle. If

5.13 EFFECT OF FIELD QUANTIZATION ON SPIN RESONANCE [7] 323

Hi <K Ho, the spin precesses through many cycles before (az) changes verymuch.

5.12 COLLISION BROADENING OF TWO-LEVEL SPINSYSTEM

The analysis in the previous section assumed that the interaction betweenthe spin and the field continues indefinitely. In a gas at finite temperaturethere are random collisions of the gas molecules that interrupt the interactionbetween the spin and field. This effect is important in a gaseous laser andresults in a broadening of the spectral lines. We have already studied twomechanisms that broaden a spectral line, namely, the natural line width andthe Doppler effect. Let us consider the effect of collisions.

If no collisions occur, the instantaneous power flow between the spins andthe radiation field is

at(5.12.1)

where |c2(0l* is given by (5.11.23). If there are random collisions, the proba-bility that no collision has occurred in a time t is exp (—tlTt), where T2 is theaverage time between collisions, while the probability that a spin had its lastcollision between t and t + dt is

The average power transferred to or from the field is therefore

7at

(co(5l22)

where we used (5.11.23). Therefore collision broadening gives a Lorentzlineshape to the spin-resonance line.

5.13 EFFECT OF FIELD QUANTIZATION ON SPINRESONANCE [7]

In Section 5.11 we considered a spin-£ particle interacting with a classicalradiation field, and we neglected the reaction of the spin on the field. Weconsider the same problem again but quantize the radiation field and takeinto account the reaction of the spin on the field.


We consider a small sample of material containing spin-£ particles locatedat a position in a cavity in which the magnetic field has a component only inthe x direction for a particular mode. As shown in Chapter 4, the energy ofthis mode of the field is

hmaâ, (5.13.1)

where w is the frequency of the mode under consideration.By (5.11.1) and (5.11.2), the interaction energy is

•/sample Z

= yH0-oz + *-\2 2 Js

ajltdr, (5.13.2)

where Ho is the value of the d-c field in the z direction and Hx is the rf fieldat the sample. If we use (4.3.9) and (4.3.30) and assume that as occupies asmall volume of the cavity, we may write (5.13.2) as

= ^- at fiK(a (5.13.3)

where all quantities multiplying ax in (5.13.2) and not appearing explicitly in(5.13.3) are grouped into a coupling constant K, which will be small. Since, by(2.8.18), <tx = <r+ + <r_, we may add (5.13.1) to (5.13.3) and write the totalhamiltonian as

H = az(5.13.4)

This hamiltonian may be simplified by the following considerations. WhenK = 0, that is, when there is no rf field, the Heisenberg operators have a timedependence given by

a(t) =a\t) = a\0)eiat

a+{t) =(5.13.5)

so that, near resonance (eo *w a>0), the interaction terms aa+ and a*a_ in(5.13.4) are practically d-c terms whereas the terms a<r_ and d*tf+ vary rapidlyat frequencies ±(to + <w0). To a good degree of approximation for times ofinterest, the high-frequency terms average to zero, and we may write (5.13.4)approximately as

H at *o_ + ao+). (5.13.6)


This approximation is equivalent to decomposing the linearly polarized rfcavity field into two opposite circularly polarized waves and keeping only theone rotating in the same sense as the spin precession. For small rf fields, littleerror is involved.*

The hamiltonian (5.13.6) is still hermitian and takes into account the effectof the spin on the field as well as the usual effect of the field on the spin.

The problem at this stage is to solve the Schrodinger equation for thehamiltonian given by (5.13.6). We present a solution due to Jaynes andCummings [7].

For convenience, we define two new operators

S+ = a+a S_ = a_a\ (5.13.7)

From the commutation relations (2.8.19)-(2.8.26) and [a, aT] = 1, it is left asan exercise to verify the following commutation relations:

[a\ SJ = [az, SJ = ±2S±

(5.13.8)

It will be found that the Heisenberg equations of motion for these operatorsare nonlinear. Rather than try to solve these equations, we resort to "trickery"in finding two constants of the motion, either by inspection or from firstintegrals of the nonlinear equations of motion. We may verify directly thatthe operators

Cx = co(Ja + \az) C2 = K(S+ + S_) - y az, (5.13.9)

where Aco = o> — <o0, commute with the hamiltonian (5.13.6) and are there-fore constants of the motion. It follows directly from (5.13.9) and (5.13.6)that

# = / K Q + Q)- (5.13.10)

Therefore, Q and C2 are not independent of H, and we may show that

[ C k . C J - O . (5.13.11)by using (5.13.8) so that Cx and C2 may be treated as c-numbers with respectto one another.

When there is no coupling between the spins and radiation field, we havea complete set of basis vectors, namely, the states |«) for the radiation fieldwhere d*a\n) = n\n) and the states | ± 1) for the spins where az\ ±) = ± | ± 1).We may use this complete set of states to expand the state vectors of thecoupled system. We see that

Ci|«, ±1) m <o(d*a + \az)\n, ±1) = co(« ± \)\n, ±1> (5.13.12)

* This is called the rotating wave approximation and is valid for weak coupling.


(5.13.13)

so that Cx has eigenvalues «(n ± $) with the corresponding eigenkets \n, ± 1 >and Cx is therefore diagonal in this representation. However, Ct is notdiagonal in this representation, but since Q and Ct commute, we may finda representation in which they are both diagonal (see Section 1.9) by takinglinear combinations ofthe eigenkets of Q . In this representation, by (5.13.10),H is also diagonal so that it may appropriately be called the energy repre-sentation. Once AT has been diagonalized, the Schrodinger equation has beensolved.

Let us choose two simple linear combinations of eigenvectors of Cx andverify that they are eigenkets of Ct (as well as H). We consider the statevectors

\<p(ni 1)) = cos 0Jn + 1, - 1 > + sin 6Jn, + 1 )

\<p(n, 2)> = - s i n 0 > + 1, - 1 > + cos 8n\n,where|n + 1, — 1) corresponds ton + 1 quanta in the field with spin "down"and \n, + 1 ) corresponds to n quanta with spin "up." There is another statethat is not included, namely, no quanta in the field and spin down, |0, —1);this state must be considered separately and is called the ground state. Theangle 0fl is a parameter at our disposal to make \f(n, 1)) and \<p(n, 2)> eigen-kets of Q , and n may have any value from 0 to oo.

We begin by observing that

2)) = 0

1)> = {f(n, 2)\<p(n, 2)) = 1. I- - >

Both these states are orthogonal to the ground state, |0, —1). We may alsonote that, by (5.13.12),

CxWiri, 1)> = co(n + l)\<p(n, 1))

2)> (5.13.15)

1),

so that the states \<p(n, 1)> and |qp(n, 2)) are degenerate eigenstates of Clt

no matter how we choose 0B [8].Let us now apply C2 to \q>(n, 1)> and \f(n, 2)>. After minor simplification,

we obtain

C2\<p(n, 1)> = \K\ln + 1 sin 0B + — cos\ 2

+ (Wn + 1 cos en - ~ sin 0flj |n, +1) (5.13.16)

C2|9>(n, 2)) = («Vn + 1 cos 0fl - — sin 0nJ \n + 1, -1>

- Ly/n + 1 sin 0» + — cos 0B) |n, +1>, (5.13.17)

(5.13.18)


where we used the relations

S+\n + 1, —1> r= Vn + l|n, +1)S+\n, +l> = 0

S l | n + 1 , - 1 ) = 0

5_|n, +1) « V« + l|n + 1, - 1 ) .In order for \<p{ti, 1)) and \<p{n, 2)) to be eigenvectors of C2, we must be

able to choose 0n so thatCt\q>(n, 1)> — Xn\<p(n, 1))

Ct\<p(n, 2)> - An|9>(n, 2)).

From (5.13.16), (5.13.17), and (5.13.19), if we let

(5.13.19)

tan0.

andXn'

(5.13.20)

*. - + J *T + *» + 1)then \<p(n, 1)> and \<p(n, 2)> are eigenkets of C2 with eigenvalues ±Xn. Thus^» — ~K- % simple trigonometry, we may show that

. . . 2 tan 0Btan 20_ = "

For the ground state,1 - tan2 0,

(5.13.21)

|0, - 1 ) , (5.13.22)

which is therefore an eigenket of C2 also.We have succeeded in finding the eigenvalues of H, for by (5.13 10)

(5.13.15), and (5.13.19) we have

H\<p(n, 1)> = h[o>{n + \) + Xn]\<p(n, 1)>

H\<p{n, 2)) = hMn + J) - Xn]\<p(n, 2)), (5.13.23a)

and for the ground state

H|0, — 1) = 10, —1). (5.13.23b)

Note that eigenstates of H are mixtures of eigenstates of i/0 , the unperturbedhamiltonian. The states \<p(n, 1)>, \<p(n, 2)) and |0, - 1 ) are complete so that

00

o «, 1)1 + \<p(n, 2))(<p(n, 2)|} + |0, - l><0, - 1 | = 1. (5.13.24)

It is useful to have expressions for |n, ± 1) in terms of | <p(n, 1)> and | y(n, 2)>.


From (5.13.13) and the orthogonality relations (5.13.14), we have (n = 0 , 1 ,2 oo)

\n + 1, - 1 ) = cos 0«|?>(n, 1)> - sin 0B|<K«, 2))\n, +1) = sin 6n\<p(n, 1)> + cos 6n\q>(n, 2)>,

(5.13.25)

and

(5.13.26)<« + 1, -l\<p(n, 1)) = cos0B = <n, +l|?>(«, 2))

</i, +l|9>(n, 1)) = sin 0n = — (« + 1, —l\<p(n, 2)>.We now calculate the transition probability between an initial state with n

quanta and spin "up" (|/i, +1)> and a final state with n + 1 quanta andspin "down" (|n + 1, —1» to compare the results of the quantized-field casewith the unquantized case of Section 5.11.

Since the hamiltonian (5.13.6) does not contain the time explicitly, we maywrite the solution of the SchrSdinger equation as

e~iCtt\f(0)), (5.13.27)

where |y>(0)> = \n, +1) for the case of interest. The probability of findingthe system with n + 1 quanta and spin "down" at time / is given by|<n + 1, -l|v(0>l*» which, on using (5.13.25), (5.13.15), (5.13.14), and(5.13.19), becomes

<n 1,-1| rap ( - Y ) | B

= sin* 20n sin2 Xnt

s i n W ( * 0 + ( + l)- (5-13.28)(Aa>)* + 4 A« + 1)

This is also equal to the transition probability from state |« + 1, —1) tostate \n, +1).

If we compare this result with (5.11.23) for the unquantized-field case, wesee that the two probabilities are remarkably similar. The square of the rfamplitude, Hf, in (5.11.23) is proportional to the number of quanta, n, in thefield. The difference is the appearance of the spontaneous-emission term, 1, in(5.13.28). If n = 0, the spin will still flip and emit a quantum that may bereabsorbed. Jaynes and Cummings [7] have shown that, if in the semidassicaltreatment the effect of the spin on the radiation field had been included,spontaneous emission could take place in the semidassical theory.

PROBLEMS

5.1 The hamiltonian for a hydrogen atom in a Coulomb field is

REFERENCES 329

where the symbols have the usual meaning. Write the Heisenberg equationsof motion for pB(t) and xH(t).

5.2 Evaluate the induced dipole moment (5.2.9) for atomic hydrogen between(a) and Is- and 2*-state and (b) the Is- and 2/>-state. The wave functions maybe found in any quantum mechanics book.

5.3 If a set of two-level atoms with energies Ea and Eb are in thermal equilibriumat temperature T, the number of atoms in state \a) is proportional toexp (—EJkT) and for state \b) the number is proportional to exp (—EJkT),where k is Boltzmann's constant. Show that, if the atoms are in thermalequilibrium with a radiation field, the average number of quanta is givenby the Planck distribution law

where Ha> = Ea — Eb.5.4 Calculate the natural line width of the 2p — Is transition in atomic hydrogen

and compare it with the observed line width.5.5 Calculate the Doppler line width for atomic hydrogen for the 2p — Is transi-

tion at room temperature.5.6 Solve the Heisenberg equations of motion for the spin operators o±(t) and

ot(f), using the hamiltonian (5.11.5). Express the result in terms of theoperators o±(0) and az(0). Hint: Let a±(t) = s±(t) exp (±imt).

5.7 Verify (5.13.11).5.8 Find the Heisenberg equations of motion for the operators a*a, az, and S±,

using the hamiltonian (5.13.6). From these equations, find the two integralsof motion (5.13.9). Finally, from these equations eliminate afa, 5 ± and obtainthe equation for oz alone. One integral of this equation is immediately ap-parent. From this integral, show that oz must involve elliptic functions.

5.9 For the hamiltonian (5.13.6), obtain the transition probability per sec for thesystem to make a transition from the initial state \n, +1) to the state \n + 1,—1> by using perturbation theory. Compare this with the exact result ob-tained from (5.13.28).

5.10 Evaluate the Rayleigh differential scattering cross-section if we assume the"atom" is a three-dimensional isotropic harmonic oscillator described by thehamiltonian

Ha hip2+f C 0 ° 2 ( a ; 2 + y 2 +

5.11 Calculate the cross-section for the scattering of a photon from a free electron.

REFERENCES

[1] E. Fermi, Rev. Mod. Phys., 4, 87 (1932).[2] V. G. Wcisskopf and E. Wigner, Z. Phys., 63, 54 (1930).[3] R. V. Churchill, Modern Operational Mathematics in Engineering,tltvi York: McGraw-

Hill, 1944, pp. 1571T.


[4] W. Heitler, Quantum Theory of Radiation, 3rd ed., Fair Lawn, N.J.: Oxford UniversityPress, 1954, pp. 66ff.

[5] L. I. Schiff, Quantum Mechanics, 3rd ed., New York: McGraw-Hill, 1968, p. 85.[6] H. Goldstein, Classical Mechanics, Cambridge, Mass.: Addison-Wesley, 1950, pp.

97ff, 107ff.[7] E. T. Jaynes and F. W. Cummings, Proc. I.R.E. 51,89 (1963).[8] That is, the states \<f(n, 1)> and \<jin, 2)> are different but the eigenvalues of C, are

the same.

6Quantum Theory of Damping—DensityOperator Methods

Damping plays such an important role in so many physical problems that wedevote two chapters to its study. In this chapter we are primarily concernedwith the density operator formulation of the problem while we devote thefollowing chapter to the Langevin formulation. We exploit the quantum-classical correspondence that we developed in Chapter 3 to separate thequantum and stochastic natures of the problem. The methods are illustratedby simple examples such as a damped harmonic oscillator and homogeneouslybroadened atoms. We also discuss a rotating wave van der Pol oscillatorwhich plays an important role in the study of lasers and optical parametricoscillators.

We begin by presenting a simple explicit model for a damping mechanismwhich allows us to give some of the properties of a reservoir. In Section 6.2we consider an arbitrary system which is damped. We discuss the reduceddensity operator introduced in Chapter 1 which describes the statisticalproperties of a system coupled to a reservoir. We derive its equation of motionunder the Markoff approximation. We obtain the reduced density operatorequation for several simple damped systems for illustrative purposes.

In Section 6.3 we develop the mean equations of motion for system oper-ators under the Markoff approximation. We again illustrate these equationsfor a damped driven oscillator and an atom with linewidth. This section isuseful for the Langevin approach of the following chapter.

The equations of motion of associated distribution functions are obtainedin Section 6.4. This allows us to study the quantum statistical properties ofsystems in a c-number domain. We work out the classical analog of a largenumber of homogeneously broadened atoms which is useful for the theory ofa laser. In this section we also obtain the operator equations of motion in ac-number domain.

In Section 6.5 we solve the Fokker-Planck equation for a damped drivenmode of the radiation field by two different methods. We also obtain anexplicit expression for a characteristic function.

331

332 QUANTUM THEORY OF DAMPING—DENSITY OPERATOR METHODS

In Section 6.6 we give a method of converting two time averages to one-time averages for Markoffian systems and obtain the spectrum of a dampedoscillator. In the final section we discuss the rotating wave van der Poloscillator.

6.1 MODEL FOR LOSS MECHANISM

The hamiltonian for a single mode of a quantized radiation field of fre-quency ct>e in a cavity is given by H = h<a<.cPa. We have shown in Chapter 2that in the Heisenberg picture, the annihilation and creation operators aregiven by a(t) = a exp (—/»„/) and af(/) = a< exp iwet, respectively. Supposethat we attempt to describe damping in this mode by introducing a phe-nomenological loss term in the cavity by analogy with a circuit resistance.Since the classical and Heisenberg equations have the same form, the oper-ators would then have the damped solutions

a{t) -

a\t) =

and the fields would decay in a time of order y1. This result unfortunatelyviolates a fundamental principle of quantum mechanics since the com-mutator

approaches zero. This implies that the uncertainty principle is violated (seeSection 1.12). For times short compared with the relaxation time (t« y~x),the violation is not serious and the model would be satisfactory.

The difficulty with the model lies in the fact that there are thermal fluctua-tions in the resistor which feed noise into the oscillator, and these fluctuationshave not yet been taken into account. In an equivalent circuit model, weshould therefore put in not only a resistance but also a noise generator. Onemight be tempted to argue that this would not be unnecessary if the resistorwere at absolute zero. Classically, this argument would be correct since thenthe oscillator would be highly excited (to be in the classical regime at all), andwe would not expect to see any quantum effects. The violation of the un-certainty principle would not concern us. However, even at absolute zerothere will always be zero point motion in the resistor due to purely quantummechanical effects and these fluctuations will couple into the circuit. At lowexcitation of the field in the cavity mode, these fluctuations are responsiblefor preventing the violation of the uncertainty principle. What we therefore'need in an equivalent circuit model is a noise generator Which has sufficientoutput even at abolute zero to preserve the commutation relation \a{i),

6.1 MODEL FOR LOSS MECHANISM 333

af(t)] = 1: a quantum noise generator. We pursue this equivalent circuitapproach in the next chapter but we work from a different point of view inthis chapter.

A similar problem was encountered in the theory of emission and absorp-tion of radiation by an atom. If the lifetime T"1 of an atom in an excited stateis very long, we may adequately describe the emission and absorption byfirst-order perturbation theory and neglect the atomic linewidth (see Section5.3). However, if the atomic lifetime is short, we must take into account thereaction of the field on the atom which gives the atom a linewidth. That is,the fluctuations of the radiation field must be taken into account (by analogywith the noise generator) to prevent the violation of the uncertainty principle.

To obtain some insight into the atomic linewidth problem, we may visualizethe process as follows. We think of the atom as a single system coupled to theradiation field in a cavity. The coupling is through the charge on the atomicelectrons and the electric field in the cavity. Quantum mechanically there willalways be zero point fluctuations in the cavity modes even in the vacuumstate (no modes excited). As we have seen, the field may be visualized as alarge number of fictitious harmonic oscillators, one for each mode of thecavity. If the cavity is very large, the modes will be very closely spaced infrequency so that they form practically a continuum. There will therefore be acavity mode of frequency eue which approximately matches the energyseparation of two atomic levels, that is,

h<oc Q± £ 2 — Ex.

The weak coupling of the atom and the field causes a splitting of the levelsinto a band and this gives a linewidth to the originally sharp atomic energylevels which were obtained by neglecting, in first approximation, the inter-action of the atom and field.

The same phenomenon also occurs classically. If we weakly couple twooscillators of the same resonant frequency, the coupled system will have twonormal modes whose frequencies differ slightly [1] (see Figure 6.1). If oneoscillator is excited initially, its energy will all go to the other oscillator at a

Figure 6.1 Normal modes of two weakly coupled oscillators.


later time because of the interference of the oscillations at the two new normalmode frequencies. We may visualize one oscillator as the atom and the otheras a mode of the radiation field. The atom in an excited state corresponds toone oscillator excited. It decays giving its energy to the field mode. In turn theenergy is reabsorbed by the atom from the field.

Similarly, if a large number of oscillators of the same resonant frequencyare coupled, the normal mode frequencies of the coupled system will spreadinto a narrow band of frequencies near the original frequency. Energy initiallyin one oscillator will distribute itself among the other oscillators due to theinterference of the various normal mode frequencies. Eventually all theenergy will return to the original oscillator and the process will repeat. How-ever, for sufficiently short times, the other oscillators absorb the energy of theoriginal oscillator and act like a damping mechanism in addition to giving alinewidth. This is the analog of an atom coupled to modes of the radiationfield. The field acts like a reservoir for the atomic energy. In addition thefluctuations present in the modes of the radiation field (reservoir) will coupleto the atom.

With the foregoing discussion as a guide, let us consider some systemdescribed by a hamiltonian H. Next we assume there is a reservoir which maybe taken as any large collection of systems with many degrees of freedom andwhich may be described by a hamiltonian R. If the reservoir consists of themodes of the radiation field, then we have

R = 2 tiWjblb,, (6.1.1)t

where the b, and b\ satisfy the boson commutation relations [bt, b%\ = d}k.These might also be the quantized modes of elastic vibrations in a solid. Thequanta of energy of elastic vibrations are called phonons and the creation andannihilation operators in this case also satisfy the boson commutationrelations.

Next we assume there is a coupling between the system and reservoir whichis described by the interaction energy V. The total hamiltonian for the systemand reservoir is given by

HT = H + R + V m Ho + V. (6.1.2)

We may assume that the interaction is "turned on" at some time t0. In ascattering experiment, for example, the system and reservoir are so far apartbefore t0 that we may neglect the interaction V so that it is not unreasonableto imagine some experimental arrangement whereby V — 0 for / < t0.

Other damping mechanisms may require different forms of R. However,the results are not very sensitive to the particular mechanism; thus we con-sider this simple model.

6.1 MODEL FOR LOSS MECHANISM 335

The statistical properties of a damped system are described by a densityoperator p(t) which by (1.20.12) satisfies the equation of motion.

do

ft V,

wherePit) = 1,

(6.1.3a)

(6.1.3b)

and where all operators are in the SP and we trace over both the system andreservoir in (6.1.3b). At t — t0, the system and reservoir are uncoupled andthe density operator factors as

P('o) =/o(*)S('o), (6-1.4)

where S(t0) describes the initial state of the system and/0(i?) describes theinitial state of the reservoir. We must therefore solve (6.1.3) subject to theinitial condition (6.1.4) and show that we have adequately described adamped system consistent with quantum mechanical principles. Actually, wemust give some special properties to the reservoir in order to accomplish thisgoal. We do this in the following section.

Usually, the reservoir will be in thermal equilibrium at a temperature Tinitially. In this case the reservoir has a Boltzmann distribution described bythe density operator

where

kT

(6.1.5a)

(6.1.5b)

If the reservoir consists of independent oscillators, then we may factor thedensity operator as

(6.1.6a)where

hcof

kT(6.1.6b)

The reader should verify that

Tr f i/o(*) = l. (6.1.7)

We leave the initial system density operator S(t0) arbitrary for the moment.Let us consider in more detail a possible damping mechanism for a mode in

a cavity. The mode is characterized by a frequency coe, polarization e, and


wave vector k. Light in this mode could be scattered by a gas atom in thecavity or from the walls, either elastically or inelastically, into other cavitymodes and be lost from the desired mode. Conversely, thermal and zero pointfluctuations in other modes could be scattered into the particular mode ofinterest. We may visualize the net effect of the scattering process from thedesired mode as the annihilation of a photon at frequency cue and the simul-taneous creation of a photon at frequency to,. Scattering into the mode con-sists of the annihilation of a photon at <ot and creation of a photon at coe. Wemay therefore write an equivalent or effective interaction energy as

V = h 2 («,&/ + K*b\a). (6.1.8)i

This is an effective interaction since we have not taken into account thedetails of the scattering mechanism but it represents the net effect. We havesummed over all modes that may scatter into or out of the desired mode.Actually, only those modes which conserve energy (co, ^ <uc) and momentum(k, *» kc) interact very strongly. The coupling coefficients K, denote the strengthof the coupling and depends on the actual interaction mechanism. We haveneglected the processes bp. + b]ar in which two quanta are annihilated andcreated since they are unimportant when the coupling is weak [1]. Thisneglect is called the rotating wave approximation (see Section 5.13). We mayargue that they are unimportant as follows. In the absence of coupling theterms in (6.1.8) vary as exp i/O*^ — coe)t which is approximately unity whenojf fas coe while the omitted terms vary as exp ±i(cot + coe)t which is rapidlyvarying. For times large compared with (o~l, they will average approximatelyto zero and we neglect them.

The interaction energy (6.1.8) also describes loss of energy from an elasticmode of vibration in a solid into other modes.

To summarize, our reservoir may be thought of as any collection of a largenumber of quantum systems (many degrees of freedom) initially in thermalequilibrium. The system is weakly coupled to the reservoir and loses energyto the reservoir. The fluctuations in the reservoir also couples back into thesystem. Further properties of the reservoir needed to prevent the excitationenergy initially in the system from returning completely from the reservoirback into the system are described in the next section.

6.2 THE MARKOFF APPROXIMATION IN THESCHRODINGER PICTURE [2-5]

Reduced Density Operator for System with Loss; Master Equation

The density operator for a system coupled to a reservoir in the SP satisfies(6.1.3) where the total hamiltonian is given by (6.1.2). In general we areinterested in the statistical properties of the system only. That is, if M is

6.2 MARKOFF APPROXIMATION IN SCHRODINGER PICTURE [2-5] 337

some function of the system operators only in the SP, we are often interestedin its expectation value given by

Mp{t) = Trs MTrRp(t), (6.2.1)

where we must trace over both the system and reservoir. Here both p(t) andM are in the SP. Since the system and reservoir variables are independent andsince M is a function of system operators only, we may first trace p(t) over thereservoir and then carry out the remaining trace over the system. We maytherefore let

(6.2.2)

which will depend only on system operators since we have traced over allreservoir operators. The S(t) is called the reduced density operator for thesystem in the SP (see Section 1.21). The one-time average (6.2.1) may then bewritten as

= TTS MS{t). (6.2.3)

Therefore, we may obtain one-time averages of system operators from S(t)without the necessity of knowing the full density operator p(t). Of course, wecould not obtain any reservoir averages from S(t) but they are usually of nointerest anyway. We would therefore like to remove the unnecessary in-formation from (6.1.3) and obtain an equation of motion for S(t) directly.For this purpose we first transform (6.1.3) to the IP to remove the rapidlyvarying unperturbed system motion from p(t). We therefore let

p(t) = c-«/««»<«-'o>z(ry</«)»»«-*o)> (6.2.4a)

where HQ = H + R and p(t) are in the SP and x(0 is the density operator inthe IP. Note that

P(to) - Z('o) = S(t0)f0(R) (6.2.4b)

so that the two pictures coincide at t = t0. We have used (6.1.4) also. Sincethe system and reservoir are independent before coupling, their hamiltonianscommute

[H, R] = 0.

In fact, since all system and reservoir operators commute at t = t0 in the SP,they must commute whenever they are in the same picture. Accordingly, if wetrace both sides of (6.2.4a) over the reservoir, we have

(6.2.5a)

(6.2.5b)

where we used (6.2.2) and have let


be the reduced density operator in the IP. We may see this as follows. In theU-representation, we have

JR|R'> = R'\R')

(R'|JR"> = « W or 8(R' - R") (6.2.6)

2 \R') (R'l = 1 or f \R') dR' {R'\ = 1.

Then we have by (6.2.4a)

R'

R'

R'

(6.2.7)

The inner term in the last line is just TtR #(f).We see by differentiating (6.2.5a) that

! [H, s(t)] + f*U# t

(6.2.8)

which relates the equation of motion for the reduced density operator in the

two pictures.We next obtain an equation of motion for %(t). If we differentiate both

sides of (6.2.4a) we obtain

ii/IDHtit-tt)

'"dt

When we compare this with (6.1.3) we see that

. e-u/a>ffo<»-t,><fydt

If we insert

(6.2.9)

Vp _ pV, (6.2.10)

between the V and p in both terms on the right above and again use (6.2.4a),

we see thatdftQ\x\, (6.2.11)f

where we have letV(t - t0) = e«/»)H»«-*o)Fc-(«/»)H.«-«o) (6.2.12)

be the interaction energy in the IP. The Ho and V on the right are both in theSP. Equation (6.2.11) is therefore the equation of motion for the full densityoperator in the IP. Since in general we cannot solve it exactly, we assume

6.2 MARKOFF APPROXIMATION IN SCHRODINGER PICTURE [2-5J 339

that the interaction is weak and resort to perturbation theory. We iterate(6.2.11) up to second order in V as in Chapter 1. This gives

X(t) = XM + 71 f W - 'o), *('„)] dt'

\ihf Jto Ju,Next we trace both sides over the reservoir. If we use (6.2.5b), (6.2.4b), and(6.1.7) this gives the reduced density operator in the IP to second order inthe interaction:

- s(t0) = 1 f Tr* [V(f - t0), s(to)fo(R)) dt'in Jto

dt" T r « \W - '•)' \y<f - ro). 4 ) / ^ ) ] l - (6.2.14)

To proceed, we assume that the interaction energy may be written as a sumof products of the form „

VhlQA, (6.2.15)

where Q{ is a function of system operators only and Ft is a function of reser-voir operators only and both are in the SP. For the example of the dampedmode of the radiation field (6.1.8) we would make the identification

Gi = af Q2 — 0r, •V 1 C X* *Lt (6.2.16)

i iIn the IP since system and reservoir operators commute, we have by (6.2.12)that

(6.2.17a)

(6.2.17b)Note that

[&(*' - '0), Ft{f - t0)) = 0, (6.2.18)for all t' and /". (Why?)

We next substitute (6.2.17) into (6.2.14). When we expand the doublecommutator and use the property (6.2.18), we obtain

V(t - t9) = h 2 Qt(t - fo)F,(r - t0)i

QXt — t) = eu/MHit-t*)Qê~<immt~to)

p.(t _ t()) —

s(0 - s(f0) - -i 2 f W sCo)]^)* dt'

- I faffWMR}, (6.2.19)


where for simplicity we have used primes to indicate t' — f0 as a time argu-ment and double primes to indicate t" — t0 as an argument. We also used thecyclic property of traces and (6.2.17b) to show that

TtRf0(R)Ft{t' - t0)

TrK

- to))R

(6.2.20a)

(6.2.20b)

TrBF<'/0(R)F';m (Ffl" - tjFtf - to))R

= (F^S - t"))R. (6.2.20c)

In Figure 6.2a we show the shaded area of integration in the double integralin (6.2.19). If we let

r = f-t" r'-ro = T + | (6.2.21a)

the area transforms into that shown in Figure 6.2b. Since

dt'dt" = dTd£,

we have one using (6.2.20) and (6.2.21) in (6.2.19)

(6.2.21b)

to i.i J0

x {[QiCr + OQtfHto) - Q/£M'o)Qi(r +- [Q*(T + S)s(to)Qtf) - s(to)Qtf)Qi($ + T X K J W ) ) , , } . (6.2.22)

We next assume that the Q, are single system operators which are given inthe IP by

It is easy to generalize when this is not true, but it will hold for the exampleswe consider [see, (6.2.16)]. In this case (6.2.22) reduces to

«*"** dt0

- 2 P'VftQ^sOo) - QAQQi] fi.l Jo I JO

li - sCOQiaiJ^^^^r))^.

f«-*o-s

(6.2.24)

'where all system operators are in the SP. We also let f = t' - t0 in the firstintegral above.


t"«f'

t-t,

Figure 6.2 Region of integration in (6.2.22).

(b)

So far we have evaluated the reduced density operator in the IP to secondorder in the interaction. We would now like to make the Markoff approxi-mation on s(t) in the IP where we have removed the rapidly varying freemotion of the system; s(t) — s(t0) represents the change in the reduced densityoperator during the time interval / — t0. By definition, a system is Markoffianif its future is determined by the present and not its past [6]. In other wordsit loses all memory of its past. This is seen to be a sufficient condition to ensurethat energy which goes into the reservoir will not return to the system. Forotherwise, the system would develop memory.

Now in (6.2.24) we have two reservoir correlation functions (F^F^R and(FJF^T)}^. In Figure 6.3 we have sketched such a correlation function.Typically, it is nonzero over some time interval re which is called the reservoircorrelation time. As long as we require that t — t0 )S> re, we may thereforeextend the upper limits on the r integrals to infinity with very little error.However, we require that t — r0 be short compared with the system dampingtime, y1. That is,

~ 'o (6.2.25)

Since the bandwidth of a system is proportional to the reciprocal of the life-time, the bandwidth of the reservoir must be large compared with the systembandwidth (see Appendix I). Physically we are smoothing out the fluctuationsof the system on a time scale during which the reservoir is correlated but noton a scale during which the system is damped and so the system will loseits past memory on this time scale. Therefore, when (6.2.25) is satisfied (6.2.24)

<F.(r)Fj>R

Figure 6.3 Typical reservoir correlation functionbehavior.


reduces to

5 ( 0 -

i.i

where we have let

= f<

The w* are reservoir spectral densities (see Appendix I).If we also require that

t — t0 » (of1 or ((Of + co,)""1,

ifi«W(«« + co,), (6.2.26)

(6.2.27)

(6.2.28)

(6.2.29)

we are integrating over a time long compared with a period of the free motionof the system and we see that the integrand of (6.2.28) goes through manycycles and averages to zero unless a>{ = 0. That is,

/(«<) = (t - /0)3(ft),., 0), (6.2.30)and

(t - t0 if (oi + coi = 0/(«,. + to,) = (f - /oWo. , , -o>,) = (6.2.31)

(0 if (Of + ft), 5"* 0.

Since none of our operators have zero frequency, we have that

/(«<) = 0,while (6.2.31) retains only the secular terms in (6.2.26). We therefore havethat (6.2.26) reduces to

K-J, (6.2.32)

(6.2.33)

At

where we have letAs = s(t) - s(t0)

re « At = / - t0 « y~

In the IP, the interval At -*• 0 as far as system operators are concerned underthe Markoff approximation. We therefore replace the exact (6.2.8) by the


approximate

dt ~

_ e-{ilh)H<t-U

- (&s(OG, - *(«)fiifi>«]}eWMJSr<Wi). (6.2.34a)

where we have used (6.2.32). If we insert between all factors

exp + (i/h)H(t - t0) • exp -(ifh)H(t - t0) = 1,

use (6.2.23), and (6.2.5a), and take account of the fact that co{ + (at = 0, wesee that (6.2.34a) becomes

dt = ih l"

X {(QiQiS(t) - CSCO&K ~ iQiS(t)Qi - SCOC&Kl}, (6.2.34b)which is the desired equation of motion for the reduced density operatorunder the Markoff approximation and we may think of all operators (onlysystem operators appear) as being in the SP under the Markoff approximation.We could call this the Schrodinger-Markoff picture (SMP). The systemoperators obey the same commutation relations under this approximation asthey did originally.

We note that the right side of (6.2.34b) no longer contains time integralsover S(/') for times earlier than the present [as they did in As in (6.2.19)] sothat the future is now indeed determined by the present. We have assumedthat the reservoir correlation time is zero on a time scale in which the systemloses an appreciable amount of its energy. Alternatively, the reservoirspectrum is infinite compared with the bandwidth of the system. We havesmoothed the system fluctuations on a time scale in which the reservoir iscorrelated. One sometimes refers to the Markoff approximation as a coarse-grained averaging.

The reduced density operator equation (6.2.34b) is sometimes called themaster equation. In the event the system hamiltonian has other terms such asW which do not depend on the reservoir, the master equation may be writtenas

X {[QiQiS - (6.2.35)


It should again be noted that all system operators in this form of the masterequation are to be thought of as being in the SP. We therefore imagine that att = t0, the system is in the SP and propagates to time t according to (6.2.35).After the Markoff approximation, we can no longer derive (6.2.35) from atrue hamiltonian according to an equation like (6.1.3).

We are free to make a transformation on (6.2.34) like

S(0 = e-«/»>ff«-««>s(Oew'»ff"-'»), (6.2.36)

which is the IP on a time scale in which the reservoir correlation time is zero.In this case we see that (6.2.35) becomes

| = itwr,s(0]- 2 *««.-»,)

wherex {[fi/fi/s - Q/sQ/W, - [QM,1 - 5f i /O«}. (6-2-37)

e-wn)mt-u)(6.2.38)

which corresponds to going to the interaction picture. We are not free to lett — r0 approach zero here because on the present time scale, t — t0 can becomparable to y~\

Driren Damped Oscillator

Let us apply the analysis above to obtain the master equation for thereduced density operator in the SMP for a driven damped mode of the radia-tion field in a cavity. The unperturbed hamiltonian is

Ho = h(oea*a + 2 Hoitfbj.

We take the interaction energy to be (6.1.8)

2i

(6.2.39)

(6.2.40)

(6.2.41)

We take the driving term to be

W = hW)cf + v*(t)a], (6.2.42)

where v(t) is a classical arbitrary function which excites the Wj-mode by an

In the notation of (6.2.16), we have

Qi = af 6* = a = Q\


external generator. In the IMP we easily see by (6.2.38) that

(6.2.43)

W1 = + av*(t)e-imclt-t'>)]. (6.2.44)

Our first task is to compute the reservoir spectral densities wfj and wjt.We calculate one in detail and leave the others as an exercise. We have by(6.2.27) and (6.2.43) that

w+ = f V ' ^ W F i ) * dr.Jo

In the notation of (6.2.43), we see that

to2 = — coe.Also by (6.2.43)

Butl.m

(6.2.45)

(6.2.46)

(6.2.47)

(6.2.48)

where R = 2 HcOjtfbj. The reader may show easily that

(b\bm) = 6lmnu (6.2.49a)where

and1 ekt - 1

Ay ^ — —

kT

(6.2.49b)

(6.2.49c)

Therefore, (6.2.45) becomes on using these results

< = 2 l«il" *i f V"1— ) r dr> (6.2.50)t Jo

where we interchanged the order of summation and integration. Since weassume the reservoir modes are closely spaced with g((ot) d(ox the number ofmodes between a>t and co, + doau we may change the sum to an integral

2 { } - * ["<*»! g(»i){ .} . (6-2.51)i Jo

346 QUANTUM THEORY OF DAMPING-DENSITY OPERATOR METHODS


w+i = f "dw, g(ca,) WcoJ\*«((«),)/, (6.2.52a)Jo

where we let

/ = ["««<»—>' dr. (6.2.52b)Jo

Nowj "giior j T _ W^(Q) ± i^1— , (6.2.53)Jo ft

where ^* is the Cauchy principle part defined by

C+i f(Q) [ f - ' / W jr»I 0>J-^-J- d£l = lim { I ——- a " •J_« Q «-»o \J-a LI

Therefore, (6.2.52a) becomes

where we have let

and

y — 2TTg(coe) \K(coe)\z

Aco = —& M ' g f l > ' »Jo 0>j — ft>c

1« =

(6.2.56)

(6.2.57)

The imaginary part of w\x represents the Lamb shift in the cavity frequencydue to the coupling to the reservoir and may be generally neglected (seeSection 5.3). The remaining w's are found by the same type arguments andare given by

- i Atoj n = wii*

(6.2.58)

= 0.

6.2 MARKOFF APPROXIMATION IN SCHR0D1NGER PICTURE [2-5] 347

If we use these and (6.2.41) and (6.2.42) in (6.2.35), we find after minoralgebra that

~ = -i(a) eot

a, S] - iv{t)[a\ S] - iv*(t)[a, S]

[2aSaf - a*aS — Seta] + yn[afSa + aSa* - a'aS - Saa*]

(6.2.59)

We therefore see that the only effect of Aw is to change slightly the cavityresonant frequency eo0. The Aco is proportional to |*c|8 by (6.2.56), which is theLamb shift. We may redefine a>e + Aco as co or neglect it. All operators in(6.2.59) are in the SMP.

The first term in (6.2.59) represents the free motion of the cavity mode.The next two terms are due to an external classical generator which drives orpumps the mode. The y terms represent the loss of energy from the system tothe reservoir, while the yn terms represent the diffusion of fluctuations in thereservoir into the system mode. Note by (6.2.57) that as the reservoir ap-proaches absolute zero, h -*• 0.

We may transform (6.2.59) to the IMP. If we let

H' = h{<oe + AeoXa s hcotfa, (6.2.60)

then (6.2.59) becomes by (6.2.36) in the IMP.

j=-fo(O[a' ,s]-i t>*(r)[a",S]ot

+ - [2asaf — a1 as — sa*a] + yii[a*sa + asa* — a*as — saa1],

(6.2.61)

where all operators are now in the IMP. We omitted the superscript / in they and yn terms for simplicity since all these terms contain both an a and an af,and we note that a*itfI = aaf and no confusion is likely to arise.

The master equation (6.2.61) may be solved by various techniques [2].We convert it to an associated c-number equation and solve it in a latersection.

Single Atom with Linewidth [7] Pauli Equations

As a second example of a damped system, we derive the master equationfor a single atom coupled to a reservoir and show how the interaction with thedamping mechanism results in an atomic linewidth. If the reservoir consists ofthe modes of a radiation field in a cavity, we obtain a linewidth in agreement


with the Wigner-Weisskopff approximation for the natural linewidth of theatom (see Section 5.3).

We may use the completeness relation for the energy eigenvectors for thefree atom to represent the hamiltonian by

(6.2.62)

where the kets |/> are the atomic energy eigehkets

H\l)(l\m) Jlm

(6.2.63)

The reservoir is described by the hamiltonian R and V is the interactionenergy between the atom and reservoir. We may use the atomic completenessrelation twice to write V as

V = h^fkl\k){l\, (6.2.64)

where(6-2.65)

are the matrix elements of V in the atom energy representation. They willobviously contain reservoir operators.

As an example, let the reservoir be the radiation field in a cavity in whichthe atom is located. In Appendix J it is shown that in the dipole approxima-tion the atom field interaction is given by

—er«E== — j t « E

* * - * £ ) ( « * • (6.2.66)

where we have used (4.3.44a). In the SP, the field operators are evaluated att — 0 and under the dipole approximation exp ±/k , • r 1. Thus

, - b+)(e,• n«) m {k\V\l), (6.2.67)

where we have let j s (/l5 lt, /3, a) and (tfet a r e the atomic dipole moment

matrix elements. We see that/fc, contains the reservoir operators bt and b]explicitly as we noted. In this case we have that

R = 2 hatfb,. (6.2.68)


In order that V be hermitian, we see by (6.2.64) that

It therefore follows that

In the IP we have thatfkl — flk-

k.l

where we used (3.10.14) and have let

Mt) =and

(6.2.69)

(6.2.70)

(6.2.71)

hcokl = ek — «„ (6.2.72)

and have let t0 = 0 for simplicity.To write the master equation (6.2.37) (here W = 0), we make the following

identifications according to the notation of (6.2.15):

(6.2.73)

(6.2.74)

In the IP the F s are given by (6.2.71) and

The coefficients w± of (6.2.27) become

We leave as an exercise to show that

wmnkl ~ (wlftnm) •

With these associations, the master equation (6.2.37) becomes

(6.2.75a)

(6.2.75b)

7 - ~ I -conn){[\k)(n\dlmS - \m)(n\s\k)(l\]<im«

- [\k) <(|s|m> <n| - s\m }, (6.2.76)


where we used the orthogonality relations (6.2.63). The secular terms arethose for which

*K« + « O = «* - «, + 6« " U - 0, (6.2.77)so that the atomic operators are the same in the IP and SP. If we assume theatomic energy levels are nondegenerate and unevenly spaced, the secularterms arise in the following three cases:

Case 1: fc = n; / = m k ICase 2: k = 1; m = n kj^rnCase 3: k = l — m—n.

In Case 1, the secular terms in (6.2.76) are

~ = 2' {[\D<k\sll)\k)(l\ - \k)WluKndt k.i

+ [\k)(l\s™\l)(k\ - s(1)|/></|[wr*«}. (6.2.78)

The prime on the sum indicates the k <= I term is omitted. We may inter-change the dummy indices k and / in the last two terms. This gives

dsw

wtm - s\k){k\wkltk}. (6-2.79)

\k){k\s™\m)(m\w-nkk)

dt

In Case 2 the secular terms are

dt fc.m

fc.m

(6.2.80)

where again we interchanged dummy indices in the second as well as thethird sums.

In Case 3, we have

= 2{[|m)(m|s (s ) |m><m| - |m)(m|s<3)]w+dt

+ [|m)<m|5(3>|m>(m| - s{z)\m)(m\]w-nmm}. (6.2.81)

We see that the k = / terms in (6.2.79) are identical to (6.2.81). If we there-fore remove the prime on the sums in (6.2.79), (6.2.81) is automatically in-cluded, and if we added this to (6.2.80), we obtain

wlk - \k)(k\sy>tnie - s\k)(k\Wk-Uk}

(6-2.82)

6.2 MARKOFF APPROXIMATION IN SCHRODINGER PICTURE £2-5] 351

where we have let

Wi* = *£« + wkm. (6.2.83)From (6.2.75) it follows that

w«* = »tm + ("&*)*, (6.2.84)so we see that the wa are real.

By inspection we see that the k = I term in the first double sum is just themissing k = I term in the last double sum by means of (6.2.83). Accordingly,we may write (6.2.82) in the alternative equivalent form

at k.i

2 {l*><fc|s|/)</|«ifc + wrlkk) - wtn*\kXk\s - wkms\kXk\}. (6.2.85)

If we take the j , i matrix element of both sides of this equation, use theorthogonality relations (6.2.63), and let

ssi a <;M0, (6.2.86)and

r«* =we obtain

- — °ii 2.Ot k

r~ l i

(6.2.87a)

(6.2.88)

which are called the Pauli master equations. Note that

I V = IV*, (6.2.87b)which follows from (6.2.75b). In the SP, the Pauli master equation becomes

^T - <5« T WikSkk ~ ( IV ~ io>u)SH, (6.2.89)dt k

as the reader may readily show.Before discussing the Pauli equations, let us consider the constants wik and

Ttic to obtain some physical insight as to their significance.

From (6.2.84), (6.2.75), and (6.2.69), the reader may show that

dr.00

We shall evaluate the trace in the /{-representation where

*fol«i> = n,\n,),andy stands for llt l2, /3 and the polarization a. We have

(6.2.90)

(6.2.91)

2 2in) in') (6.2.92)


where |{»}> = \nlt n^,..., O .

2=200 00 CO

and we have inserted a completeness relation before/a. We also used (6.2.71).Since

we have that

xwhere

"}> = g|2 *»*Li

= 2 2{({n}|/«l{n'})<{«'}|/«|{«}>

FlT (1 - c"At)«"*'"'l I « p i\(oki + 2 (om(nm - n'J \r dr, (6.2.95)Li J J-« L TO J

A, - HcoJkT. (6.2.96)If we note that

f"e iXT dr = 2ird(x), (6.2.97)J—00

and use (6.2.69), we obtain

{ r n

Li J „

H+"2 » - ( » - . - O i l (6-2-98)The ^-function is just a statement of energy conservation between the atomand field:

*< + 2 K 4 = «* + 2 *<»»«,»• (6-2.99)m m

If we next insert the interaction (6.2.67), we obtain

d[n (i - «-*

- V«, + 1 n't • • • • n, + 1 • • -

(6-2.100)

6.2 MARKOFF APPROXIMATION IN SCHR0DINGER PICTURE [2-5] 353

If we use the orthogonality relations, the reader may show that this reduces to

"7^2 2 2 «>> l«* *e^JL {n> (n7) I i

• • • n} -

IT (1 - f*)*-i™d[ahu + 2 «•(«» - «;)])- (6.2..101)

Next we may carry out the sums on all the {»'}. This gives because of theorthogonality relations

X {M(a>w+ toy) + (n, + l ) i K - a . , ) l ( l - e-*>-*""}. (6.2.102)

We may next carry out the sum on all {n}. When nx ^ njy we have

(1 - e~x>) 2 e~Xini = 1, (6.2.103)

while for theyth term, we have

(1 - e"A') 2 n ^ " * B h, = [eXl - 1]~*.

Therefore,2 (6.2.104)

2-nh-

(6.2.105)We have restored the sum over modes and polarizations.

Since cot > 0, we have two cases: <ohi < 0 which corresponds to the ab-sorption of radiation by the atom and the atom goes from state k to state 1 ofhigher energy and <oki > 0 which corresponds to the atom going from state kto a state of lower energy 1 with the emission of radiation. Thus for ef > et

(6.2.106a)

is the transition probability per second for the absorption of radiation whilefor €t < ek

~ 2 «>« l«u (6.2.106b)

gives the transition probability per second for spontaneous and inducedemission. These should be compared with the results of Section 5.2.


Consider next the constants i y . These will in general be complex and wemay write them as

i y = Ttj — i Ao)M (6.2.107)where we separate the real and imaginary parts. We see by (6.2.89) that theimaginary part will cause a shift in the atomic frequencies. Since A<ott issmall of order K2 in the perturbation, we either neglect it or define newatomic frequencies including the shift as we did in the case of the dampedoscillator. This is just the Lamb shift.

If we use (6.2.87a) and (6.2.75a) we have

V - f Kdr {-( /«(Jo

+ 2 . (6.2.108)

If we again evaluate the traces in the /{-representation and insert a com-pleteness relation in each term, we have

I V = f"drY 2 (-<{«}|/«|{n'})<{n'}|/«|{n}>rexp i 2 <»,(«< -nfr + cclJo M {«') I L i J

+ 2 [K{"}l/«l{n'}>l2] exp i[coit + Y wjnt

+ KWdMM'exp i[coK - 2 <»/«<-»

X \lJ0- - e-x")e-imn'\ . (6.2.109)

where cc means complex conjugate. Since

Jo

fJo

(6.2.110)

we have for Re iS

2 i( n m ~ n'J~\

<6-2- in>


while the Lamb shifts are

Aeo,-, = — i If-)« [a)H + 2 <»Jnm - n'J

l

If we combine the / = y term from the first /-sum in FM above and the / = *term from the second /-sum with the first term and note that a>ti = msi = 0,we may rewrite F o as

2 2{nl {n'

2 I<WI/*I{«'}>I2

+ 2 K{«}l/«l{«'})l2«[»« + 2 «»„(",» -

X n (1 - e~*r>~"*"*-r

When / =y, we have since//, —fu

ru = r , = 2TT2 2 2 K{«'}l/«l{«}>l2i { } {'}

(6.2.113)

x «L« + 2 ">w(«m - O l IT (1 - «-*'>-*"*. (6.2.114)L m J r

If we compare this with (6.2.98), we see thatr < s r « = 2wjl, (6.2.H5)

That is, Tf is the sum of the probability per second for the atom going fromstate |i) to all other states. We may use this to rewrite (6.2.113) as

(6.2.116)where

r,i = *(rf + r,) + i y \

' 2 2 (l<{»'}l(/«-/«)!{«}>l2

(nj In') V

x ($[2 «>».(».» ~ O l IT (1 - e-x')e-x""}. (6.2.117)

For the particular interaction we have chosen i y * = 0. If we went to higherorder in the interactions and took into account elastic scattering (see Chapter5) we would obtain a i y * 0. A phenomenological term of the form

bn (6.2.118)

1V


would correspond to the absorption of a photon of frequency con and theemission of one at frequency <om with the atom remaining in state |/). Thiswould correspond to the elastic scattering of a photon by an atom through avirtual (intermediate state). If we let

K, (62.119)

then

and

l.m.n

(6.2.120)m.n

V * = 2 2 [KM 2(««« - *iln)b\b j{n»|8{nJ (n'}\ l.m

5I.m

1) <M*

- o] na - (6.2.121)

(6.2.123)

which is not zero in general. Such inelastic scattering interrupts the phasecoherence in that the atom goes through some virtual state which interruptsits phase. Atomic collisions could also contribute to F,/*.

By inspection of (6.2.116) and (6.2.117) we see thatr,, = r ,<>0 . (6.2.122)

Let us return to the master equation (6.2.89). When i j£j, we have

^* = -(i\, - toys*,ot

where(o'u = mi} + L<oif. (6.2.124)

The solution of (6.2.123) isSH{t) = e-(r"^<o«'»S«(0), (6.2.125)

so that the off-diagonal elements of the density matrix decay with a relaxa-tion time P^-1 by atoms leaving both state/ and i as well as by elastic proc-esses which interrupt the phase of the atomic motion.

The diagonal elements i —j of (6.2.89) satisfy the Pauli or rate equations

dt -TtSu +

(6.2.126)


where we used (6.2.115). This states that the time rate of change of the prob-ability of finding the atom in state i equals the rate of loss of atoms in statei going to all other allowed states plus the rate at which atoms in all otherstates are entering state i. Unfortunately, these represent an infinite set ofcoupled equations which we cannot solve in general.

In the steady state (superscript ss), the rates in and rates out of each levelare equal. Then

Btso that

*This will have a nontrivial solution if the determinant

| r ,d a -w a ( i -That is,

o.

(6.2.127)

(6.2.128)

(6.2.129a)

(6.2.129b)

However, we may easily show that this determinant is automatically zero andtherefore does not represent any new relation between the F's and w's. To seethis we note that since F, = Y wu, if we add every row to the first row and

leave all the other rows the same, every element in the first row equals zero.In the steady state we expect the atom to come into thermal equilibrium

with the reservoir at temperature T. We then have that

V = \l *~fu I V". (6.2.130)where ft = (kTy-\ That is, the atom will be distributed over its possiblestates with a Boltzmann distribution. Since Tr S — 1, we see that Y, £«M = 1.If we use (6.2.128), we see that

T w, (6.2.131)

Driven Two Level Atom. The Bloch Equations

To gain further physical insight into the meaning of our master equationfor a damped atom, let us specialize to the case of a damped two level atom inthe presence of an externally applied electric field. The energy of interactionbetween the atom and field by (6.2.66) is given by

W=-er-E(r,t), (6.2.132)where ex is the atomic dipole moment operator and E(r, t) is the applied


electric field: •E(r, 0 - i / ^ #[be«k-k-*"> - « ] . (6.2.133)

We assume the field is so highly excited in the cavity that we may treat itclassically. Also we assume that the wavelength is long compared with theatomic dimensions so that we may replace exp ±ik • r s= 1. If we use thecompleteness relation twice for the two levels,

UXU + |2X2| - 1, (6.2.134)and assume that the atom has no permanent dipole moment

<l|r|l> = <2|r|2) = 0,then (6.2.132) may be written as

= -ti12.E(0|l><2|where

and

The b and 6* are classical amplitudes.In the IP, we have by (6.2.74) that

where

e<l|r|2> = e<2|r|l>*,

(6.2.135)

(6.2.136)

(6.2.137)

e-to»tit

(6.2.138)—ha)lt — hcoa ss e2 — «x > 0.If we therefore retain only the terms in (6.2.135) of the form ±(a> — cujj)and neglect those of the form ±(et> + WJI) (rotating wave approximation),W reduces to

Ws* -h[v*{f)\\){2\ + P(OI2><1|), (6.2.139)where we let

CO „-*»*voe~"". (6.2.140)

If we use (6.2.35) and (6.2.85), the density operator equation becomes

»*«[|1><2|,S]

iv(t)[\2Xl\fS] + Jt'Wn\lKk\S\k)(l\

-wr l t tsifc><fei}.(6.2.141)

2


Note by virtue of (6.2.74) and the retention of secular terms that the form of(6.2.85) is unchanged in the SP except for replacing s by S. If only two levelsare involved the sums go from 1 to 2 only.

If we take they, / matrix elements of (6.2.141) we obtain the equation Sn

d 5and

»21

dt

iv*(t)Sa - iv(t)Sia - T ^

r2js21 -(6.2.142)

Here we have used (6.2.115) which for only two levels reduces to

r 2 = w12.(6.2.143)

(6.2.144)

In addition we have thato2 1 = S12

1 = S u + 522.There are three operators of interest: the population difference;

az . |2><2| - UXH, (6.2.145)

and the dipole moment operators;

(6.2.146)<r_ = |1><2|.

We showed in (3.10.17) that these operators could be put in one-to-onecorrespondence with the Pauli spin-J- operators.

The mean value of these operators is given by

(<r+> = Tr 5(012X11 =

= Tr 5(0[|2><2| -= Tr 5(0[|2)<2| +

(6.2.147)

Sn(t).

If we use these and (6.2.142) and (6.2.144) above, we obtain the mean equa-tions of motion

- r21)<a+>

- iv(t)(at) (6.2.148)

Y,


These should be recognized as the Bloch equations. If we let

<*± - lip* ± toy) or

we may write them in the more familiar form

(6.2.149)

x H], -O - cr.)

(6.2.150)

if we make the identifications

(6.2.151)

W12

w- * . - '21 "I" Wjl

pt is of interest to compare these with (3.17.30).] We therefore see that a twolevel atom in the presence of a classical driving field coupled to a reservoir isequivalent to a spin-£ particle in the presence of a d-c field along the z axisand an rf field in the x-y plane. The T2 is commonly called the spin-spin ortransverse relaxation time which contains damping due to phase interruption(IV*). The Tt is called the spin-lattice or longitudinal relaxation time whenthe spin interacts with phonons. These equations have been discussed verythoroughly in many places. The main point that we wish to emphasize is thatthey are a special case of our present treatment of damping and we haveobtained the relaxation constants explicitly.

63 THE MARKOFF APPROXIMATION IN THE HEISENBERGPICTURE [7]

Equation of Motion for System OperatorWe have shown that the mean value of a system operator M may be

evaluated in either the SP(6.3.1)

where we used (6.2.3) and where 5(0 is the reduced density operator

(6.3.2)

6.3 MARKOFF APPROXIMATION IN HEISENBERG PICTURE [7]

or in the HPTrB , s P(t0)M(t)

361

where M(t) is in the HP, and we have let

K'o) = S(to)fo(R)and

{M(t))R = TrR fo(R)M(t).

(6.3.3)

(6.3.4)

(6.3.5)

In the prior section we obtained an equation of motion for S(t) under theMarkoff approximation. We may use this as follows to obtain the meanequation of motion for {M(t)) under this approximation. From (6.3.1) and(6.2.34b) we have

dt * dt

\ [H, S(t)] -

(6.3.6)

where all operators are in the SMP. We may next use the cyclic properties oftraces to move S(t) to the left in each term. This gives after slight rearrange-ment

(6.3.7)X ([Af, jwhich is the equation of motion for (M(t)) under the Markoff approximationin the SP.

Let us next obtain the mean equation of motion for <M(0) directly from(6.3.3) or (6.3.5) where we make the Markoff approximation in the HP. Inthe HP, MB(t) obeys the equation of motion

where by (6.1.2)V.

If HT is time-independent, it follows that

W B zi 8

(6.3.8)

(6.3.9)

(6.3.10)


That is, the total hamiltonian in the HP is equal to the total hamiltonian inthe SP. Therefore, (6.3.8) becomes

(6.3.11)

(6.3.12)

dt ih

If we next make the transformation

MB(f) =we see that

dt \ih

J- \MH(t), Ho8]

ih1 w dt

If we use (6.3.12) and (6.3.13), we see that (6.3.11) reduces to

whereT^TlMO-n'o-O]dt ih

V(t — t) = g-««/»>JEr»S('-*»)Vsre(*/a)Ho('~'').

) ( 6 3 1

(6.3.14)

(6.3.15)

We therefore see that the transformation (6.3.12) is analogous to transformingto the IP in that it has removed the high-frequency free (unperturbed)motion [MH, Ho

8] term from (6.3.11). We would therefore like to make theMarkoff approximation on the exact equation (6.3.14) in the same way we didfor the reduced density operator 5(0 in the last section.

We begin by writing the iterated solution of (6.3.14) up to second order inFas

m(t) - 01

[[m(t9), V(t0 - T ) ] , V(t0 - O]. (6.3.16)

(6.3.17)

As in the prior section, we assume that

Tl » t - t0 » TC

* - to » wf1,

as far as the behavior of m(i) is concerned. At time tQ, we have that

m{to)-M8, (6.3.18)

6.3 MARKOFF APPROXIMATION IN HEISENBERG PICTURE [7] 363

which is the system operator in the SP. Therefore, in (6.3.16) we have

m(i) — m(t0) _ Am

t — ta ~ At

+ \t)t\dt'\ df[[M8,V(t0-niV(t0-t')], (6.3.19)\ih/ AtJt Jitwhere as far as the system is concerned

At = t-t0 (6.3.20)

is a small time by virtue of (6.3.17). Under this approximation we may re-write the first form of (6.3.13) as

^ e , . ( I [Mt HQ] +dt [ih At)

since / -*• t0 as far as the system behavior is concerned and in the [m(t), Ho8]

term m(t)^m(t0) = M8 [the corrections are give by (6.3.19)].For use in the next chapter, let us define gm(t) by means of

gjt) = - [Ms, V(t0 - 0] = ~ [M8, e-<i"l>B''lt-MV8eu/*>H''<t-«)],ih ih

(6.3.22)which we shall tentatively call the Langevin force. With this notation (6.3.18)may be written as

tr - i &«dt'+{TJ i i[M8-(6.3.23)

Since Ho<= H + R and [H, R] = 0, we see that if we multiply both sidesof (6.3.21) by the reservoir equilibrium density operator fo(R), trace over thereservoir and use the cyclic property of traces for reservoir operators, weobtain

since

± [M8, H8]ih At/s

(6.3.24)

and TTli{ }fo(R), (6.3.25)

TrB [M8, R)fo(R) = [M8, Tr^ Rfo(R)] = 0. (6.3.26)

We therefore need the thermal average (Am/At)R. If we use (6.2.15), (6.2.17),and (6.2.23) we see that

V(t0 - 0 = - t), (6.3.27)


so that by (6.3.23) we have

\A*/jR

X <[[MS, Q<8Ffi, QfF'ftn exp iK(f0 - f) + a>,(*o - O l (6-3.28)

and we have let ft = /"<(*„ — t") and F'j s F,(f0 — *')• Also we have

(6.3.29)

(6.3.30)

(6.3.31)

If we use this, expand the double commutator in (6.3.28) and use argumentsas we did in (6.2.20), we readily obtain

fAt

so that on using the cyclic property of traces, we have

<*•.«>* = - * 1 1 * * 8 ,i

sinceF,(f0 - t) = TrB

- i f , \*df f rff {[Ms,

X exp

If we use (6.2.21), we obtain

- O + w,(f0 - 01- (6-3.32)

" 2 f H t t f <~'°" dr{[Ms,i.i At Jo Jo

exp - i(6.3.33)

If we now use (6.3.17), we see as in the argument leading to (6.2.32) that

(6.3.33) gives

%T) = - I « ( » « . -«>,){[M, ejQ,<, - QAM, fijwrj, (6.3.34)At/


where all operators are system operators in the SMP and w± are defined by(6.2.27), since

(6.3.35)

This follows since

if fi>, = 0, *. dx=\~

&*J* 10 if ft),Ô,when At»(of1. Since no cof — 0 for system operators in the IP, (6.3.35)follows.

If we next substitute (6.3.34) into (6.3.24) and insert

between appropriate factors, we obtain

AJ <MH{t))R e* A

.-Awhere we have let

» (6-3.36)

*> (6.3.37)be any time-dependent operator appearing in (6.3.36). These operators arethe Heisenberg operators under the Markoff approximation and we could, forconvenience, say they are in the Heisenberg-Markoff picture (HMP). Theyobey the same commutation relations as the original Heisenberg operators.

If we multiply both sides of (6.3.36) by S(t0) and trace over the system, weobtain

dt _ Tr

(6-3.38)

where all operators are in the HMP.Just as we may evaluate means exactly in the SP or HP

(Af(0> - TrsS(t)M = Tr s S(to){MH(t))R, (6.3.39a)we may also evaluate under the Markoff approximation

<M(/)> as Trs S(t)M = Tr S(to)M(t), (6.3.39b)where 5(0 satisfies (6.2.34b) and M(t) here satisfies (6.3.36). Also the readershould compare (6.3.38) with (6.3.7) and note that the functional forms of


these two equations are identical just as they would be under the exact trans-formation from the SP to the HP. This is the analog of thinking of a fixedcoordinate system and rotating vectors as being equivalent to a rotatingcoordinate system and fixed vectors. This invariance of form will be im-portant in our later work.

Driven Damped OscillatorAs a first example, let us consider again the damped driven oscillator. The

total hamiltonian by (6.2.39)-(6.2.42) is

HT = H0+V+W. ^ (6.3.40)

In this case (6.3.36) reduces to

^ ^ = <-toe[M, afa] - iv(t)[M, a+] - iv*(t)[M, a) - [M, aâwât

+ a[M, a V S ~ W, a ] a V i + ^[M, a]WZ>& (6.3.41)where all operators are in the HMP (Heisenberg-Markoff picture) and where

M^(MB(t))R. (6.3.42)

The w's are given by (6.2.58). If we use this and the identity[M, a*a] = [M, af)a + a*[M, a], (6.3.43)

then (6.3.41) reduces to

\dt\= (-Ht)[M>flt] ~ lV(0[M' fl] " [K(O° + Ao)) + ^\[Mt °t]a

- [i(«c + A«)) - ^\a\M, a] + yn[a, [M, a ^ , (6.3.44)

and we have used the fact that[a,[M,J]] = [a*,lM,a)]. (6.3.45)

If we make use of the commutation relations

~(6.3.46)

we may rewrite (6.3.44) as

r yl *dM . 92M


We emphasize again that all system operators are in the HMP which repre-sents a reservoir thermal average under the Markoff approximation. If con-fusion is likely, we use a superscript M to represent the HMP or ( ) s .

In case M — a, we see that

Jt (")R = ~Ht) - [i(coe + Aca) + R, (6.3.48)

where we have emphasized that these are Markoff operators. Similarly, if welet M — a*a, we see that

j (aa)R iv(i)(a% + iv*(t)(a)R - y[(a'a)R - n]. (6.3.49)at

The solution of (6.3.48) is easily seen to be

< ' > (6.3.50)where we have let

co' = coe + Aw

g(t) (6.3.51)

If we substitute (6.3.50) and its adjoint into (6.3.49) and solve, we obtain

tft' v\f) exp {y(r' - 0 - (to' +

^t' v(t') exp {y(*' - 0 + (to'

+ \*df e«*-*[u(f)s?(n + v*(t')g(t% (6.3.52)

for the thermal average of the mean number of photons in the cavity mode.

Atom with Linewidth

Let us next apply (6.3.36) to the damped atom considered in the last section.If we again use (6.2.73), (6.2.62), and (6.2.75), we obtain

\ 2 2 mn)in I Jelmn

X <{[M, \kXl\]\>n)<n\wtlmn - \m)(n\[M, \k)(l\]W-nkl)R. (6.3.53)


If we retain only the secular terms (6.2.77) for which (a) k = n, I = m(k ?* /)» (P) k — h vn. — n (k # m), and (c) k — I = m = n, this reduces to

ih h.l

2 {(*«*Jc.l

, (6.3.54)- wk+

l(k(M\k)(k\)R -

where we used (6.2.84).If M s \i)(j\ and we use the orthogonality relations (6.2.63), we obtain

<IOU

+ [«, +or when we use (6.2.87a)

\}<»u ~

+ w^jdi)01>«, (6.3.55)

(IOU. (6.3.56)

These results are useful when we present the Langevin method in the nextchapter. All operators are in the HMP.

6.4 ONE-TIME AVERAGES USING ASSOCIATEDDISTRIBUTION FUNCTIONS [8-10]

In the prior section, we have shown that under the Markoff approximationwe may evaluate one-time averages of system operators in either the SP orHP. In the former case, we must solve the master equation (6.2.34b):

x {(&Q,S - QjSQJwt, - {QtSQ, - SQ&dwTt}, (6.4.1)where all operators are in the SMP. That is, the H, Qif and Qf are systemoperators at t0 and S is at time t.

A formal solution of this operator equation/may be written as

(6.4.2)

solution

Xttt[S(t0)],

where Xttt is a linear functional of S(t0) which plays the role of the exact

p(0 - . V(t, to)P(to)U-\t, f0), (6.4.3a)

6.4 ONE-TIME AVERAGES USING FUNCTIONS [8-10]

where

U(t, t0) = Uth = exp - { HT(t - Q,ft

369

(6.4.3b)

when HT is time-independent.The operators in the SMP obey the same commutation relations they did

before we made the Markoff approximation.Once we have a solution for S(t) in the form (6.4.2), the mean of a function

of system operators alt a2,... ,at s a evaluated at t0 in the SMP is given by(6.3.39b)

{M{at t)) = Trs S(t)M{a) « Trs Xth [S(to)]M(a) (6.4.4)

In the HMP, we must solve the operator equation (6.3.36)

^f = 1 [M, H] -at in ,e,]*v,l}, (6.4.5)

where all operators are in the HMP at time t and

M[qW^{MB[a{i)])R- (6A6)The mean of M by (6.3.39b) is then given by

<M(r)> = Tr S(te)M[a(t)]. (6.4.7)

Just as we may use the cyclic property of traces to show in the exact casethat

<M(0> = Tr P (t)Ms(t0) = Tr UttopH(to)U^M8(to)

= Tr PH(t0)U£M8(t0)Uttt = Tr P

H(t0)MH(t), (6.4.8)

we may use the cyclic property of traces to rewrite (6.4.4) as

(M(a, 0> = Tr Xtto[SB(to)[M

s[a(to)}

= TrSH(t0)2tto{M8la(t0)]}

= Tr SH(t0)MH[a(t)], (6.4.9)

where the last step follows from (6.4.7). Therefore, a formal solution of (6.4.5)may be written as

M*[a(t)] = Xtt9{Ms[a(t0))}, (6.4.10)

where XiU is the "transpose" of XtH. Actually the solution of (6.4.5) definesXtw

In both the SMP and the HMP we must solve operator equations. In thissection we would like to take advantage of the quantum-"classical" corre-spondence that we developed in Chapter 3, Section 3.9 to transform theoperator equations above to equivalent c-number equations so that classical

370 QUANTUM THEORY O F D A T O M N G * - £ > B N S I T Y OPERATORMETHODS

mathematical methods of solution become available. We begin in the SMPwith the density operator equation.

Equation of Motion for Associated Distribution Function

If au a2,..., af = a are a complete set of system operators in the SMPand we choose the ordering 1, 2 , . . . , / a s in Section 3.9, then by (3.9.3),(3.9.4), (3.9.22), and (3.9.23), (6.4.4) becomes

(Me[a(t0), t]) = Tr S(t)Me[a(t0)} = <#{i«?c[a(f0)]»

(6.4.11)

where we have let a0 = a(f0), rfa0 = I I rfa^f,,) and

Pc(So, 0 = Tr SiOWcco - fl0) = (W«t - flo)>. (6.4.12)

and a0 = a(t0). The <5c(oo — o0) is the product of 5-functions in c-order. Aswe have noted, Pc(a0, i) is the associated distribution function when systemoperators are in c-order. If M is in c-order, the quantum one-time average in(6.4.11) looks like a classical average if Pe were a classical probability distri-bution function. We would therefore like to obtain directly an equation ofmotion for Pe.

If we differentiate (6.4.12), we obtain

^ (oc0, 0 - Tr ^ <5at at

If we use (6.3.7) and let Mc(a0)

!T(ao,0 = 7<5c(aoot at

ao - a0) « - j (d'^ - ao)>.at

<5c(2o — ?o). we obtain

(6.4.13)

Tr ^ [<5c(a0 - a0), H]-in ~

. (6.4.14)

We easily see that we may express the <5-function as

i r°°a(a - a) = -

2ir J_o

\—oo

± d f f ;27T J-CO ntg 71! 9a* lit J-00

(6.4.15a)

€A ONE-TIME AVERAGES USING EUNGTIONS [8rl0]

where the steps are self-explanatory. Then since

*e(«o - So) = I I e-*"'** <J(a<0) = e-*

(6.4.14) becomes

^(ao ,0 = Tr5(0f^K^/'~"),/f]

371

(6.4.15b)

Tr , go) *Bo) . (6.4.16)

We show by several examples below that we may use the commutationrelations to put all operators in L' into chosen order so that L' — L'e. Thenwe may proceed, to rewrite (6.4.16) as follows.

Tr

From the form of U in (6.4.16), we see that we may always rewrite L'as

Thus

But

so that

- aao

dPedPt r— = Tr S(OJ

We illustrate below how the intermediate steps above may be taken. If wethe definition (6.4.12), this reduces to

(6.4.17b)

use

~ * (oo, 0 - l \ ^ , Z o, 0, (6.4.18)


which is the desired c-number equation for the distribution function. Ingeneral, all derivatives with respect to a,(/0) occur. In many cases of interestthose derivatives higher than the second are missing or are small and (6.4.18)then reduces to the Fokker-Planck equation

(6.4.19)

where the a's refer to f0 and we sum over repeated indices from 1 to/ . This isthe c-number equivalent equation of the master equation for S(f).

Damped OscillatorTo illustrate this procedure we consider the first term on the right side of

(6.4.16) for a damped oscillator where chosen order is normal order.Then

J = -ico Tr

= -ico Tr S(r){*(6.4.20)

We put the terms in the curly bracket into normal order. Since [a, F] =dF/da* and [a \ F] = —dFjda, we see that

(6.4.21)

If we use these in (6.4.20), we obtain

= +ito Tr | - a - -^-9a 9a*

), (6.4.22)

which is now in normal order. We may therefore use (3.9.3) and (3.9.4) to

write this as

- a

By arguments like those in (6.4.15a) it follows that

. (6.4.23)

(6.4.24)


and we have

373

3aa - a)

~ af*aaf* *] P c ( a ' a *' °' ( 6 A 2 5 )

which is in the required form.In the case of a damped oscillator, there is a close connection between the

associated distribution function Pe when the chosen order is normal order andthe antinormally ordered function associated with the density operator. Thisconnection may be obtained as follows. The 5-function in normal order is

<5<"> = 6(OL* - a^dioL - a). (6.4.26)

If \fi) is a coherent state where F(a)\fi) = F(fi)\p), then the normally orderedfunction associated with din) is by (3.2.12)

</3|(5(a* - a*) <5(a - a)\p) = (5(a* - $*) <5(a - /?). (6.4.27)

If we assume the density operator is in antinormal order, we have by (6.4.12)

P (nJ(a, a*, 0 = Tr Sla\a, a\ t) <5(a* - af) <5(a - a)

JJ S*, 0 <5(a* - (6.4.28)

where we used (3.2.34) to evaluate the trace. The Sla)0, 0*, t) is the functionassociated with the reduced system density operator after it has been put inantinormal order. We also used the normally ordered function (6.4.27)associated with the <5-functions in normal order. If we carry out the ft integralsin (6.4.28) we see that

IT(6.4.29)

which shows the desired connection between the associated distributionfunction for normal ordering and the function associated with the densityoperator in antinormal order. We therefore see that

5<°)(a, a\ t) = ^{^«>(a, a*, t)} = ^{/> ( f l )(a, a*, t)}. (6.4.30)

By (3.2.23) we see thatP(B)(a, a*, /) is identical to the P-representation of thereduced density operator.

Rather than go through the formalism above, we obtain the equation ofmotion of P(n)(a, a*, t) directly as follows. By (6.2.59), the reduced density


operator obeys the equation of motion

i, S]} - Ht)[<*\ S] - iv*(t)[a, S]

- Sa'a] + yn[a1Sa + aSa1 - a'aS - Sao1],

(6.4.31)

where we used the identity

[^c, S] = [a\ S]a + a*[a, S]. (6.4.32)

Let us now assume we have somehow managed to put S(a, af, t) into anti-normal order and proceed by means of (3.3.9)

as ( o >

f£ = -i(coe + Aco){[a\dt

S(a)a

(6.4.33)

to put all terms in (6.4.31) into antinormal order. We have for example that

= aSia)a' -a^p-- S(°\da

(6.4.34)

and it follows that all terms on the right of (6.4.34) are in antinormal order.If we proceed in this way, (6.4.31) may be rewritten in the equivalent form

_ = _ I f t ) e J _ f l _ + _ f l J + _ | _ ( f l S ) + _(S a)J> ( 6 A 3 5 )

where we have letco'e = (oe + Aw. (6.4.36)

Since now every term is in antinormal order, we may apply the sSf-1 opera-tor to both sides and use (3.2.10) which gives

yh dado

6.4 ONE-TIME AVERAGES USING FUNCTIONS [8-10] 375

where we used (6.4.30). An equation of this type is known as a Fokker-Planck equation in agreement with (6.4.19). In Appendix K we discuss a fewof the properties of such equations and we solve (6.4.37) in the followingsection. It is of course identical to (6.4.19) for the damped driven oscillatorby virtue of (6.4.30).

We again remind the reader that if M(a, a1) is an arbitrary function of aand af in the SP, then by (3.2.34), the average under the Markoff approxi-mation is

(M(a, a\ 0> = , a* (6.4.38)

where Mln) is the function associated with M in the chosen (here normal)order.

Homogeneously Broadened Three Level Atoms. We wish to obtain the equa-tion of motion for the associated distribution function for homogeneouslybroadened atoms.

The equation of motion for the reduced density operator for an atomunder the Markoff approximation in the IP is given by (6.2.85). If we use(6.2.36) and (6.2.74), this equation in the SP becomes

2

+ 2 {\k)(k\S\l)(l\(wtlkk + «w) - w+kllk\k)(k\S - wkllkS\k)(k\}.

(6.4.39)The imaginary parts of the w^'s cause small shifts in the energy levels of the

atom which we may neglect for simplicity and retain only their real parts. Ifwe use (6.2.75b) and (6.2.84) we see that

km(6.4.40)

. (6.4.41)

Re *W = Re wBy (6.2.87a) we see that

Re (*,+ * + wfw) = - R e T^ + Re 2 (w+k

m

If we use (6.2.107) and (6.4.40), this becomes

Re (wf«* + MW) = -Tlk + i 2 (",»* + O . (6-4-42)m

If we next use (6.2.115) and (6.2.116), we see that (6.4.42) reduces to

Re (M&» + wTlkk) = - i y * + \Wkk + iw,,. (6.4.43)


If we use (6.4.40) and (6.4.43), (6.4.39) becomes

- = lî1 IMHdt i ih

Since

+ 2 ' Wm\lHH\S\k)<!\

2

we may simplify the wM and w,, term. Also we have that

Ttt*h - I V * - 0

by (6.2.117). Accordingly, we may write (6.4.44) as

Jt - 2««^ [MH S] + J w

(6.4.44)

(6.4.45)

(6.4.46)

- 2' 2 ***

In the term J 2 ^ H ^ I M I w e m a y change

and combine the last four terms. This leaves

(6.4.47)

dummy summation index to k

f- 2'

(6.4.48)

If we use (6.2.115) this may be written in the alternative equivalent form

dt ih i ' *i

+ 2 ' wi*!*) <felslfe> < l - 2 — [lfc> <fcls + slfe> <fcN' (6-4-49)

Suppose next that we have a large number N of identical independentatoms, each coupled to its own reservoir. We assume the damping of allatoms is the same and we say the atoms are homogeneously broadened. Inthis case if we let (\k)(l\)x be an operator for the Ath atom, (6.4.49) becomes

(6.4.50)2 ' [w

€.4 ONE-TIME AVERAGES USING FUNCTIONS [8-10] 377

The S is of course in a Hilbert space corresponding to N atoms so that a statevector would be written as |«i) \n2) • • • |/ijv)» a product of state vectors foreach atom.

The distribution function by (6.4.13) and (6.4.50) for the JV atoms obeysthe equation of motion

dtTr [(HI

2 ' ["•k.l

where we used the cyclic property of traces and <5C is the d-function productin a chosen order which we must yet specify.

Let us now specialize to the case of N three level atoms which we laterneed for the theory of a laser. In this case the sums above will run from theground level which we label 0 to the upper level which we label 2. In theapplications we use the ground level only as a source of atoms which cansomehow be pumped up to levels 1 and 2 as well as a sink into which atomsmay decay. We are primarily concerned with the interaction of these atomswith radiation of frequency a> ^ (c2 — e^fh which induces radiative transi-tions between levels 1 and 2. Accordingly, we are interested in the populationsof levels 0, 1, and 2 and the dipole moment of the atoms induced by theradiation field between levels 1 and 2. The operators of physical interest aretherefore the level populations

where it is observed that2 (6.4.52)

- t f , (6-4.53)(=0 A-l (=0

by virtue of the completeness relation and the dipole moment operators

(6-4.54)

These operators have the property effectively of "annihilating" an atom inlevel 2 and "creating" one simultaneously in level 1 for M and vice versa forMf and summing for all N atoms. These transitions are physically necessaryto induce a dipole moment. For our problem a complete set of operators istherefore M\ Nu JV2, M since No — N — ATj - Nt and we choose the order-ing defined by

de = b{Ji* - M^dlsV^ - N^8{jri _ N*)d{J[ - M). (6.4.55)

378 QUANTUM THEORY OF DAMPING—DENSITY OPERATOR* METHODS

The script variables play the role of the various a's in the formal theory above.By (6.4.15) we have

d(«. - a) = r*UM 3(a), (6.4.56)where a is a c-number and a is an operator. Accordingly, we may write (6.4.55)in the very useful form

(6.4.57)Note that Jt*y Jt, JTX, and JTX are now considered as independent c-number variables.

We next write out (6.4.50) explicitly for three level atoms. If we measureenergy from the ground state, then e0 = 0. Also we use the completenessrelation (6.4.53). This gives on using (6.4.52)-(6.4.54) after minor algebra

)Ad°(\2)(2\)x - ri2ph(|2)<2|)/ld<(|1><1J|]J. (6.4.58)

We have used the result (6.2.117) that rMph — Ta

v\ Also the terms propor-tional to r V h = I V * vanish since for example

fllXlD* *(|0><ODa - (UXH0X0I), 6* = 0.This follows since from the definition of the operators M, M*, Nlt and Nt

and the orthogonality relationsMDx = «« (6.4.59)

for the Ath atom, we see that (|0)<0|)A commutes with all operators in dc and<l|0> = 0.

In the first term in (6.4.58) we should have

- ^ {de No + No 6e} = - -Nt- Ns) + (N-N1-Nt),

In our problem it is reasonable to assume that iVi «JV and JV2 « N withoutrequiring that JV and Nt be small. We have made this -approximation.Physically we are assuming that there is such a large supply of atoms in theground state that we may neglect depletion of the ground state when atomsare pumped into levels 1 and 2.


The next step in the procedure is to put all terms on the right side of (6.4.58)into the chosen order. We must first bring terms like (|2>(1|X, and (UX2IX, inthe wlt terms together and reduce them to bilinear form by means of theabove orthogonality relation (6.4.59). This process is quite tedious and wecarry out the details for two terms and refer the reader to Ref. 8 for furtherdetails.

Consider the term

x 6(J?*) (J(^"i) M Q K-#)> (6-4.60)where we used the representation (6.4.57) for d" and the fact that Nt com-mutes with the c-number ^-functions. We wish to commute Nt through theM and N2 terms to its left. Since

we may write (6.4.60) as1,

Consider the term

(6.4.61)

( 6 A 6 2 )

(6.4.63)

To prove this, we first note that since operators for different atoms commute,we have the commutation relation

x

m(6.4.64)

(6.4.65)

(6.4.66)

d£

= e^Me5* = M, (6.4.67)where we used (6.4.64). If we integrate this and use (6.4.66), we obtain

/ « ) - JVi + W. (6A68)

Consider the function

where

If we differentiate /(£) with respect to f, we have


If we identity f with dfiJC, (6.4.63) follows. Accordingly, (6.4.62) reduces to

I = Tr s(t)e~ml>/ijr*}e~Iflli/i-ri)e~NtWijr*)\St + M —~\—~\e~mil9Jl)

X W) Wd K-^z) K"#)- (6-4.69);The M(d{dJif) term is now in chosen order. Also it is easy to show that

[Nlt JVjJ -> 0. (6.4.70)Therefore,

x d(je*) diJ^J d(JT2) 6{JT), (6.4.71)

and we have put all operators into chosen order. We may therefore use(3.9.3) and (3.9.4) to write this as

/ = Tr S(f) f• • • f[e-^'w***Je-*1'ld'*^l)Jrte--"t'{m-"'

x d(J(*) XJVJ at/T8) d(JT)] HJl*1 - M1) Wi -

x W - M) &J(' dî dJT'i + ^- Tr S(t) f• • • f [e-**'""-**1

(6.4.72)where we replaced the operators by c-numbers designated by primes andmultiplied by ^-functions in chosen order. Now by (6.4.15), it follows that

e-.'O/a«) a ) = ^ _ a ) . (6.4.73)

If we also use the definition (6.4.12), we see that (6.4.72) reduces to

I - f- • Wai*' - Ji*) diyV', - J"x) Wt - ^^

*', Jî, î, JT, t) d*JC dJT'x

dJT'

i + ~ Jl, t), (6.4.74)


which is the desired result for this term.Consider next the term

381

= Tr S(r

X (11X21), <J(^*) K^d K^z) K-^)- (6.4.75)

Since operators for different atoms commute, we may use (6.4.52)-(6.4.54) towrite this as

X JJ [e

(6.4.76)

where we have let( 6 A 7 7 )

Our first task is to commute |1>(2| and |2)<1| through the various terms un-til they come together at the middle. Now / may be obviously rewritten as

Tr S(t) IJ' T 2

since, for example,

and

Consider the term

(6.4.80)


where we expanded the exponential operators and used the orthogonalityrelations. Similarly, we see that

c-|2><«|<a/^,)|1)<2|e|2><«|«/M',) = ea/a^,|1)<2 |# ( 6 4 g l )

If we use these (6.4.79) reduces to

/ « Tr S(

x e,-MO/dJT) (6.4.82)

where the intermediate steps should be self-explanatory. Since this is now inchosen order, we see by arguments like those used in (6.4.72>-(6.4.74) that

fxfJrstj(ti), (6A.Z3)

which is the desired form of the term.If we continue in this way with all terms, the result is

. ry / a a 1+ WaLlexp fcr" ~ W*

(expexp

J


(expa^) a*

where we have let

Xt^Nwa i = l , 2 (6.4.85)

be the rate at which atoms are pumped by the reservoir from level 0 to level/. If we note by (6.2.116) that

and by (6.2.115) that(6.4.86)

= w01

and if we add and subtract the terms(6.4.87)

to (6.4.84), we find after minor rearrangement that the equation for theassociated distribution function becomes

t[fa ~£r) ~[fa -Ur) ~


-\h*

where we let

(6.4.88)

(6.4.89)

and we have again neglected Jfx and JVZ compared with N which means wehave neglected depletion of the ground state. The Jfx and Jft may still belarge compared with unity. Since derivatives of all orders are involved, wesay this is a generalized Fokker-Planck equation. Since Jfx, Jf* Jf, andM* are large, we see, for example, that

a \ , a , I a* I a3 ,

and the higher derivative terms may be neglected since they are down byorder JTX~X. If we retain only terms up to second order, we obtain theFokker-Planck equation [see (6.4.19)].

dt


a ,„ n //. .

385

l +

a8

a2

a2

2 dJl*a2

(6.4.90)

The first derivative terms give the drift or mean motion while the secondderivatives give the diffusion as shown in Appendix K and below.

We have therefore shown that the equation of motion for the c-numberdistribution function obeys a Fokker-Planck equation for a damped harmonicoscillator and for TV three level atoms when we neglect depletion of atoms inthe ground state.

Equation of Motion for System Operator

We may also use our ordering techniques to convert the equation of motionfor system operators in the HP under the Markoff approximation to c-number equations.

We begin by considering again (6.3.7) and (6.3.38):

7at

- 2 <*(«\->2i.iwhere all system operators are in the SMP at t0, and

w7i))> (6-4.91))

R

- I*«„ -coXIM,QJQ,wtt - Qt[M, (6.4.92)


where all system operators are in the HMP at time t. That the form of thesetwo equations is the same is of the utmost importance. In evaluating thesemeans in the SMP, we may visualize the operators as fixed and the coordi-nates (state vectors) rotating while in the HMP we visualize the systemoperators as rotating and the coordinates as fixed just as we do in the case ofa true SP or HP.

In the HMP, because the forms above are the same, we may order theoperators at time t, rather than at t0, as we did in the SMP. Thus we have01(00,(0 • • • a,(0 s a{t) or at. The functional form of M" ordered at t andt0 will obviously be the same. Therefore, we see that

{M\a, »)> - Tr

Tr S(r0) JMX&, t){de(%t - aJ)B da,, (6.4.93)

where we let a, s a(/) be our independent variables. If we use (6.4.92), wehave

t - at), H]Tr S(l0) 0 d*

1i.i (6.4.94)

The expression in the curly brackets now has the identical form as in (6.4.14)except the variables here are a, and qt instead of a,, and q0. If we thereforeproceed exactly as in (6.4.15)-(6.4.17) we see that

^ = TrS(f0)at j wo,

- Tr S(* o)^ V{m«u t)}^R. (6.4.95)

In case Lc contains derivatives only up to second order, it follows from (6.4.19)that

d{M(t)) d

dt

(6-4-96)

= Tr S(t0)


where we again sum from 1 to/over repeated indices and a(t =. a,(/). If wenow integrate the first derivative terms by parts once and the second deriv-ative terms by parts twice [the integrated parts vanish because the d-func-tions are zero at the limits], we obtain

itoa.jt)(6.4.97)

We should next observe that the integration over the ^-function has the effectof putting the quantity in the curly bracket into c-order; that is,

(6.4.98)

Therefore, (6.4.97) may be written as

.itd«.st)(6.4.99)

Next it follows directly since S(t0) is completely arbitrary that the argumentsof the traces must be equal so that we have

toxit)Under the Markoff approximation we may interchange the ordering and

differentiation operation

As a result, since both sides of (6.4.100) are in chosen order, we may operateon both sides with (&'~1 and obtain the c-number equations

.itd<xjf/Rwhere we sum i andy from 1 t o / Note that Me{x^ may still be an operator asfar as the reservoir variables are concerned. This will be discussed in moredetail in Chapter 7.


If we let M = ak we have Mc = cnt(t) then it follows from (6.4.102) that

T <«*«>« ==<<(a«)>*. (6.4.103)at

"We therefore see that the sfk first derivative terms in the Fokker-Planckequation give the mean drift motion of the corresponding operator ak.

If we let M = akat and these operators are in chosen order, then M" =a t(0«i(0 and by (6.4.102) we obtain

7 WW0) B - <fl4fr*tet) + *4fc*&d + 2@M)R. (6.4.104)at

So we see that if we know the mean equation of motion for a* and a,, weknow s#k and s/t and if in addition we know the mean equation of motionfor afca,, we can obtain the diffusion coefficients from (6.4.104). If we applythe & operator to both sides of (6.4.104) we obtain

Sf(ocka0) - <?{«*»/, + a , ^ } ) * . (6.4.105)4t / R

Damped Oscillator. As a simple example, consider the Fokker-Planckequation for the damped oscillator for normal order given in (6.4.37). Wehave by inspection

(6.4.106)

««a - yh\ 0.

Then (6.4.103) becomes

and the solution is

<«(0>* - o(0)e-*(>/2)+toi*

or on applying the ^ operator

<«(<)>« - o(0)e^"m + <" ; ) t+ i pAlso if M == a*a, we have that

£dt

(6.4.107)

i', (6.4.108a)

W-*i dt; (6.4.108b)

(6.4.109a)

6.4 ONE-TIME

or

•rV(0a«at

The associatedd , *It{*1

AVERAGES USING FUNCTIONS [8-10]

•>a = -Yia\t)a(t))R + yn - iv(t)W(t))R A

c-number equation is

*>* = -y<«*«)ie + yn- KtW)* + '»*('

38S

h iv*(t)(a(t))R.

(6.4.109b)

>)(x)R. (6.4.109c)

Three-Level Atoms. As a second example, the Fokker-Planck equation fora set of iV-damped three-level atoms (6.4.90) gives by inspection the followingdrift vectors and diffusion coefficients for the ordering specified:

i +(6.4.110)

We therefore see by (6.4.103) that

dt

dt(6.4.111)

dt; = - ( r u

The first two are simple classical rate equations for the populations in level1 and 2. Since

F H>21

(6.4.112)w01

we see that (djfjdt)]^ gives the net change in the average population of level1. The Rx — Nwl0 is the rate at which the reservoir "pumps" atoms from theground state to level 1. The T1(J

r1)R term gives the rate of loss of atoms

from level 1 to levels 0 and 2 while the wli{Jri)R term gives the rate of


atoms entering level 1 from level 2. The w(] give the transition probability persecond for atoms to go from level j to level i as we have shown earlier. Asimilar interpretation can be given to the equation for {dJT^dt)^

The solution of the Jt equation is readily seen to be

(Jt(i))R - Jr(P)e-ir"¥ia»u, (6.4.113)which shows that the dipole moment decays in a time of order IV"1.

The populations approach steady state (d/dt = 0) values given by

P

1 1 21

The populations will be inverted when the population difference satisfies theinequality

MW ) + ^ W ) 0. (6.4.115)1 il 2 ~

When (J^tyâs — (-/Î)K.BS» the average populations in the two levels areequal and we say the transition is saturated.

We obtain the transient solution of the rate equations later.

6.5 SOLUTION OF THE FOKKER-PLANCK EQUATION

Damped Oscillator [5]In this section we solve the Fokker-Planck equation for the damped

harmonic oscillator in two different ways and obtain characteristic functions.

Eigenvalues and Eigenfmctions. The Fokker-Planck equation (6.4.37) fora damped driven oscillator under the Markoff approximation is

\fi, (6.5.1a)9a*

(6.5.1b)

where we have let

a*,n

and P is normalized so that

Tr Sia\a, a\ t) - Tr

since Tr |a)(a| = (a|a> «= 1.

rr

ffp(a, a*, 0 <**«=!, (6-S.lc)


We begin by removing the high-frequency behavior from (6.5.1a) which isequivalent to going to the IP. For this purpose, we let

a = pe-**'

Then

9a 9/5P

3a 9a*

9a 9/3dP _dpd8 dp 80* dp

dt dpdt dp* dt dt

9/89/8*

If we use these we see that (6.5.1a) becomes

+If we restrict our driving term to

then (6.5.4) becomes separable in the time. If we let

then (6.5.4) reduces to the eigenvalue equation

LQ = -XQ,where the operator L is

(6.5.2)

(6.5.3)

; • (6-5.4)

(6.5.5)

(6.5.6)

(6.5.7)


To simplify we may introduce the real variables x and y denned by

It is easy to show that

fdy

9/3*

3,8 9/3* 8aL2\8x" r dy

4\9* dylXdx dy

(6.5.9)

(6.5.10)


s + 2 )We would like to change variables to eliminate the first derivative terms.

When this is done, the equation is said to be self-adjoint. To attempt thiselimination the standard procedure is to let

(6.5.12)Q(x, y) - e-xl"^R{x, y).

If we substitute this into (6.5.11), we obtain

dyldy~-+-+2{x-^-

_ ( - + 2iR. (6-513)

Vy Vij.iiA h


We choose % to eliminate the first derivatives dR/dx and dR/dy:

393

so that

dxdxdy

With this choice, (6.5.13) reduces to

where we have let

and

4A . <,= — + 2,

(6.5.14)

(6.5.15)

(6.5.16)

(6.5.17)

% /?*, 0 = *-**€-"lB ** 'R(x, y). (6.5.18)

Equation (6.5.16) is now self-adjoint (see Appendix K) and must be solvedsubject to the boundary conditions that R(x, y) -»• 0 as x -*• ± oo and y ->± co. This equation with these boundary conditions is well-known since it isjust the Schrddinger equation for a two-dimensional isotropic harmonicoscillator. The eigenvalues are given by

«»,.n, = ~ K.n, + 2 = (2n, + 1) + (2«, + 1), (6.5.19)

where nx and «„ = 0, 1, 2, 3 , . . . . The eigenfunctions are just the oscillatoreigenfunctions given by

(6.5.20)

(6.5.21)

where Hn(x) is the hermite polynomial and

is a normalizing constant such that

(6.5.22)


Because (6.S.16) is self-adjoint, the eigenfunctions are orthogonal so that

00

, V)dxdy = aB,.m.aB,tTOr (6.5.23)

since

F<-'HJLu)HJLu) du = dnm2»n! JZ. (6.5.24)J-co

The eigenfunctions Rnx.n (x, y) form a complete set so that

S |i?n,.n,(^y)«n,.n,(^y') = ^ - * ' ) % - y ' ) . (6-5.25)n*—0 n y 0

To state this another way, if the eigenfunctions form a complete set, we mayexpand/(*, y) as

/(*,y) = t icBl.nA,.B,(*,y). (6-5-26)ni-0 n»-0

If we multiply both sides by Rmm.mJ(x, y), integrate over all * and y and usethe orthogonality relations (6.5.23), we see that the expansion coefficients aregiven by

00

cn,.n, =jjdx' dy'f(x>, ?')*„,.„,(*', y')- (6-5.27)— 00

If we substitute this into (6.5.26), we have

00

f(x,y) = !!dx'dy'f(x',y')f i*n,.B,0> ', y'X (6-5.28)

where we interchanged the order of integration and summation. We see that(6.5.25) must follow in order that / (z , y) = f(x, y).

If we use (6.5.18), (6.5.19), and (6.5.20), we see that the eigenfunctions aregiven by

*, V, 0nx ^X) y) ( 6 > 5 29)

We choose A so that each eigenfunction is normalized to unity. By (6.5.1c)and (6.5.2), we have

00

JjW, 0*. 0 *t = J]p«,.»,(*, y, t)n dx dy = 1, (6.5.30)


since d*0 = n dx dy by (6.5.9). We have by (6.5.29) that

395

00

((dxdynANn,.n,e-wmn*+»*>te-*HJix)e-*'Hn,(i/) s X. (6.5.31)—00

If we note that H0(u) — 1 and use (6.5.24), we see that

f" e^H^x) dx (6.5.32)Therefore,

X = Ayfn h — 1,

where by (6.5.21) Nw — it~XA. With this choice of A, the eigenfunctions be-come

P»«.«,(«» y, t) - T7=riyjir

, y).( 6 . 5 .3 4 )

The steady-state solution is the lowest order eigenfunction

Poo(*> y) = —: exp - ( * * + y2) = -L eXp _ I 2ive. (6.5.35)

All other eigenfunctions vanish as t -*• oo.Next we obtain the conditional probability or Green's function solution for

(6.5.7). This is the solution which satisfies the conditions at t = 0 that x = x'and y — y/. We therefore take a linear superposition of eigenfunctions

P(x, y, t)

At / = 0, we require that

P(x, y, 0) = 6(x -

>n,(*, y, t). (6.5.36)

\ (6.5.37)n».nr

where we used (6.5.34) at t = 0. If we multiply both sides by Rma,mt(x, y) Xexp +$(x* + y \ integrate over all x and y and use (6.5.25) we have

împ(x, y) 8(x - x') S(y - y') dx dy

or3«,.», (6.5.38)

(6.5.39)


Therefore, (6.5.36) becomes when we use (6.5.34) the Green's function

p(x, y, */*', y', 0) =

nz.n* c(6.5.40)

Clearly, at t = 0 we see by (6.5.25) that (6.5.37) is satisfied. This solution isthe conditional probability of having x and y at time t given they had thevalues x' and y' at t = 0. If we use (6.5.34) and the steady-state solution(6.5.35), this may be written as

' f •<•<». (6.5.41)1)

If p{x', y') is the probability the system has the values x' and y' at f = 0,

then we have

K*, y, 0 - ffpC*. V, tlx', y\ 0)p(*', y") dx' dy\ (6.5.42)

as the total probability of finding the system at x and y at time t. If the sys-tem is initially in the steady-state, then p{x', y') = />oo(*'» y')so that by (6.5.41)we have

J**> V, t) - f [dx1 dy' n | pnxtn,(x, y, t)pntin,(x', y', 0)

eoJJ—eo

n«.n»-0

CO

—oo

*„.„. (6-5.43)

where we used (6.5.24). Therefore, the system will always remain in the

steady-state.General Solution for Arbitrary Driving Term. The eigenvalue technique usedabove is not applicable when the driving term v(t) in (6.5.4) is arbitrary sincewe cannot then separate out the time dependence. In this section we derivea general solution when v(t) is arbitrary.

We begin by finding the Green's function or conditional probability offinding the system at j3 and P* at time t given that it had the values P' andj8'» at / s= 0 when the system-reservoir interaction was turned on. This initialcondition may be represented by the 6-functions

<3Q? - lim -e-Ko IT

(6.5.44)

6.5 SOLUTION OF THE FOKKER-PLANCK EQUATION 397

[see (3.5.6)]. The representation of the ^-functions by a gaussian is partic-ularly appropriate for the method of solution that we use.

We now look for a solution of the equation (6.5.4)

d§

p(P,p*,t/P',P'*,O) = (6.5.46a)

dt 2ldp

of the form

where

We could include higher powers of p and P* but, as we shall see, we can, bysuitable choice of r\, if, £, and v, solve (6.5.45) subject to the initial condition

p{P, p*, OIP', P'*, 0) = lim - e-«l>-W-r*\ (6.5.47)«->00 tf

with only the terms given. As we have shown in (3.5.5), since sf{p(P, P*, 0/P', P'*, 0} *= 1/TT \P')(P'\, this is a coherent state. Comparison of this with<?(0) shows that

G ( 0 = ~lh P ~ nmifl* ~ V*(t)) + In v(t). (6.5.46b)

£(0) = - ; = - . (6.5.48)it

If we put (6.5.46) into (6.5.45) and equate the coefficients of equal powers ofP and p*, we obtain after minor algebraic simplification the set of equations

dt= —yC + yn

ldvvdt

leiidt

(6.5.49a)

(6.5.49b)

(6.5.49c)

together with the conjugate of the drijdt equation. The solutions of thesewhich satisfy the initial conditions above are easily seen to be

£-*00

(6.5.50)

Ht) =


where

w(0 = -i (*v(t - tWVm**iimfr df. (6.5.51)Jo

Therefore, the conditional probability is given by

(6.5.52)

Clearly, as / -»• 0, w(t) -*• 0, and n(l — er*1) = 1/c -»• 0 so that p >.

d(p - P') <5(/S* - 0'*). Note that

8/? J>(ft /**, */£', /?'*, 0) « 1, (6.5.53)

so that the Green's function is normalized to unity. Furthermore, the effectof diffusion is clearly illustrated by this solution. At t = 0, the distributionfunction is a ^-function but as time goes on it spreads to a gaussian centered

at

and variance

In the special case

we see that

andp(P,p*,tip',p'*,o)

p

t 2 .

V

(6.5.54a)

(6.5.54b)

(6.5.55)

(6.5.56)

exp -

(6.5.57)which is another representation of the Green's function given in (6.5.40).

For the case above, the steady-state solution is

«-•»

which agrees with (6.5.35).The general solution of the Fokker-Planck equation (6.5.45) is

(6.5.58)

(6.5.59)


where/>(/?, /?'*) is the probability the system has the values /?', /?'* at t — 0.Another interesting limiting case occurs when the reservoir is at absolute

zero. Then

and the Green's function (6.5.52) approaches

p(P, p\ t/P', /?'*, 0) —> 6[p - pro

X dtf* - P'*e-fr/iU + MÔe-*8"']. (6.5.61)Therefore, if the system is initially in a coherent state, it will always remain ina coherent state whose center of gravity moves according to (6.5.54a). Inother words, when the reservoir is at absolute zero, there is no diffusion tocause the distribution function to spread. This may also be seen directly from(6.5.45) since when n —*• 0, the diffusion term vanishes.

As / -»• oo, (6.5.52) approaches

where «-»» wn

l im eia>ct I v(t — t'<-»« Jo 'dt\

(6.5.62a)

(6.5.62b)

This corresponds to a coherent signal in the presence of gaussian noise. Toshow this we argue as follows. At t = 0 the system is in a pure coherent stateand this corresponds to a coherent signal. As time goes on, the reservoirintroduces thermal noise and the resultant distribution corresponds to acoherent signal plus thermal noise. If the signal is zero (v = 0), then (6.5.62)corresponds to pure thermal noise.

To obtain the distribution function in the SP, we use (6.5.2) so that theGreen's function (6.5.52) becomes (/?' = a', /?'* == a'* at t — 0)

P(a,a*,r/a',a'*,0) = - 7 ^ i

exp -- g'e-<r/*>«

expH (6.5.63)

Characteristic Function. Let us calculate the characteristic function definedby

TTB (6.5.64)


where we traced over system and reservoir and used (6.2.3). The S(f) is thereduced density operator in the Schrodinger picture. Since the exponentialsare in normal order, we may evaluate the trace by means of (3.2.34):

CM(£, t) « JJ ~ S(e>(a, a*, t) exp [»(fa + f*a*)]. (6.5.65)

But

IT

= fJp(a, a*, t/a', a'*, 0) P(a', a'*) dV. (6.5.66)

If we use this and (6.5.63), the characteristic function (CF) becomes

x exp

If we let

I \n _ w'p-K«M-(r/»]« i

-'iL_^£ ±I n(l - e-*)

z = <x— U;

(6.5.67)

(6.5.68)

then when we complete the square, (6.5.67) becomes

f, 0 - Jp(a', a'*) dV J ^ exp {-* [z* - if

x exp {-|£l*£ + i[£l7 +where

Since

the CF reduces to

C(n)(f, {*, 0 « Jp(a', a'*) â' exp {-If |2£ •

where we used (6.5.68). From (6.5.64) we see that

(6.5.69)

(6.5.70)

(6.5.71)

(6-5-72)

(6.5.73)


As a special case, let the initial distribution correspond to a signal plusgaussian noise. Then

P(a', a'*) = — exp - (W ~ "H. (6.5.74)

If we use this in (6.5.72), let z' = a' — u, complete the square and integrate,we obtain

C(">(£, I*, 0 - exp {-|f|2£ - i[

The mean number of photons in this case by (6.5.73) is

W(i)a(t)) = |M|2 e-»* + KOI2 + [«If no signal is initially present (u = 0), then

W(0a(t)) - K0P,while if there is no driving term (w — 0)

- w(0)]}. (6.5.75)

*(0 + cc]. (6.5.76)

(6.5.77)

(6.5.78)so that the initial coherent signal decays to zero.

Damped Atoms

So far the Fokker-Planck equation (6.4.90) for N homogeneously broad-ened atoms has not been solved. However, it is of some interest to solve it forthe case in which we may neglect all diffusion (all second derivative terms).In this case it reduces to

OJr i 2

- (2I\2 + I \ + T8)PC = 0. (6.5.79)We may solve this equation by the method of characteristics (see AppendixA). The characteristic equations are

dJK*APe icoa)J? - ( I \ 2 -

+ t f i ^ - T,^*! Rz + yfilJr1 -

(6.5.80)


whereA s 2T

lt

(6.5.81)

The reader will quickly note that these characteristic equations are identicalin form with the mean equations of motion (6.4.111). In other words whenwe neglect diffusion, the "motion" of the associated c-number variables isjust the "classical" motion of the atoms.

The solution of the first three characteristics equations are easily seen to be

Pe(0)

(6.5.82)

where ^(0), Jt^, and Jt% are constants of integration. The solution of thedjV^dt and dJT^dt equations may be obtained in a form useful for presentpurposes as follows. We multiply both sides of the dJT-^dt equation by a and

Jfddt equation by B and add. This gives

1 * < * * • • •1 We choose

• ] • •

|; ort

1 '

f- BJT1) + (riOt - w21B)jr

a and B by requiring that

I\a - ws

r 2 j8 - H>,

(I\-A)a

-Wl2* + (r

t + CV

!i? = Aa

«<* = XB

- wtl8 =

* - X)B =

— i

- 0= 0.

(6.5.83)

(6.5.84a)

(6.5.84b)

In order for these two equations to have anontrivial solution, the determinantof the coefficients of a and B must vanish. This will be true if A takes on eitherof the two values Xx or A2 given by

, r i + r » AAa = 2 — ^>

where we define

* +

(6.5.85)

(6.5.86)

6.5 SOLUTION OF THE FQKKER-PLANCK EQUATION

In this case, we see from the first of (6.5.84) that403

w21

w21We are at liberty to choose ax and a2 in any convenient way. We therefore let

<*i — *a = W2i (6.5.88)so that

A = fl - A

A - « + A, (6.5.89)where we have let

r — rQ = * ' . (6.5.90)

If we next substitute (6.5.84a) into (6.5.83), we obtain

7- (cu/Tx + pJ^t) + XidJr1 + /?^r2) = *i« + -R - (6.5.91)dt

The solution is easily seen to be

' - 1)],(6.5.92)

where JTu and ^"20 are constants of integration. This may be rewritten as thetwo equations

a

a

(6.5.93)

where we solved for */f10 + Bj^"2o/X a n d w e u s e d the two solutions for AtA corresponding to the eigenvalues Xx and A2. These equations may besolved easily for ^ " M and ./fô

A — A

A —A

, 0(6.5.94)


It and f, are defined by (6.5.93). These give the two remaining integrals whichare required. The general solution of the distribution function equation(6.5.79) is therefore

PJWi. *, *, 0 = ^"g lu fe^***^ 1 , urV1*-1-*1 , u, v], (6.5.95)

where g is an arbitrary function. If at r = 0, we know the initial values Jto,Jl*, JV'U, and ^Vm, we obtain the Green's function solution

?> 0 )

rtf t) _ ^ t) _ (6.5.96)

where we have determined the functional form to agree with the initial con-ditions. This shows that the variables follow their "classical" trajectories forall time. This is the course to be expected since we have neglected all diffusion.

If we use the definition of A (6.5.81) and use the well known property of^-functions

*(«*) = -i- 6(x) (6.5.97)M

then the Green's function solution may be written at

, 0 -= b\J( -

0 -(6.5.98)

6.6 TWO-TIME AVERAGES, SPECTRA [10]

Up to this point we have developed the theory in such a way that we maycalculate the average of a function of system operators which are all evaluatedat the same time under the Markoff approximation. However, a completestatistical description requires mean values of operators at different times.For example, in the case of a mode of the radiation field, the fluctuationspectrum is defined by

t. (6.6.1)rJ—

The intensity spectrum is given by

f" e-"»%a\O)a\t)a(t)a(fi)) At,J— 00

(6.6.2)

6.6 TWO-TIME AVERAGES, SPECTRA [10]

while the photon number spectrum is given by

e-*°\aXi)a{f)a\<S)a<S$))dt

405

(6.6.3)

(see Appendix I). These are related directly to measured experimentalquantities which the theory must be capable of predicting. In this section weshow that under the Markoff approximation, such two time averages may beconverted to one time averages.

We have shown in several cases of interest that the distribution functionPe obeys the Fokker-Planck equation (6.4.19)

OJ , *)]• (6.6.4)

We may write the solution of this equation subject to the initial distributionFe(§> h) as

d§ Pc(a, tl§, to)Pe(§, t0), (6.6.5)

where Pe(%, tl§, t0) is the conditional "probability" or Green's functionwhich obeys (6.6.4) and has the property that

P/X. t/§o, to) > <*(« - §). (6.6.6)

We then see by (6.6.5) and (6.6.6) that

Pe(«,t) >Pe(<x,t0) (6.6.7)as required.

By (6.4.11), we have that the one-time average of M(a) evaluated at /0under the Markoff approximation is

(*, tj§, to)Pe(§, t0) d§.

<M(0> (Me(x)Pe(a, t) da


<M(0) s J3?c(a) d*

From the definition (6.4.12)

Pe(§, t0) « Tr S(t0) &>(§ - a),

we may write (6.6.9) as

^ Tr S«0){jd*jd§ M°(*)Pc(cc, t/§, t0) d<(§ - g)}. (6.6.11)

(6.6.8)

(6.6.9)

(6.6.10)


If we compare this with (6.4.7) we see that the solution of (6.4.S) for M(t)under the Markoff approximation is

S& M(t) =jd*jd§ ^(s)P e(a, r/0, t0) d<(§ - a). (6.6.11)

That is, if we know the Green's function solution of the Fbkker-Planckequation, we have a solution for (6.4.5).

Consider next the exact two time average

<M(l)N(ro)> = TrB.s P(t0)MH(t)N(t0), (6.6.12)

where N(t^ is a system operator in the SP at time t0 and MH(t) is the fullsystem operator in the HP. The time f0 is any time at which the densityoperator factors as S(to)f9(R) when the system has lost its prior memory.Then, we have

(M(t)N(t0)) = Trs S(toXMH(0)«N(to). (6.6.13)Under the Markoff approximation, we may replace (MH(t))n by (6.6.11) sothat

<M(f)tf (*„)> Si Tr s S(to)M(t)N(t0)

=jdzjd§ tt\%)Pt{*, tig, t0) Tr [S(t0) 6\§ - a)N(t0)].(6.6.14)

We have therefore reduced the average to a trace of operators all evaluated atthe same time f0. All that is needed is the same Green's function solution.If N = 1, this reduces to (6.6.9).

Let us illustrate this result for the damped oscillator. The Green's functionis given by (6.5.63) with t0 = 0

P(B)(a, a*, tlfi , P, 0) = -~^ exp - | g " , I 7 ( 0 1 2 , (6.6.15)

where

(6.6.16)t/(0 =

W(t) = - i f W ~Jo

If we let — cfi and N = a, we have by (6.6.14) [using normal order]

- a)a].

(6.6.17)

6.6 TWO-TIME AVERAGES, SPECTRA [10] 407

If 5(0) is in antinormal order, we see that since the other terms are in normalorder that

Tr S(o)(0)d(0* - «f)<5(/5 - a)a = (s(a)(a'ta'*,0)a'd(P* -a'*)6{fi - a ' ) ~

(6.6.18)Therefore,

(aâfO)) = p « J ^ «*/5P<»,(«, a*, <//?, /5*, 0)P(n)(^, /?*, 0). (6.6.19)

We have shown earlier in (6.4.30) that

(6.6.20)

If we use (6.6.15) and carry out the integration over a, we obtain

- (}w*(}w*(t)}PM(0, P, 0)

(0. (6-6-21)So long as / is greater than a reservoir correlation time, we may take theinitial distribution to be the steady-state distribution for thermal noise,namely,

P(n)(|8,/?*,0) = - i : e x p - ^ . (6.6.22)rrn n

When we carry out the integrals in (6.6.21), we obtain

(a\t)a(0)) = neia"e-(r/*m. (6.6.23)Since the process must be stationary, it must also decay if / ->• —/. Therefore,

(af(t)a(0)) = /ieê-W*!". (6.6.24)

When we take the Fourier transform, we obtain the fluctuation spectrum

J-cyh

which is seen to be Lorentzian with half-width given by y/2 centered at o> =Oic.

We leave as an exercise the evaluation of the intensity and photon numberfluctuation spectra.


6.7 ROTATING WAVE VAN DER POL OSCILLATOR

It has been shown [11] that the equation of motion for the associateddistribution function for the reduced density operator in antinormal order forthe radiation field of a single mode laser is the Fokker-Planck equation

*, 0 = -§- [(g -9a

- ^ [(g -dor

p] + 4r^(«, «*, 0,da dor

(6.7.1)

where g is a numerical pumping parameter and p(a, i) refers to the inter-action picture. It describes* a radiation field mode at frequency a>e which isdamped for g < 0 and starts to grow exponentially when g > 0; g = 0 iscalled the oscillation threshold. For g > 0 the oscillation builds up butbecause of the nonlinear term |a|8, the oscillations will stabilize at a fixedamplitude since g — |«|* eventually goes to zero as |a|* increases. An equationsuch as (6.7.1) describes a classical Van der Pol oscillator which is the asso-ciated "classical" description of a laser. The quantum density operator is

given bys(a, at, r) = vs/{p(*, a*, *)}, (6-7.2)

where s/ is the antinormal ordering operator.Because of the |a|z terms in (6.7.1), we cannot solve it analytically. How-

ever, there are two interesting regimes where it may be solved, namely, farbelow threshold (|a|8 « |g|) and far above threshold (|a|* » |g|). We solve itin these two regions and obtain the spectra of the laser.

Far Below Threshold (|a|* « \g\)

In this region, (6.7.1) reduces to

dot.(6,7.3)

which by (6.5.4) is the equation for a damped oscillator if we make theidentification v = 0

yh = 4y (6.7.4)

The Green's function associated with the Schrddinger picture can be obtaineddirectly from (6.5.63), namely,

P(a, a*,!/«', «'*, 0) = -~^ exp - '* " L L . (6-7-5)


where

gFrom (6.6.25), we see that the fluctuation spectrum below threshold isL t i i h h l f idh | | d

g<0.

409

(6.7.6)

Lorentzian with half width |^| and is given by

J— 0(6.7.7)

" (coe - co)* + g2 '

Par Above Threshold |a|2 » \g\

Before we attempt an analytic solution in this regime, it is advantageousto convert (6.7.1) to polar coordinates. We let

a = re*9 d2x = r dr dtp.

By the usual rules of differentiation, we have

(6.7.8)

a =

9a * 2L9r2d, Or dr = 1. (6.7.12)Jo Jo

Equation (6.7.11) is still exactly equivalent to (6.7.1). We may obtain thesteady-state solution which is independent of <p very easily. Since

it becomesd~r° di = 0' (6.7.13)

(6.7.14)


or if we integrate once and take the constant of integration to be zero we have

dpdr

(g - r*)rp

orp,(r)

(6.7.15)

(6.7.16)

-ff/2

(6.7.17)

where we normalized the steady-state solution so that

rdrp,(r)-\Jo

We sketch this solution in Figure 6.4 for g = —5 (far below threshold), atg = 0 (threshold) and g = 5 (far above threshold).The mean number of photons in the steady-state is given by

r)r dr/•oo

J-a/2+ 2x)e-' dx

J-t/2

(6.7.18)dx

This is sketched in Figure 6.5 as a function of g. Far above threshold, we maytake the lower limit to — oo so that

(â)t^g. (6.7.19)

Far above threshold, the mean number of photons will have small fluctuationsabove this steady-state value. Let us expand r as

r^Jg + rx(0; rdrg^Jp' drx (6.7.20)

where rx(r) <C \g- If we put this into (6.7.11) and neglect quantities of orderrf, we have

i- -i2_ a 1 as,,

(6.7.21)

This equation is now separable and may be solved. We therefore let

(6.7.22)


3r411

The steady-state photon probability distribution function for three values of thenet pump rate^. g = 0 corresponds to threshold. (Reprinted from reference 11.)

where we normalize so that

or since £ is large, we require

1

i - l .

(6.7.24)

r-

-8 -6 -4 - 2 0 2 4 6 8 10NET PUMP RATE , P

Figure &5 Mean number of photons in the steady-state as a function of the pump param-eter £•. (Reprinted from reference 11.)


When \tre put (6.7.22) into (6.7.21) and separate, we obtain

dR _ d*R cdt drf Bi

The;are

steady

_

dt g dq>* '-state solutions of these (djdt = 0) such that <J>(y + 2n) = <b(q>)

_1_lit'

(6.7.26)

so that the amplitude is a gaussian and the phase is uniformly distributed.Thus we see that in the steady-state

(r\ - g + W) = g + f2g

(6.7.27)

<r4), - g* + 6g(r18> + <r/> = g* + 3 + - ^ .

4g8

We next look for the conditional probability solutions of (6.7.25) such that

lim R(rlt r/r10) 0) lim / I

1M/?>o>O) =

<-»0 A / I —«u

In the radial equation we look for a solution of the form

(6.7.28)

,0)

where

= e;

exp |

r1 0;16

(6.7.30)

6.7 ROTATING WAVE VAN DER POL OSCILLATOR 413

If we substitute (6.7.29) in the radial equation (6.7.25) and equate equalpowers of rlt we obtain

(6.7.31)

ldv = _ldtv t t,dt

When we solve these subject to the initial conditions (6.7.30) and let e -> GO,we obtain

V(t) (6.7.32)

To obtain the Green's function solution for the phase equation, we let

Ofo tf<p0,0) - Zcn(t)e<**. (6.7.33)—00

If we put this in (6.7.25), we obtain

org

(6.7.34)

(6.7.35)

If we use the initial condition (6.7.28), we have

®(<P, tin, 0) - fcMein* exp (-5-* t) —->-^ f e'«<~->. (6.7.36)

Therefore,

<*>(?>, */?„, 0) = f f2ir-oo

exp ( - - «).\ g /

The field correlation function is given by (6.6.19) as

<af(f)a(0)> = J J W , P*) ^/sJJa*P(a, a*, 0, /?*,

(6.7.37)

(6.7.38)


where by (6.5.2), we must replace a by uP* and a* by a**-** in our distri-bution functions or

Therefore,re

-«(6.7.39)

In the steady-state, by (6.7.32) we see that

g

so the steady-state distribution is

since only the n = 0 term contributes as / -* oo. Also

while in the integrandP = U/g +

When we use these, (6.7.38) becomes for large g

x exp - in( = eia

The fluctuation spectrum is given by

2

—2g J

[2 + (1/g2)]

415

(6.7.47)

(« . -

(6.7.48)

The spectrum far above threshold consists of two Lorentzians. The first hashalf-width g-1 which is extremely narrow. The second has half-width 2gapproximately and is very broad.

We leave as an exercise the calculation of the intensity and photon numberspectra for this oscillator.

For a more rigorous discussion of some of the topics treated in this chapterthe reader may consult Ref. 12. Reference 13 will be of interest for presentinga different approach to a quantum-classical correspondence. A few otherrecent examples of the use Fokker-Planck equations are given in Ref. 14.

REFERENCES

[1] W. H. Louisell, Coupled Mode and Parametric Electronics, New York: Wiley, 1960.[2] W. H. Louisell and L. R. Walker, Phys. Rev., 137, B204 (1965).13] M. Lax, / . Phys. Chem. Solid, 25,487 (1964).[4] W. H. Louisell, "Quantum Theory of Noise", in International School of Physics

"Enrico Fermi" XLU Course in Quantum Optics, 1967, Varenna, Italy.[5] W. H. Louisell and J. H. Marburger, / . Quantum Electron., QE-3,348 (1967).[6] F. Reif, Fundamentals of Statistical and Thermal Physics, New York: McGraw-Hill,

1965, Chapt. 15.[7] M. Lax, Phys. Rev., 145,110 (1966).[8] M. Lax and H. Yuen, Phys. Rev., 172, 362 (1968).19] J. P . Gordon, Phys. Rev., 161, 367 (1967).

[10] M. Lax, Phys. Rev., 172,350 (1968); W. H. Louisell and J. H. Marburger, Phys. Rev.,186,174 (1969).

[11] M. Lax and W. H. Louisell, / . Quantum Electronics, QE-3, 47 (1967).[12] J. R. Klauder and E. E. G. Sudarshan, Fundamentals of Quantum Optics, New York:

W. A. Benjamin, 1968.[13] G. S. Agarwal, Phys. Rev., A2,2038 (1970) and 3,1783 (1971).[14] T. von Foerster and R. J. Glauber, Phys. Rev. A3,1481 (1971); R. Brambilla and M.

Gronchi, Lett. Nuovo Cimento, 2, 511 (1971); H. Hubner, Z. Phys., 239,103 (1970);R. Graham and H. Haken.Z.Phys., 235,166 (1970); Y. M. Golubev, Optics Spectrosc,28, 528 (1970).


PROBLEMS

6.1 Derive (6.2.37) from (6.2.35) using the transformation from the SP to the IP

(6.2.36).6.2 If the reservoir is at absolute zero and there is no driving term, solve (6.2.61)

as an operator equation.6.3 Evaluate the diagonal matrix elements of both sides of (6.2.61) in the co-

herent state representation (a|oc) = a|a».6.4 Evaluate the m, n matrix elements of both sides of (6.2.61) in the number

representation (a^n) = n\nj).6.5 In Problem 6.3 above, the density operator is in the IP. Transform it to the

SP.6.6 Find the eigenvalues and eigenfunctions in Problem 6.2 in the coherent state

representation when v(t) = vêr*"'*.6.7 Find the conditional probability solution of Problem 6.3 when « = a' and

a* = a'* at f = 0. Note that- e |a - a'|2}.<—l

6.8 Verify (6.2.75b).6.9 Can we use the master equation (6.2.85) when the "atom" is a harmonic

oscillator? Why?6.10 Verify (6.2.87b).6.11 Solve the Pauli equation (6.2.126) for a two-level atom.6.12 Show that at in (6.2.151) is given by

( )

6.13 Solve the Bloch equations (6.2.148) when hot = «2 — ex = hcon and v(t) isgiven by (6.2.140). Find the steady-state solutions also.

6.14 Solve (6.3.48) and (6.3.49). Obtain <o(r)> and <pt(f)a(r)>.6.15 Solve(6.5.11) in polar coordinates * + iy = re**. Show that the energy eigen-

values are given by « = 2(|m| + 2n — 1) where m <= ±1 , ± 2 , . . . . andn — 1 , 2 , 3 , . . . and that the eigenfunctions are

where L,*(K) is a generalized Laguerre polynomial and N is a normalizing

constant.constant.6.16 Solve (6.5.45) directly when h = 0 subject to the initial condition that/>(0) =

a(a - «') 6{a* — a'*). Hint: Use the method of characteristics.6.17 Obtain the density operator from (6.5.57) by applying the antinormal order-

ing operator to a and a*.6.18 If there is no driving term [v(t) = 0] in problem 6.17 show that the diagonal

elements of the density operator are (as / -* oo)

PROBLEMS 417

where X = Ha/kT. This is just a Boltzmann distribution corresponding togaussian thermal noise which is approached when the system started in acoherent state.

6.19 Evaluate the intensity and photon number spectra defined by (6.6.2) and(6.6.3) for a damped driven mode of the radiation field.

6.20 Obtain the steady-state solution of the Fokker-Planck equation (6.4.90).Hint: Use the method of Appendix K.

7Quantum Theory of Damping-Langevin Approach

In this chapter we present the quantum theory of damping from a Langevinviewpoint [1]. Essentially the method consists of replacing the reservoir bydamping terms in the Heisenberg equations of motion for a lossless systemand adding random forces as driving terms which add fluctuations to thesystem. The forces must be chosen in such a way that the system has thecorrect statistical properties to agree with those obtained in Chapter 6.

We begin by considering a damped harmonic oscillator in Section 7.1 inthe Heisenberg picture. The results we obtain here will allow us to visualizebetter the general quantum Langevin theory of noise sources which we presentin Section 7.2. We illustrate this formulation in Section 7.3 and 7.4 for asingle multilevel atom and N homogeneously broadened three level atoms,respectively. In the final section we give the c-number formulation and relatethe drift vectors and diffusion matrix of the prior chapter to our c-numberLangevin equations.

7.1 LANGEVIN EQUATIONS OF MOTION FOR DAMPEDOSCILLATOR

In this section we obtain solutions of the Heisenberg equations of motionfor a damped oscillator under the Wigner-Weisskopff approximation. Theseallow us to obtain the Langevin equations of motion for a damped oscillatorand are useful when we present the general Langevin approach to dampingin the following section.

The damped oscillator is described by the hamiltonian

H H 2 (Kt

K*b}a). (7.1.1)

418

7.1 LANGEVIN EQUATIONS OF MOTION 419

Let M(a, a*) be an arbitrary function of a and a*. It satisfies the Heisenbergequation of motion

1 [M, HT) = -io)e[M, a'a] - i £ K*b)[M, a]in t

where all operators are in the HP. If we use the identity

[M, a*a] = [M, a*]a + d*[M, a],

and the commutation relations

... , dM[M, a] = -

(7.1.2) reduces to

[M, ar] = — ,da

(7.1.2)

(7.1.3)

(7.1.4)

_ . ( 7 . L 5 )

where we again emphasize that all operators are in the HP. Similarly, wesee that

— - - u o A - lKia. ( ? j 6 a )

We may write this as the integral equation

b,(t) '-') dt', (7.1.6b)

where 6,(0) is in the SP and a(t') is in the HP. Although £,(*) and b}(t)[obtained by taking the adjoint of (7.1.6)] commute with all functions of a(t)and af(t) in the HP, the two separate parts of bt(t) and b}(t) in (7.1.6b) donot. We must therefore be careful of order when we put bf(t) and bj(t) backinto (7.1.5). When we do this, we obtain after minor algebra

dM . dM , . .dM

fJo I da1 da

'-") dt' + GM, (7.1.7a))

where we have let

(7.1.7b)

420 QUANTUM THEORY OF DAMPING—LANGEVIN APPROACH

Again all operators are in the HP except bt(0) and bj(O) which are in the SP.So far (7.1.7) is exact. Let us consider two special cases. First, let M = a.Then we have

*« = _i(Oea - 2 |,c,|* f oOV*9 '" '-0 dt' + Ga (7.1.8a)dt t Jo

Ga=-»-2*A(0K-to''. (7.1.8b)

If we leta(t) = Atfe-**' (7.1.9)

to remove the high-frequency behavior from (7.1.8), we see that[a(0, a\t)] = [A{t)t A\t)) - 1, (7.1.10)

and (7.1.8) reduces to

X " - 2 M" [d* AV) «P'(», " «>e)( - 0 + G , (7.1.11a)af j Jo

where we have letGA = - i 2 »cy6,(0) exp -i(a», - me)t. (7.1.11b)

Unfortunately, this exact integrodifferential equation cannot be solved andwe must resort to the Wigner-Weisskopff approximation. If we take theLaplace transform of (7.1.11) and use the results of Section 5.3, we have afterminor algebra

-<0)+0-<<;> , (7.1.12.)A\S) —

s + i(o)} — a>e)where

and

(7.1.12b)

Under the Wigner-Weisskopff approximation (see Section 5.3) we let

(7.1.12c)

o — ft>e) — is i-*o J (w, — coe) — is

= - i I dm, g |*c|* (

(7.1.13)

7.1 LANGEVIN EQUATIONS OF MOTION

where by (6.2.56)y = 2irg(o>{

"I

421

(7.1.14)

Thus (7.1.12a) becomes

— o ( 0 ) ;. (7.1.15)s + \y + i Ao> 'f[s + i(co,- we)][s + Jy + i Aw]The inverse transform gives

a(t) = «(0a(0) + 2 t>,(f)fr,(O) = e-*°"M(0, (7.1.16a)where

«(0 = exp -[\y + i(coe + Aco)]t

m -*,e-*»"[l - exp i(co, - <oe - AQ))^-1"72] (7.1.16b)a)e — to, + Ao) — iy/2

If we neglect the small frequency shifts, we see that the effect of the Wigner-Weisskopff approximation is to replace the exact (7.1.11) by the Langevinequation

where

4d =dt

GA(t)=-i

GA(t), (7.1.17a)

(7.1.17b)

since the solution of this equation is given by (7.1.16) as the reader mayreadily verify. In this case GA is the random operator Langevin noise sourceand the —(y/2)A gives the drift motion. The noise sources are always chosenso their reservoir average is zero. If the reservoir is in thermal equilibrium,we see that

<GA(t))a = TxBf0{R)GA(t) = 0, (7.1.18)

when we use (7.1.17b). Therefore, we see from (7.1.17) that

(7.1.19)atso that

<X(0>fi = ^ ' / 8 f l (0 ) , (7.1.20)

since (A(0))B = (a(0))R = o(0). This result also follows from (7.1.16)directly.

422 QUANTUM THEORY OF DAMPING—LANGEV1N APPROACH

As a second example, let M = a*a s A*A. In this case (7.1.7) becomes

dt t Jo

- -yat(0a(0 + Got8, (7.1.21)

where we have used the Wigner-Weisskopff approximation as we did ingoing from (7.1.11) to (7.1.17) and where

(7.1.22)

where ha means hermitian adjoint. If we use the solution (7.1.16), Goto

becomes

+ 2 ^ _ -(y/2) - i(«)k -

Since for the reservoir in thermal equilibrium

+ ha- (7.1-23)

(b](0)bk(0))B =1 (7.1.24)

we see that

,, . {y -

(7.1.25)

Since l^l2^ is slowly varying and the summand is so strongly peaked atID, = ft)e, we may convert this sum to an integral and remove the slowlyvarying factors. This gives

ft*+X*(7.1.26)


where we let x = a>t — <oc and dx = d(af. We extended the limits from — ooto + oo since the main contribution comes at x — 0. Since

(7.1.27)

we see that (7.1.26) reduces to

(&*u(t))s - yn, (7.1.28)

where we used (6.2.56). If we therefore take the thermal reservoir average ofboth sides of (7.1.21), we obtain

which has the solution

(a\t)a(t))R

yh, (7.1.29)

(7.1.30)

Let us now rewrite (7.1.21) and include the reservoir average of (?0to sothat (since A1 A = d*a)

where we have let

• j A*A = -at yn

— Gau — yn.

(7.1.31)

(7.1.32)

Since now (GAtA(t))R — 0, we see that the thermal average over the reservoirof (7.1.31) gives the identical result as the thermal average of (7.1.21). TheLangevin force GAtA is chosen so that its reservoir thermal average is zeroand the remaining terms in (7.1.31) give the thermally averaged drift motionof the operator on the left side. It is the Langevin equation of motion forafa — A*A, the photon number. The Langevin force and the damping termin these equations replace the reservoir in an equivalent circuit representation.The damping term plays the role of a resistance and the operator force playsthe role of a noise generator which puts fluctuations into the circuit. Since itis a quantum noise generator, it maintains the commutation relations andtherefore insures that the uncertainty principle is not violated under theapproximations (see Problem 7.1).


SpectraOnce we have the approximate solutions of the Heisenberg equations, we

may calculate the various spectra. The fluctuation spectrum defined by(6.6.1) is

f °° e-*V(0«(0)> dt = f °° e-4itt-m'u(AXf)A(0)) dt, (7.1.33)•'—00 •'—00

where we used (7.1.9) and its adjoint. This spectrum is just the Fouriertransform of the correlation function

(7.1.34)

(7.1.35)

where the initial density operator is

p(0) = S(0)/,(i?)S(O)e->B

and we trace over both system and reservoir. If use the adjoint of (7.1.16) wehave for the two-time correlation function

Tt8

or« cH'r/*"<l<at(0)a(0)>

where we used the adjoint of (7.1.18) and have let

'*«>)}(7.1.36)

(7.1.37)

(7.1.38)

We have used the absolute value of t since for a stationary process [2](7.1.39)

(7.1.40)

which is Lorentzian centered at <o = toe with half-width y/2. This resultagrees with (6.6.2S) if at t — 0 the cavity is in thermal equilibrium with thereservoir so that

« = <flt(O)a(O)>. (7.1.41)

The fluctuation spectrum (7.1.33) reduces to


Note that (6.6.25) was calculated under the Markoff approximation whereashere we have used the Wigner-Weisskopff approximation which to ourpresent accuracy is equivalent.

Consider next the intensity spectrum (6.6.2)

dt

The correlation function is

(7.1.42)

(7.1.43)From (7.1.30) we see that

K(t) = Tr5S(O)at(0){at(q)a(q)e-'< + n[l - **l}a(Q)

- ( a ^ o y ^ ) ) ^ " ' + nfc'CqMQMl - e-»|f|]. (7.1.44)

The intensity spectrum is

T2rrn(a\0)a(0)) d(co),

(7.1.45)

which consists of a Lorentzian of half-width y centered at a> = 0 plus a <5-function at a> = 0.

Diffusion Coefficients. Fluctuation-Dissipation Theorem [2]

In this section we obtain two time correlation functions for the Langevinforces and derive the fluctuation-dissipation theorem.

We have shown in (7.1.18) that

= 0. (7.1.46a)

It also follows since G*A(t) = GAt(t) that

{GAi(t))R = 0. (7.1.46b)

Similarly, one may easily show that the autocorrelation functions are zero:

TrB MK)GAh)GAtz) = 0

^ ) 1 ) ^ ( 2 ) ) B = 0 ( 7 ' L 4 7 )

(see Problem 7.4).


Consider next the cross-correlation function defined by

j, (7.1.48)

where we used (7.1.24). Note that this is a function of the time differenceh ~~ h o n ty which is characteristic of stationary random processes. In asimilar way we see that

(7.1.49)f °°dcof g(o>y) |K(CO,)|2 [1 + «(

We therefore see that(7.1.50)

as would be the case with a classical cross-correlation function.Rather than pull out the slowly varying terms in (7.1.48) and (7.1.49) and

replacing the integral of the exponential by a <5-function, let us proceed alittle more carefully to show that this approximation corresponds to theMarkoff approximation made in the last chapter.

We expect that as 1^ — tt\ -*• oo, the rapid oscillations in (7.1.48) and(7.1.49) will cause the correlations of the Langevin forces to vanish so thatfor a sufficiently long time difference, they will become uncorrelated. Let re

be the correlation time of these forces. If we multiply both sides ofKAiA(ti — t2) by dtx and dtt and integrate both from t to t + At whereAt y> re but short compared with the system damping time y~x and divideby At we obtain the diffusion coefficient defined by

r1 ft+At r«+A«

=r dtAToo f«+At ft+At

d«>igK)K«>,)l2S(«>i) dtA dJo Jt Jt

At)Jo

da>ig(coi)\ K(tt),)l* n(cof)

(7.1.51)


The integrand is now highly peaked at <of — <ac so that we may withoutserious error let the lower limit extend to — oo and remove the slowly varyingterms g\x\2n from the integral. If we use (6.2.56) again

then

where we let 2x = otf — toe. Then since

•" sin* x Atdx = irAt,

(7.1.52)

(7.1.53)

(7.1.54)

the diffusion coefficient becomes

provided

t+i.t

y-l»At»re.

(7.1.55)

(7.1.56)

We see that we could have obtained this identical result if we replaced (7.1.48)by

KA'AOI - h) - <

= yh d(tx - (7.1.57)

Therefore the Markoff and Wigner-Weisskopff approximations to the presentaccuracy give the same diffusion coefficients, since

i r<+A« /*<+Af^- dtAAt Jt Jt

dtt yn b{tx (7.1.58)

We always assume that the random forces are 6 correlated provided that thesystem is Markoffian—its future is determined by the present and not thepast. The reservoir becomes uncorrelated long before the system has changedvery much and the system therefore cannot develop any memory of the pastthrough its interaction with the reservoir.

Similarly, we see that

- h) = y{n + 1) d(tt - t2)yin + 1).

(7.1.59)


From (7.1.57) we see that on integrating both sides we obtain

dr. (7.1.60)1

This states that the system damping y is determined by the reservoir fluctu-ating forces which introduce fluctuations into the system. This is oneformulation of the fluctuation-dissipation theorem. Similarly, by (7.1.59) weobtain another form of the theorem

(7.1.61)

are easily seen to be zero.

n + 1 •'-The diffusion coefficients (DAA)R and

Langerin Equation for Photon NumberWe have already obtained the Langevin equation of motion for the photon

number (7.1.31). To obtain more insight into the nature of the Markoffapproximation and to aid us in formulating the Langevin method for moregeneral systems, let us obtain the Langevin equation for the photon numberby another method.

By the usual rules of differentiation, we have that

— A A = A — + A. (7.1.62)dt dt dt

If we use the Langevin equation (7.1.17) and its adjoint, we obtain

- A*A = -yA'A + AGA + AGA,dt

where we have been careful about order. The reader should convince him-self that this equation is identical to (7.1.21) and (7.1.22). We need the reser-voir average to give us the drift motion

f {A\t)A t))R = -y(A\t)A(t))Rat

(GA,(t)A(t))R. (7.1.64)

We have already evaluated i[A\f)GA(t) + GA*(t)A(t)))R = yn. We usedthe solution for A(t) and A*(t). The method we now use relies more directlyon the Markoff approximation and does not require knowledge of thesolution for A(t) and A\t). Consequently, it is more general.

We begin by writing the identity

0 + GA,(t)A(t)])S

)(7.1.65)

7.1 LANGEVIN EQUATIONS OF MOTION

where t > te and y 1 » / — te» TC. Clearly,

<AKte)GA(t))B = 0(GAt(t)A(te))R = 0.

429

(7.1.66)

This result must be true under the Markoff approximation for if the systemoperator and reservoir Langevin force were correlated over this time interval,the system would develop memory. Since we happen to know the solutionfor /4t(/(,)J let us verify that (7.1.66) is indeed true before proceeding. We haveby (7.1.11b) and the adjoint of (7.1.16) that for t > te > 0 with Aoa = 0

<AXQGA(t))R

(0)(y/2)

J l K) +1(«>, - (oe)i k

TpUo>i-ae)tc _

|2 n

= 2™ f °27rJ_< --dx, (7.1.67)

where we used (7.1.52), (7.1.24), and let x = <o} — coe. Since for a > 0

fj—<

dx = 0 (7.1.68)(y/2) + ix

(7.1.66) follows directly. A similar argument verifies the second relation of(7.1.66). Again (7.1.66) is a direct consequence of the Markoff approximationand is not peculiar to the damped oscillator. Accordingly, (7.1.65) reduces to

f)jds

If we next use (7.1.17) and its adjoint, we obtain

By the Markoff approximation again since t > s, we have that

(A'(s)GA(t))R « 0

{GA,(t)A{s))R = 0.

(7.1.70)

(7.1.71)


Also we have shown that over the interval t — te that

<GAt(s)GA{t))R - (GAt(t)GA(s))R - 2{DAU)B 6{f - s) - y n 6(t - s).(7.1.72)

Therefore, (7.1.70) reduces to

<G9\(t))B = [ds 2yn 6(t - s) = yn, (7.1.73)

since

F(s) 5(* - s) (7.1.74)

This agrees of course with our prior calculations. Therefore, (7.1.64) becomes

dt(A*A)B = -y{A*A)B + yn.

We may take as our Langevin equation

where the Langevin force is

G ^ ( 0and has the property that

yn + GAtA,

GA(t)A(Q,

0.

(7.1.75)

(7.1.76)

(7.1.77)

(7.1.78)

Again the drift motion is explicitly included and a random force is added toretain the correct quantum fluctuations. The first term in (7.1.76) is the rateof loss of photons into the reservoir while the second gives the rate at whichphotons enter from the reservoir. The force causes fluctuations from themean photon number.

Eiiistein Relation

Suppose we know the thermal average equations

2{A)*(7.1.79)

dt


We may then write the Langevin equations

^f » JL Af + GAi (7.1.80)

dt 2

^ = -yA<A + yn + GAU.dt

In a time interval / to / + At where y~l )^ A/ )J> re, we may integrate these as

21 f'+A*hl G

At Jt

At At

where

(7.1.81)

(7.1.82)AA = A(t + At) - A(t)

AA'A = Af(t + At)A(t + At) - A*(t)A(t).

That is, during At, A, A1, and A*A do not change whereas the forces do.We see that

\ At /B \/[A\t) + AA*][A(t) + AA] -\ At

VAKt)M + A^\ At At At /B

(7.1.83)

If we use the first two of (7.1.81) on the right above, we obtain

2 Ar

r«-At

\ A» /R

ds'(GAt(s)GA(S'))B

By the Markoff approximation, we know that

(AHt)GA(s))B = 0 s > t<GAi(s)A(t))B = 0 s > t,

(A\t)A(t))RAt

(7.1.84)

(7.1.85)


so that as A* -»• 0 as far as system operators are concerned the (y/2)8 kt(AfA)R

term vanishes and we have

d_dt

= -y(A<A) + 2(DAU),

since<GAt(s)GA(s'))R = 2(DAU)B 6(S - s').

W th the aid of (7.1.17) and its adjoint, we may rewrite (7.1.86) as

(7.1.86)

(7.1.87)

(7.1.88)

which is called the Einstein relation to determine the diffusion coefficient.If we use (7.1.80), we see that (7.1.88) reduces to

2 < / W « = yn (7.1.89)

so that the diffusion constants may be obtained from a knowledge of themean equations of motion according to the Einstein relation (7.1.88).

7.2 QUANTUM THEORY OF LANGEVIN NOISESOURCES [1]

A quantum system experiences damping and fluctuations when it interactswith a reservoir as we have seen in the previous chapter. The Langevinapproach adopts the philosophy that the reservoir may be completelyeliminated provided that the frequency shifts and damping caused by thesystem-reservoir interaction are incorporated into the equations of motionand provided suitable quantum (operator) noise sources are added as drivingterms to the equations of motion, The moments or correlation functions ofthe Langevin noise sources must give the correct statistical behavior to thesystem (correct diffusion coefficients).

According to (6.3.36), the thermal reservoir average of a system operatoris given by

d(M)R = - - <[M, H])B ~ 2 &(<»i>h

fc - (Q&M, 0J)aw5}. (7.2.1)

We have taken into account the frequency shifts and damping caused by thereservoir in this equation. We may remove the reservoir averages if we add arandom Langevin force GM which will be a function of the system operators

12 QUANTUM THEORY OF LANGEVIN NOISE SOURCES [1] 433

as well as reservoir operators which has the correct statistical properties.We therefore write the Langevin equation

,*£ - Q,[M>^ = - [[M, H] -dt h i.j

sAM+ GM. (7.2.2)

In order for (7.2.2) to be consistent with (7.2.1), we must first require that

(GM(t))R = 0. (7.2.3)

Furthermore, since we have retained terms up to second order in the system-reservoir in (7.2.1), we see from the derivation of (7.2.1) given in Section6.3 that GM be only first order in the interaction. We therefore let GM begiven by

GM(0 = -i 2 dM> Qi1)n{Fi(t) ~ <*"«>*}. (7-2.4)where

pst) = eWMR(t-tt)f se-u/H)RU-u) (7.2.5)

We have obviously not "derived" (7.2.4). We show that it gives the correctresults for a damped oscillator and that its two time correlation functionsgive the correct diffusion coefficients in special cases.

We first note that the choice (7.2.4) satisfies (7.2.3). Consider again thespecial case of the damped oscillator. If we let M = a, and use (6.2.41) and(6.2.58), we see that

da

If we use (7.1.9), we see that this agrees with (7.1.17).If M — a*a, we have on algebraic simplification of (7.2.2)

d t t— a1 a = —ya;a + yn + GAiA

at= -i([afa, a^Frit) - fta'a, a])RF2(t)

(7.2.6)

(7.2.7a)

(7.2.7b)

which clearly agrees with (7.1.31) if we use the solution of the reservoiraverage of (7.2.6) and its adjoint.


Consider next a set of system operators a = {alt Oj , . . .} . Then theLangevin equation for aM is

da. i . „ .

For simplicity assume

i.i

(7.2.8)

(7.2.9)

It is always imperative that this high-frequency free motion be removed. Wetherefore let

*„(<) - e<a>'(<-'o)a;(0. (7.2.10)Then (7.2.8) becomes

where

=P--2 *«*-•*«[«;s ^; + G;,

G;=-»*2<W>

(7.2.11)

(7.2.12)

Of course, the Q{ and Qf will be functions of the a's and they should also betransformed according to (7.2.10). It then follows that

(7.2.13)dt

After removing the high-frequency behavior, we may repeat the analysisof Section 7.1 to derive the Einstein relation

i+At fM-AI

j ds'(G',,(S)G'v(s'))B

- T <«;«;>« - tow* - <4&)»at

while the random forces are ^-correlated and given by

(7.2.14)

Also, we have a statement of the Markoff approximation that

<F[a'(0]G;(s)>B - 0 if * > « , (7.2.15b)

where F is an arbitrary function of the system operators. The reader mayverify these results for the damped oscillator.

7.3 LANGEVIN EQUATIONS FOR A MULTILEVEL ATOM 435

7.3 LANGEVIN EQUATIONS FOR A MULTILEVEL ATOM

Let us apply the Langevin Theory to a multilevel atom coupled to areservoir. For simplicity let us adopt the notation of Equations (3.10.38) and(3.10.39) in the following equations. According to (6.3.56) and (7.2.8) if

(7.2.8) becomes

/ll/T. =

at

where by (7.2.4)

(7.3.1)

(7.3.2)

(7.3.3)

since

We must first remove the high-frequency terms from (7.3.2). We let

where to^ includes the imaginary part of Ffie. Then (7.3.2) reduces to

(7.3.4)

(7.3.5a)

^ ) - (fmn)R}. (7.3.5b)mn .

The reader should note that the b's obey the same orthogonality relations asthe a's. For example,

dt ~ ~w*^

where Yti is the real part of Ttie and

g«(0 = -

whereas (7.3.6)

(7.3.7)

in order for these two to give the same result.To convince the reader the random forces git give the correct diffusion

coefficients, we calculate them first from the Einstein relation

d_dt

- {b\b,AH)R - (7.3.8)

436

where

QUANTUM THEORY OF DAMPING—LANGEVIN APPROACH

Ai} - -Tublbt + 8it T wivb%. (7.3.9)V

As an alternative we use the random forces to calculate

l rt+At m(7.3.10a)

and show that the results agree. Note that the Einstein relation does notinvolve the random forces directly. Equation (7.3.10) implies the forces are^-correlated in this time interval so that

- *')• (7.3.10b)

By the orthogonality relations, (7.3.8) becomes

h)R - « « r *«,If we use the reservoir average of (7.3.5) and lety = /, we obtain

<- dik(-rit + r« + rwx&!&,>*blb^ - difwtk(blbt)R, (7.3.12)

where the wfi and w,f terms are missing.Let us next use (7.3.10). First we must obtain g(k(t) in a more useful form.

From (7.3.5), we see that

(7.3.13)

JW* tJ. (7.3.14)When i =y , we have an infinite set of coupled equations which we cannotsolve exactly. However, gH(t) is already of first order in the system reservoirinteraction and the correction to (7.3.14) for i = j due to the wi9 terms wouldbe of second order. We therefore may use (7.3.14) in gif when i = / .

If i ^y , the solution is

7.3 LANGEVIN EQUATIONS FOR A MULTILEVEL ATOM

By (7.3.10) and (7.3.5) we have437

Now (7.3.15)

(7.3.16)

We may therefore remove the slowly varying system terms from (7.3.15)evaluated at t0. This givest0. This gives

- 2 Mbt.mnpq

X {<fmm(.s)/W))x ~However, as we saw in Chapter 6,

If we therefore let

(7.3.17) reduces to

2{D'im)R

s - s' = T

(7.3.17)

(7.3.18)

(7.3.19)

i tbib^bibMKbibj^0 •/-{

* - </„„)*(/„)*}. (7.3.20)Again because ( / ^ W / J O ) ^ = 0 for T > re we may replace the upperlimit on the r integral by + oo and the lower by — oo. Also isince

we see that we may let A/ -»• oo in

Since eo'mn = —<o'PQ, we see that (7.3.20) reduces to

"•" ' & I . O .where

rfr.

(7.3.21)

(7.3.22)

(7.3.23)

(7.3.24)


(See Problem 7.6 to show why the last term in (7.3.20) does not contribute.)If we break this up from — oo to 0 plus 0 to +oo, use the fact that o)'VQ =—G>'mn and let T -*- —T in the integral from — oo to 0, we may easily show that

r w - _ (7.3.25)vmna»a W

where w± are given by (6.2.75a).Next we may take the limit as Af

becomes

-O(to-*t) in (7.3.23) which then

2(Dim)R = - £ <[&>„ blbn][btbt, bX])Rd((0'mn> -(7.3.26)

where the b's are now evaluated at time t. Finally, if we use the orthogonalityrelations and pick out the secular terms for which (a) m = q,n = p (m •? n),(b) m = n, p = q (m 5* p), and (c) m = n = p = q and use the definition ofthe T's in terms of the w±, we see that (7.3.26) reduces to (7.3.12).

Although the replacement of [bib,, blfin][blbi, b%bQ] evaluated at f0 by itsreservoir average at time t as to-*t may seem strange, the reader shouldnote that this is exactly the same procedure we used in obtaining the equationfor (dMldt)R in Section 6.3.

7.4 LANGEVIN EQUATIONS FOR N HOMOGENEOUSLYBROADENED THREE-LEVEL ATOMS

In this section we obtain the Langevin equations of motion for N identicalthree-level atoms which are independent, each coupled to its own independentreservoir. The atoms ultimately will be coupled to a radiation field of fre-quency Co, ^ (e2 — €^)jh = a>21 which coincides approximately with theenergy separation of the upper two levels of the atoms. This field will inducethe atoms to make radiative transitions between these two levels. The groundstate, 0, will act only as a source of atoms which may be pumped incoherentlyfrom the ground state to the upper two levels and a sink into which atoms inthe upper levels may decay. Therefore, we shall be interested in the operators(ft]a^x where i — 0,1,2 which give the probability of finding the Ath atom inlevel i and the dipole moment or radiative transition operators (a\a2)x and(«I°I)A for the Ath atom between the upper two levels.

The equations of motion for the Ath atom for these operators follow from(7.3.2):

7 (afo)* - (-ton* - T12\a\at)x + Gu\t)at

4at

7at

„VX«I)A "

(7.4.1)

(7.4.2)

(7.4.3)

7.4 LANGEVIN EQUATIONS FOR N THREE-LEVEL ATOMS 439

together with the adjoint of (7.4.1) and the completeness relation

(alao)x + (afoXi + (4a2)x = 1. (7.4.4)The Langevin forces are given by

Gti\t) = - i I i<[(aja,)a, (alan)x])R[fmn\t) - (fmnx)R], (7.4.5)

m—l n = l

for i, j — 1 or 2. The m = n — 0 terms are not included since these wouldintroduce operators other than those of interest. Again we are actuallyconsidering two level atoms which have another level as a source or sink.

Again we wish to remove the high-frequency behavior from (7.4.1). We,therefore, let ^ _ ^ ( , , , ( ^ ( 7 A 6 )

(ajai)x = (blbjx,where

= °>aX- (7.4.7)When we couple the atom to the radiation field, co0 will only be approximatelyequal to coa as we shall see.

If we now use (7.4.6) in (7.4.1)-(7.4.3) and assume that all atoms areidentical so that the transition probabilities, H1,-/, damping constants, I\-/Sand frequencies are the same for all atoms we may omit the A and we see that

T(blb2)x - +[Kcoo - coa) - rn](blb2)x + gl2\t) (7.4.8)at

at

7at

>v2 0 g22\t) (7.4.9)

+ wMbz)x + gn\t). (7.4.10)

In this case we say the atoms are homogeneously broadened. Here we haveused (7.4.4) to eliminate (bo

fbo)x and have let

w20

w10

(7.4.11)w21 = w21 - w20

w{2 = w12 - w10.

gi*\t) = e*«"*Gu\t) = I&Aor gtt\Q = Gu\t). (7.4.12)For homogeneously broadened atoms, thefkl

x =fki in (7.4.5).Since the atoms are independent and the reservoirs are independent, the

Langevin forces are independent. If we define

ff (b\b2)x = f (ala (7.4.13)

J = 0,1, 2


then for N atoms, (7.4.8)-(7.4.10) become

- «„) - T12]M + gu(r)dt

dt

dt

w20

W 1 0 -

(7.4.14)

From the orthogonality relations

and the commutation relations

we see that

(7-4.16)

[M, JVJ = -M [M\ JVJ = AT* (7.4.17)

[M, iVJ = M \M\ Nt] » - M f .

The random forces for different atoms are uncorrelated so that- «**<ft/(*)&iV)>». It therefore follows that

2 2X X' (7.4.18)

or f^+A«<fc'<g

From the above, we may write the explicit forces as

= -i(M(t))B[fM(t) ~ / u ( 0 - ((/«>* -

(7.4.19)

(7-4.20)

i(M(t))B[fu(t) - ha

7.5 LANGEVIN THEORY OF NOISE SOURCES

From (7.4.14) the reader may obtain the solutions

<*r(r)>« = M(t0) exp [/(«„ - (oa) - T12] t0)

441

(7.4.21)

'IP

wherea2 -

I 1 1 - 2 \ , » /

(7.4.22)

(7.4.23)

If 1 and 2 subscripts are interchanged everywhere in (7.4.22), we obtain(WI(0)JR- These solutions may be needed to obtain equations of motion foroperators such as MNt.

7.5 LANGEVIN THEORY OF NOISE SOURCES; ASSOCIATEDFUNCTION FORMULATION

We may also convert the operator Langevin theory to an associated c-number theory. If we write A,, and G,, in c-order in (7.2.8) as far as the systemoperators are concerned at time t, we have

(7.5.1)

Then since A,,* is in chosen order, we may write

^ " <V>* • (*W(fc)}>* (7-5.2)

If we compare this with (6.4.100) when M = a,,, we see that

.*,(&)-4K&). (7.5.3)

We may therefore obtain the drift vectors for the c-number Fokker-Planckequation by ordering A,, (7.2.8) and replacing at by a,. We may therefore


write a c-number Langevin equation for <*„

^ - * / „ + * / . (7.5.4)at

where ^ has also had its system operators put into chosen order in GM andat is replaced by at.

Similarly from (7.2.2) we may write

d (7.5.5)

where we assume <*„<*, is in chosen order. If we put A^ and (?„, systemoperators in chosen order we see that

Adt

We may write the c-number equationd_dt

If we use (7.5.4) we see that

d

dxy da. (7.5.7)

(7.5.8)

Under the Markoff approximation we clearly see by the techniques used inSection 7.1 in the operator case that

<«„<?, + *&}n = (2^V>B> (7.5.9)so that

at

= (aM< + a , ^ + 2 ^ , ) B . (7.5.10)

If we apply the # operator to each side, this agrees with (6.4.105). Therefore,

^ = aM.< + « , < + 2 ^ y + 9^ (7.5.11)al

We have for simplicity neglected removing the high-frequency free motion inthis section.

REFERENCES

The reader should show that in general443

M) (7.5.12)Since no ordering is involved in the operator Langevin equations for the

N damped atoms, we may replace the operators directly by their associatedc-numbers.

M-^Jt\ M*-+J?*; N^JTi. (7.5.13)

This also applies to the random forces as may be seen by inspection of (7.4.20).Thus quantum and c-number Langevin equations in this case are identicalin form. If we also use (7.3.12) for atom X and the result (7.4.18),

2{9'im) = N12<^1>je, (7.5.14)

we see that again no ordering is involved since only M, A/f, and Nt occursingly so that the classical and quantum diffusion coefficients also haveidentical forms in the cases in which i,j, k, and / equal 1 or 2.

PROBLEMS

7.1 Verify that the solution (7.1.16) and its adjoint satisfies the commutation re-lation [a(0, <>*(')] = 1-

7.2 If S(0) = (1 - r*yuU, evaluate the means in (7.1.45).7.3 Calculate the photon number spectrum for a damped driven oscillator using

the techniques of Section 7.1.7.4 Verify (7.1.47).7.5 Show that (7.3.26) reduces to (7.3.12).7.6 Show that the last term involving </mn>B</M>ie in (7.3.20) vanishes identically

when Ar -+ 0. Hint: Because </,nn(0)>B</i*(0)>je is not peaked, we cannotextend the limits on the r integral in this term to °o; A/ » o>mr^

x.7.7 Verify the commutation relations (7.4.17).7.8 Give a physical interpretation to each of the terms in (7.4.14).7.9 Obtain the diffusion coefficient 2<Z>2222>.R directly from the random force

^ M ( 0 in (7.4.19).7.10 Verify (7.5.9) under the Markoff approximation.

REFERENCES

HI M. Lax, Phys. Rev., 145,110 (1966).[2] F. Reif, Fundamentals of Statistical and Thermal Physics, New York: McGraw-Hill,

1965, Chapt. 15.

8Lamb's SemiclassicalTheory of a Laser [1]In Chapter 5 we considered stimulated and spontaneous emission andstimulated absorption of radiation by an atom. We showed if the light offrequency <w was approximately equal to the energy separation between twoatomic energy levels (hco ^ e,, — co = hw^ that absorption occurred if theatom was in lower level \a) and stimulated emission occurred if the atomwas in the upper level \b). Furthermore, we showed that the transition proba-bility per second for induced emission wai and absorption wt0 were the same.If we have a large number of atoms in a cavity consisting of two mirrorsand the gas is in thermal equilibrium at temperature T, there will be moreatoms in the lower state, JV0, then in the upper state, Nb, given by the Boltz-mann distribution

where k is Boltzmann's constant. If therefore light of frequency co^ («-6 — €„)/kT passes through the gas there will be a net absorption of radiation since

If, on the other hand, by some means we can invert the populations so thatthe number of atoms is greater in the upper level than in the lower level, anet amount of energy will be supplied to the radiation field by the atoms. Ifthe amount supplied is greater than the losses, the field in the cavity will grow.The process by which we achieve population inversion is called pumpingand is the source of energy for the field. As the field builds up to a large value,nonlinear effects come into play which tend to equalize the populations.When the populations are equalized, the atoms are said to be saturated andno further gain occurs. Then we must supply additional atoms with invertedpopulations to maintain the laser oscillation.

In Figure 8.1 we show a diagram of the first He-Ne gas laser [2]. The tubewhich is approximately 100 cm long is filled with a He-Ne gas mixture(1 torr pressure of He and 0.1 mm of Ne). The "cavity" is a Fabry-Perot

444

LAMB'S SEMICLASSICAL THEORY OF A LASER [1] 445REFLECTINGENDPLATE PLATE-ANGLE

ADJUSTER

REFLECTINGENDPLATE

RADIOFREQUENCYEXCITER

Figure 8.1 Sketch of He-Ne gas laser. (Reprinted from A. L. Schowlow, Sci. Amer.,June, 1961.]

interferometer consisting of two parallel plane semi-transparent mirrors [3].The pumping which we describe briefly is provided by an rf discharge asshown in the figure.

In Figure 8.2 we see a rough energy level diagram of He and Ne. The firstexcited level of He is a metastable level which means that the atom cannotbe excited in first approximation by an electromagnetic wave as we have justdescribed. The transition is forbidden by selection rules which means thedipole moment vanishes for these two states. However, an electron movingthrough the tube because of the discharge can collide inelastically with theHe atom and excite it to the metastable level. A metastable level also cannotradiatively decay spontaneously to the lower state because the dipole momentmatrix element is zero. However, the He ion can collide with a Ne ion in thedischarge and transfer its energy to the Ne by an inelastic collision. Thishappens because the Ne 2j-level is in approximate synchronism with the 23Smetastable He level. When energies are approximately the same, a resonanceexchange of energy on collision is very likely. Since the 2/>-levels of the Neare practically empty, we obtain a population inversion with more atoms inthe 2s-levels than in the 2/vlevel of Ne.

The oscillation starts with a spontaneous decay of a Ne atom from a 2s-to a 2/7-level. If the emitted radiation is emitted in a nonaxial direction in thedischarge tube, it escapes and is lost. If it is approximately axial, there is alarge probability that it will collide with another Ne atom and induce it toradiate. Since the probability for induced emission is proportional to theradiation present, it is greater than for spontaneous emission and the induced

446 LAMB'S SEMICLASSICAL THEORY OF A LASER [1]

20

19

<uc

UJ

17

16

23S-He

i

Ner= = 2 p

Losingtransition

Figure 8.2 Energy level diagrams of He and Ne.

field is in phase with the incident field. We therefore obtain a cascade buildupof the wave. At the mirror most of the light is reflected so that the mirrorsprovide a feedback mechanism for the buildup of energy. Light going in thewrong direction is lost by diffraction, some (say 5%) is emitted through thesemi-transparent ends (which act like an antenna) and some is lost in themirrors. The oscillations will build up from spontaneous emission if we havepumped enough atoms into the upper state to overcome these losses.

It is easy to see that the emerging laser light will be very well collimated,since it would escape by diffraction otherwise. Also the intensity will begreater than with conventional sources of light, since each atom emits inphase. In an ordinary source, the atoms emit their radiation by spontaneousdecay (rather than by stimulated emission) which is a random process.Consequently, the fields sometime add in phase and sometime cancel. Infact the laser intensity is orders of magnitude greater than ordinary sources.Furthermore, since the frequency is determined by the atomic energy levelseparation, it is a resonant phenomenon and will be very monochromatic.Laser light is also coherent.

It is of interest to discuss the resonator briefly [4-6]. We consider it inmore detail in the next section. Since the mirrors are assumed to be perfect

8.1 MODES IN "COLD" SPHERICAL RESONATOR 447

conductors, each mode will have an integral number / of half-wave lengths

where d is the mirror separation. For d = 100 cm and X = 6000 A, thiscorresponds to approximately 10* half-wavelengths. To obtain these normalmodes, one would have to solve Maxwell's equations subject to boundaryconditions on the mirrors and in free space, since there are no mirrors on theside. Fox and Li [4] circumvented this difficult boundary value problem byusing Huygen's principle. They assumed a given field distribution on onemirror and used Huygen's principle to calculate the field on the other mirror.After allowing for reflection loss, they used the calculated field on mirror 2to recalculate that on mirror 1. Each time some energy was lost out the sidesby diffraction. After several hundred such calculations, the field energydistribution did not change (except for diffraction losses), and this distri-bution was taken as a normal mode.

With this brief introduction, we present the semiclassical theory of alaser, neglecting all unessential complications [1]. We first discuss cavitymodes for spherical mirrors. We then proceed in two steps. First, we assumethe atoms in the cavity act as radiating dipoles which act as a source ofradiation in the cavity. Next, we obtain the dipole moments of the atoms whichare induced by the electric field in the cavity which is treated classically. Wethen solve for the field in a self-consistent way. Pumping and losses are takeninto account in a phenomenological way.

8.1 MODES IN "COLD" SPHERICAL RESONATOR

Instead of using Huygen's principle as Fox and Li as well as Boyd andGordon did to obtain the resonator modes, we present a simpler treatmentdue to Kogelnik and Li [5]. Although the latter is not as general, it has theadvantage of being analytically tractable.

We would like to solve the wave equation

(8.1.1)

inside a spherical resonator as shown in Figure 8.3. Each mirror has a radiusof curvature b and the mirrors are separated by a distance d. It is notnecessary for both mirrors to have the same radius. When b — d, theresonators are said to be confocal. The equation of the right mirror is

(8.1.2)


(8.1.4)

Figure 8 3 Diagram of symmetric spherical resonator. The radius of curvature is b andthe separation is d.

wherer2 = *« + y*. (8.1.3)

In the event r « 6, a standard approximation, (8.1.2) reduces to

Z~2 2b'In this approximation, we therefore cannot distinguish between a sphericaland paraboloidal mirror. The equation for the left mirror is under the sameapproximation

•«-f+s- < 8 1 - 5 )

We expect the laser beam to propagate approximately as a plane wave inthe z-direction but to have a nonuniform radial intensity distribution con-centrated near the axis of propagation. We also expect the phase fronts to beslightly curved. We would like to obtain solutions of the wave equation inwhich the mirrors coincide with a surface of constant phase.

As a zeroth approximation, let us look for a solution of the form

£. = rt*,y,*ytt~0, (8.1.6)where k = 2TT/A and

(8.1.7)

(8.1.8)

Then the wave equation becomes approximately

dy*where we neglected 92y/3z2 compared with dipjdz.

In an attempt to find solutions in which the surfaces of constant phasecoincide with the mirrors, we look for solutions of the form

y, *) = 4w(z


where for convenience we let449

L' R(z) kw\z)' (8.1.10)

and R and w are real. If we combine this with (8.1.6), we see that the constantphase surfaces are determined by

k\z + -~\L 2K(z)J

+ ReP(z) = constant, (8.1.11)

where Re is the real part and we used (8.1.10). This is very close in form to(8.1.4). The w(z) is a scaling factor and R(z) is effectively the radius ofcurvature of the wave front for a given z. The ReP(z) will not alter the phasefront very much as we shall see. The -v/2 is used for convenience.

If we substitute (8.1.9) and its derivatives into (8.1.8), we obtain after somealgebra

-{—. — ik\w— — — \S } + ~ {—: — ik\w « — }

gWf 2 L dz q j d e l hldrf L dz q \ ' drjl

where we have let

rj =w(z) w(z)

If we are to have any luck in solving this equation, we must choose P,w, and q or equivalently P, w, and R so that

dw w2

w— = Cl

dz q(8.1.14a)

(8.1.14b)

(8.1.14c)

Then the equation will be separable in f and rj at least. Unfortunately, theseequations are so nonlinear that the solutions are extremely awkward. If welet c3 =s 0, the solutions are then quite easy. In this case

\q dz)w*(dq \

7>\7z ~ V "Cs-

or?dz

q(z)

(8.1.15)


By (8.1.11) we see that R(z) plays the role of the radius of curvature of thewave front at a given z. In the present case the symmetry would suggest thatat z = 0, the wave front is a plane so that R(Q) = oo. By (8.1.10) and (8.1.15)we see that

kw *q = * + qo = *-i*J-, (8.1.16)

where w0 ~ w(0), a parameter still at our disposal. If we then use this and(8.1.10), we find after minor algebra that

R(z) = (W\2 [1 + (2z/fcw0t)8]

(8.1.17)

Thus by choosing c8 = 0, q, w, and therefore R are completely determined.By happy coincidence when we use (8.1.16) and (8.1.17) we see that

dw w8 2idz q k

which is a constant. Therefore, (8.1.12) becomes

ht

g US*where we have let

(8.1.18)

(8.1.19)

(8.1.20)

or when we use (8.1.16) and (8.1.17) we obtain on integrating

The solution then is of the form

Ex(x,y,z,t)

exp iiklz + - ~ - j - (1 + €4 + e,)tan"11, - a>*J exp -\(J-% + if),

(8.1.22)


where we have let

kw0*and also

451

(8.1.23)

( 8 > 1 .2 4 )

H>.2(1 + O- (8-1-25)The constant c2 = ef + e, is still at pur disposal. One can show that thesolution of

(8.1.26)

(8.1.27)

blows up faster than exp (££*) for large £ unless we let

2/M ( )where m = 0,1,2 Since Ex must be finite as £ -*• oo, this is the onlyphysical solution. Then the solutions are

g -where Hm(g) are hermite polynomials. Thus we have

(8.1.28)

x exp if fcz + —^-r —. - (m + n + 1) tan"1 rf. (8.1.29)r+1,

If we next require that at z = d/2, R(d/2) — b, the mirror radius of cur-vature, we see by (8.1.24) that

2d(b - d)2 (8.1.30)

Then both mirrors coincide approximately to surfaces of constant phaseaside from the phase term

(m + » + Dtan-1 ± - = L = = = * ± f (m + n + 1), (8.1.31)

which for moderate m and n is small compared with the ±kd/2 = ±ird[Aterm.

Finally, the phases at z — ±dj2 and r = 0

kd . . .

- d/2)(8.1.32)


To determine k and therefore the resonant frequencies we require that the

round trip phase change

«2*rJ, (8.1.33)= 2M-4(m+ „ + !).»• ^ m z r m

where / is an integer. Thus the normal mode frequencies are given by

irl + 2(m + n + 1) tan"1, . 2ird dkd = -— = - o>mnl

Km cwhere c is the velocity of light. In the confocal case b — d and

a W - ^ P + Km + n + D]-a

For a CO8 laser resonant at 10.6 n and 100 cm long we see that

l+$(m + n + 1 ) ^ 2 x 105.Since m = n — 0 is the lowest order transverse mode

l~2 x 106 (8.1.37)and we see that the resonant frequencies are primarily determined by thelongitudinal quantum number / in agreement with the statement made inthe introduction. Since H0(x) = 1, the fundamental TEMW, mode is given by

(8.1.35)

(8.1.36)

. V> z> 0

exp

where

and

(8.1.38)

(8.1.39)

(8.1.40)

(8.1.41);2d(fc - d\2)

It should be remembered that we have assumed the mirror radii are smallJnpared with their radius of curvature so that these modes are approx,maTdycorrectforlowordermodesinwm^Tear the centers of the mirrors so that diffraction losses are qu.te small.

com

8.2 THE CAVITY FIELD DRIVEN BY ATOMS 453

8.2 THE CAVITY FIELD DRIVEN BY ATOMS

In this semiclassical theory of a laser oscillator we treat the field classically.The field obeys the Maxwell equations (4.8.1)-(4.8.4). Since there are nofree charges, we let p — 0. In the laser medium we take the constitutiverelations

B = ^0H; D = c0E + P; J = oE; (8.2.1)P is the polarization of the medium and will act as the source of radiation todrive the field in the cavity. The third relation above accounts phenomeno-logically for the cavity losses including diffraction. If we take the curl ofboth sides of (4.8.2) and use (4.8.4) and (8.2.1), we obtain the wave equation

curl curl E + ~ + Wjj = ~f*o^, (8.2.2)

where we see that the polarization drives the electric field. We are interestedonly in the solutions to the inhomogeneous equation, since the loss term willcause all solutions of the homogeneous equation to damp out. As we havenoted, the driving term will be provided by the dipole moments of the atomswhich radiate into the cavity mode. In this part of the analysis, we assumethat the polarization is known and calculate the field it produces in the cavity.

We assume the field is plane polarized in the x-direction and we neglectthe transverse field variation for simplicity. Then (8.2.2) reduces to

dt2 • eodt " dz2 ~~eo a / - ' ; ( 8-2 '3 )

When we neglect the losses and driving term, we obtain the wave equation

= c dz2 ' (8.2.4)

The solutions which satisfy the boundary conditions Ex(z, t) = 0 at z = Qand z = d for all time are

(8.2.5)

(8.2.6)

(E^+EV*1*) sin Knz,where the eigenfrequencies are

„ v tin 2-ncd An

n is an integer of order 108 when d — 100 cm in a typical laser. The modesare separated by

— = — An ~ 150 MHz, (8.2.7)2v 2d

for d = 100 cm and An = 1.


To attempt to solve (8.2.3) let us expand the field in normal modes

Es(z, 0 = 2 An(t) sin Knz, (8.2.8)n

since Ex must still vanish at the perfectly conducting mirrors. We assumethe losses are due to diffraction. If we substitute (8.2.8) into (8.2.3), multiplyboth sides by sin Kmz and integrate from z = 0 to z = d, we obtain

dt c0 dt c0 dt

where we have let

P«(0 - \d Jo

, 0 sin Kmz dz. (8.2.10)

If the losses are small, the resonant cavity frequencies will not change verymuch from the loss free frequencies given by (8.2.6) and the driving term willexcite these frequencies. We therefore look for solutions of the form

Am(t) = g(t)e-"" + e*(tyni, (8.2.11)

where we expect co to be approximately one of the loss free frequencies Q m

and that &(t) will be slowly varying:

(8.2.12)dt

«

The polarization is expected to oscillate at frequency

— «s co;<*>„ —(8.2.13)

we therefore assume we may expand the polarization as

PJf) = P(t)e-iai + P*(t)eiat, (8.2.14)

where if the atomic lifetimes in states \a) and \b) are long compared with the

atomic period we expect that

\dP\dt

«<o\P\. (8.2.15)

If we substitute (8.2.11) and (8.2.14) into (8.2.9) and use the approximations(8.2.12) and (8.2.15), we obtain

(8.2.16)dt 2(o

8.3 THE INDUCED ATOMIC DIPOLE MOMENT 455

together with its complex conjugate. Here we have assumed small loss so that

and have let

a2e0co

Since we expect co «* fim on coa, we have

2co


dt

2co

& s l^ P.

(8.2.17)

(8.2.18)

(8.2.19)

(8.2.20)J Ûnder the approximations above (essentially small damping and long atomiclifetimes) this is the equation for the field amplitude driven by an atomicpolarization whose frequency is approximately in resonance with a naturalcavity mode frequency. We have removed the high-frequency field andpolarization oscillations by the expansions (8.2.11) and (8.2.14) so that £and P are expected to change very little during a period of the high-frequencyoscillation. It is the first fundamental equation of a laser oscillator. In thenext section we study the effect the field has in inducing the atomic polariza-tion.

8.3 THE INDUCED ATOMIC DD7OLE MOMENT

In the previous section we derived an equation for the electric field in acavity which was radiated by an atomic polarization. In this section we derivethe polarization for a large number of atoms which is induced by a givenelectric field in the cavity.

Consider a two-level atom in a cavity in which a polarized electric fieldexists. The Hamiltonian is

H=HA+V, (8.3.1)

where HA is the free atom hamiltonian and

V = —a • E(r, 0 = — exEx(z, t)

= -esWO*-"" + £*i(tf*\ sin Knz, (8.3.2)where we have assumed that only one mode of the field is excited and that itis polarized in the z-direction. The field is evaluated at the position of the"atom."


We now look for solutions of the Schrddinger equation

f)> (8.3.3)•" at

of the formIV& 0) - ca(z, 0etot/l|a> + cb(z, 0 r * * » , (8.3.4)

where we shall find later that hoi PH cb — ea. The two atomic states are orthog-onal and assumed complete in the present context. The exponential timefactors were introduced for convenience and cause no loss of generality. Ifwe insert (8.3.4) into (8.3.3) and use the eigenvalue equations

#J*>=»««tf> ' - « , * , (8-3-5)we obtain

dca ico 1 Jdcb ico 1Idt 2 J

= €acaeiat'*\a) + €bcbe-iat'i\b) + Viz, Ofc/"""!*) + c5r*>w |t>]. (8.3.6)

If we next take the scalar product of both sides of this equation with (a\ anduse the orthogonality relations (a\b) = (b\a) = 0, we obtain

17 " -*(? + l)c« + ^ <fl'F<z' W>e~"% (8.3.7)at XL n/ in

We have assumed as is usually the case for most lasers that the diagonal matrixelements (a\V\a) and (b\V\b) are zero. In the integration

(a\V(z, t)\b) = ~e(a\x sin Knz\b)[£e-iat + cc], (8.3.8)the atomic wave functions are strongly peaked near the center of mass ofthe atom, and we may remove the sin Knz evaluated at the atomic center ofmass.j If we let

fi = e<a|*|6> = e{b\x\a)*, (8.3.9)then (8.3.7) becomes

1 ? = -» ( f + €f)ca + i£ sin Knz[*(t)e-*iat+ *\t)]cb. (8.3.10)at \2 n/ n

Under the rotating wave approximation, &(t)exp (—2icot) is very rapidlyvarying compared with £*\t) and wiD average approximately to zero. Wetherefore neglect it and obtain

f sin KnzS*(t)ch. (8.3.11)

t In this case we do not make the "dipole approximation," since atoms are spatiallydistributed throughout the cavity.

8.3 THE INDUCED ATOMIC DIPOLE MOMENT

In a similar way we obtain457

s i n

These equations give the time development of the probability amplitudes forfinding the atom in levels \a) and \b) in the presence of a driving electric field.

So far we have not taken into account the finite lifetimes of the atom inlevels \a) and \b). We have assumed the energy levels are infinitely sharp.Based on our prior work, we assume there is a lower energy level to whichatoms in the two levels may decay whose energy separation is large com-pared with et — €a so that the driving field will be far out of resonance andwill not induce transitions to the ground state. We then introduce the atomiclifetimes TV"1 and Ff1 so that (8.3.11) and (8.3.12) become

i^sin Knz**(t)cb (8.3.13)

(8.3.14)

We are interested in obtaining the polarization induced by the field in theactive medium. This is defined by

P(z, t) = N(xp(zt t)\ex\w{z, t)), (8.3.15)where there are iV atoms per unit volume in the cavity. If we use (8.3.4) andlet {a\x\a) = (b\x\b) = 0, we have

P(z, 0 = N{c

where we have let(8.3.16)

Pniz, t) = JV>c*(2, t)cb(z, t). (8.3.17)

We therefore see that we need the bilinear combinations c*cb and cac* ratherthan ca and cb separately. Also we need to know the mean number of atomsper unit volume in the two levels given by N\ca(z, t)\2 and N\cb(z, r)|2.

If we multiply both sides of (8.3.13) by c* and add it to its complex con-jugate, we obtain

| cX " ~TacX + l]£h £*cX - ^ #cac*j sin Knz, (8.3.18)which gives the time dependence of the probability of finding an atom inlevel \a). The first term represents the loss of atoms out of level \a) due tosuch incoherent processes as spontaneous decay to lower levels, collisions,

Vi

LAMB'S SEMICLASSICAL THEORY OF A LASER [1]

d such. The second term causes the population to change by the inducedion of the radiation field present. To have laser action, we must in addition

provide some pumping mechanism to pump atoms incoherently into levels\a) and \b); thus we add a phenomenological pumping term and we obtain

458

andactio

at Z, 0 Z, 0sin Knz fr\

(8.3.19)

where we multiplied both sides of (8.3.18) by N, used (8.3.17) and have let

,t)\*. (8.3.20)

Also Ra(z, t) — NwM(z, f) is the rate at which atoms are pumped from theground state to state \a) seconds per unit volume.

If we next multiply both sides of (8.3.19) by (2/tf) and integrate over the

cavity from z = 0 to z = d, we obtain

where we have let

d Jo(8.3.22)

Pn(t) = -. \dpn(z,t) sin Knzdz.d Jo

The reader may proceed in a similar fashion to obtain

-7[i '»*(*K*(0--P (8-3-23)

together with the conjugate to (8.3.24). We have let

(8.3.25)

4598.3 THE INDUCED ATOMIC DIPOLE MOMENT

Figure 8.4 Pumping and decay rates for three-levelatom. The ground state acts as a reservoir for atomsand the laser action is between levels a and b.

The physical meaning of these equations should be quite clear. Equation(8.3.21) gives the net rate of change at which atoms are entering and leavingstate \a). The Ra term gives the rate at which atoms are being "pumped"into level \a). The — TaNa term represents the incoherent decay of atoms fromlevel \a) to lower levels. We could also add a term +wabNi to represent in-coherent transitions from \b) to \a), but we omit this for simplicity. TheF,,'1 is the lifetime of the atom in level \a) in the absence of driving field.These first terms are incoherent since they contain no phase information[see Figure 8.4].

The last term i{P£* — P*S) represents the net induced population changein level \a) due to the presence of a driving field. A similar interpretationapplies to (8.3.23).

Equation (8.3.24) gives the net rate of change of the induced polarizationdue to the presence of a driving field. The first term allows for the decay ofthe induced polarization at the average rate Tab. This decay can arise fromspontaneous decay, collisions, and such. We could redefine rBft as

nPhaselab (8.3.26)

to allow for phase interrupting elastic collisions as we have seen from ourprevious work.

The term — i(coa — co)P shows that we have removed the high-frequencybehavior from P since it will turn out that o> &* coa and P(t) is slowly varyingin a time of order coa~K The deviation of to from «„ represents a detuningeffect due to the coupling of the field to the atoms.

The last term |/|2(JVS — NJtf represents the induced polarization of theatoms by the field. The field causes the atom to oscillate between levels \a)and \b) which induces a dipole moment.

Equations (8.3.21), (8.3.23), (8.3.24), and its conjugate together with(8.3.15) and its conjugate represent the complete set of coupled equationswhich describe a laser oscillator. To solve these nonlinear equations exactly isimpossible. We therefore proceed to make some approximations which arevalid for a gas laser. We should note that we have neglected such things asthe inhomogeneous broadening effect of Doppler motion. The reader shouldrefer to Lamb's classic paper for such a discussion.

it


8.4 ADIABATIC ELIMINATION OF THE ATOMIC VARIABLES;PROPERTIES OF THE OSCILLATOR

In a gas laser the "cold" atomic linewidths To and Tb are large comparedwith the "cold" cavity linewidth y:

Tb»y. (8.4.1)

If we neglect the weak coupling between the atoms and the field in (8.3.21),(8.3.23), and (8.3.24), the polarization Pn decays to zero and the populationsapproach the steady-state values Rjrb and /?B/ro in a time of order T^-1.From (8.2.20) the field & approaches zero in a time of order y1 in theabsence of the atomic driving term. Since F » y we see that the atomicvariables will decay so quickly that the field has not had time to changeappreciably. The atomic variables will therefore follow the electric fieldadiabatically when the coupling is weak. This allows us to eliminate theatomic variables as we shall show.

Consider (8.3.24). The polarization and populations are changing veryrapidly compared with the field. Let us therefore integrate both sides over atime tx such that

-»*i»^-. (8.4.2)

This givesLa»

dt>- < 8 A 3 )

We removed &(t') from the integral since its value does not change muchbetween t' = 0 and t' = tx « y~K Since tx» Tah-\ the first term will bevery small and we neglect it. The second term gives us a "coarse grained"average of P(t'). If we let

(8.4.4)

and

then we have

1 f'»- - P(f) dt',

tx Jo

= a,b

w -

(8.4.5)

(8.4.6)

8.4 ADIABATIC ELIMINATION OF THE ATOMIC VARIABLES 461

A similar "coarse grained" average of (8.3.21) and (8.3.23) gives uponusing (8.4.6)

0 = Rb - VbNb - i g i - p t j _ £ _ ^ W ( ^ - #„) (8.4.7)

rr2l<*r (#>-#»)• (8.4.8)

We here assume Ra and J?s are time independent. Since N} — Nc*ct (j = a, b),we see that the quantity

wL a 6

t2 ]i 2 -tW2 (8.4.9)

is the probability per second that the field induces an atom in level b to goto level a as well as an atom in level a to go to level b. When detuning isneglected, <o = coa, and we have

•W = ffl!jf-. (8.4..0)n 1O6

which is the maximum transition probability.We may now solve (8.4.7) and (8.4.8) for the "coarse grained" average

populations where we averaged over times large compared with the lifetimesof the excited states of the atoms. We easily find that the population differenceis

- Na0

where we define

and*>o = 7T J = a, b,

^ + (<oa-<o)r(8.4.11)

(8.4.12)

The population difference adiabatically follows the instantaneous fieldintensity |«^(0l2. As we know, the field will build up when Nb(t) - ffa(t) > 0.For fixed pump rates, Rb and Re, we see that as the intensity builds up, thepopulation difference will start to decrease. This nonlinear saturation effectcauses the laser to come to a steady state intensity. Physically, we start withmore atoms in level b than level a. They decay spontaneously and are inducedto go from level b to a and a to b. Since the induced probabilities are equal,more will come down than will go up with a net emission of radiation. Thisprocess tends to equalize the populations with a reduction in the build up of


Figure 8.5 Real and imaginary parts of nonlinear susceptibility of atoms induced by aresonant radiation field in the steady state.

the field. A dynamic equilibrium is reached between the pumping and thelosses and gives a steady-state oscillation.

The polarization (8.4.6) which is induced by the field with the help of(8.4.11) reduces to

(NM-NaB)[co-coa-irah] | /

(8.4.14)

where we have defined a complex nonlinear susceptibility of the atoms whichdepends on the field intensity /(/) s 1^(01*-In Figure 8.5 we sketch %' andX" versus the detuning when the intensity has its steady-state value andNw > Na0. The polarization increases as the field first builds up but willalso saturate when the nonlinear term becomes dominant.

We have therefore succeeded in obtaining the polarization in terms of thedriving electric field alone under the assumptions that Fa and I \ (the atomiclinewidths) are large compared with y (the cavity linewidth). Part of theinduced polarization is in phase with the inducing field and will add to itwhile part is out of phase due to losses (decay of atoms to the ground state).The field radiated by the dipole must also be the field causing the dipole forself-consistency. We therefore substitute F(t) for P(t) in (8.2.20). This givesthe equation for the field

dt

where

2 2 I V + (a) - ©„)* + pl

o.

- *«*

(8.4.16)

The y/2 term is the cavity loss while the second term with &0 gives the gain.

4638.4 ADIABATIC ELIMINATION OF THE ATOMIC VARIABLES

Let us first study the steady-state behavior of the oscillator. In this casedS\dt — 0. Since & 3* 0, we see from the above that each square bracketmust vanish separately. That is, ;

Lo»(8.4.17)

a> , -G B = - ^ ;

These two equations determine the intensity in the steady-state, /„ and theoscillation frequency, cot. If we use (8.4.17) in (8.4.18), we see that

or the steady-state frequency is given by

where we have let

ab(8.4.19)

(8.4.20)

which is the ratio of the cavity linewidth to the average atomic linewidth. Wehave assumed that ro + I \ )g> y so that «$" « 1. Therefore, to first orderin Sf, the steady-state frequency is

S"o>tt)

• • •. (8.4.22)The cavity tends to oscillate at the cavity resonant frequency rather than <oabecause the 'Q' of the cavity mode is high.

The steady-state intensity from (8.3.17) is

>. - coaf = Ta6 (—° -

We sketch this detuning curve in Figure 8.6. Since / , =negative. There will be no steady-state oscillation unless

or by (8.4.16). ^ * **»

(8.4.23)

?,|2, / , cannot be

(8.4.24)

(8.4.25)

464LAMB'S SEMICLASSICAL THEORY OF A LASER [1]

M

Figure 8.6 Detuning curve of oscillator which is the steady intensity versus the detuning.

as we see from Figure 8.6. Therefore, the population in level b must exceed thepopulation in level a by this amount to overcome the losses and build to asteady-state value. The equality is the "start oscillation" condition for thelaser. We obtain maximum steady-state intensity if <ot ?& QB «* o>a. Lesspump power is needed if JVa0 = 0. It is therefore desirable to have Fo largeand Tb small so that the lower state decays much more rapidly than the upperstate. Then since Tab fa TJ2 the pump threshold reduces to

(8.4.26)

When to, = e»o, the maximum steady-state intensity by (8.4.23), (8.4.13),and (8.4.16) becomes

r,r» |"*9a-™r _ K .1 - — 1 (8.4.27)

Far above threshold, the second term here may be neglected.We may obtain the steady-state population difference from (8.4.11),

(8.4.13), (8.4.16), and (8.4.23):

(8.4.28)

(8.4.29)

( 8 A 3 O )

If we let the threshold value by (8.4.25) be

then the steady-state population is

i J !

8.4 ADIABATIC ELIMINATION OF THE ATOMIC VARIABLES 465

This varies between the limits (Nw — Ngo)th when cot = <wo to JVM — N& =

(RbIF) when to, = co0 ± Vl*(flr0 - yTab)jy.If we use (8.4.29) and (8.4.16), we may rewrite (8.4.23) as

t*I, + (« . - «>«) - raft — —— - 1 . (8.4.31)

Let us next consider the transient behavior of the oscillator. For thispurpose let us write

^ ( 0 - v ^ W * * * " ' (SA.32)

where J(t) = 2 is the intensity and q> the phase. Then since

d l _ ,d<p-i<f (8.4.33)

if we put this into (8.4.15) and equate real and imaginary parts, we obtainthe two equations for the intensity and the phase

17, = -r +

dcp— = co-Q.dt

(co — co,)

2 rab*

(8.4.34)

(8.4.35)

We may integrate the first equation exactly. Let

(8.4.36)

Then it may be written as

dl (fi +

I fr-P-ydt,

or as partial fractions

rft a


so that on integrating we obtainI . (,_0_,j)^( ^ ( g A 3 7 )

where Io is the initial intensity. Unfortunately, this is a transcendentalequation so we cannot obtain I(t) explicitly. It may be rewritten as

L Jo - /J - (il(t)(8.4.38)

It is actually easier to visualize the behavior of the intensity in two limitingcases.

CASE 1/«/«/? = Tal* + (o> - to,)* (far below threshold)

In this case the solution of (8.4.34) reduces to

(8.4.39)

which shows that if

or(8.4.4Q)

the intensity will grow exponentially as the oscillations start to build up from theinitial intensity (see Figure 8.7). Note that the intensity will build up only if radiationis initially present. This arises from our semidassical treatment in which fluctuationsdue to the spontaneous emission of the atom are neglected when the field is treatedclassically. We correct this in the next chapter.

The phase equation (8.4.35) in this case reduces to

r

».

?>(O - KO) • -n« + (8.4.41)

Figure 8.7 Buildup of laser intensity from belowT threshold to its steady value.

PROBLEMS

CASE 2

In this case, by (8.4.34) we have

467

(for above threshold)

Idtor

t-»0O' y *

which is the approximate steady-state value by (8.4.23) when ftl, » ro62 +

(a — «o0)2 (see Figure 8.7).

We therefore see that the intensity starts to grow exponentially and as it builds up,it saturates and approaches a steady-state value.

The phase approaches a steady-state value also under these conditions.

REFERENCES

[1] W. E. Lamb, Jr., Phys. Rev., 134, A1429 (1964); 1963 Varenna Summer SchoolLectures, Course XXXI, p. 78; M. Scully, 1967 Varenna Summer School Lectures,Course XLI. '

[2] A. Javan, W. R. Bennet, Jr., and D. R. Herriott, Phys. Rev. Lett., 6, 106 (1961);A. L. Schawlow, Set. Amer., June 1961.

[3] A. L. Schawlow and C. H. Townes, Phys. Rev., 112,1940 (1958).14] A. G. Fox and T. Li, Bell Syst. Tech. J., 40,453 (1961).[5J H. Kogelnik and T. Li, Proc. IEEE, 54,1312 (1966).[6] G. Boyd and J. P. Gordon, Bell Syst. Tech. /., 40,489 (1961).

PROBLEMS

8.1 If we define a density matrix by

P - IvWXvWIwhere |y(0> is given by (8.3.4) and where its matrix elements are given by

Pa(0 - <»l P(t)\f> i,j = aotb

show by using (8.3.13) and (8.3.14) with sin K& = 1 that p satisfies the equa-tinntion

%=]£&, pl


where we have let

and [A, B] m AB - BA and {A, B}+ m AB + 2M. This is just a model for atwo-level atom with linewidth. r • * .

8 5 Expand the nonlinear terms in (8.4.15) in powers of 1 - h where / , is thesteady-state value. To first order in (/ - /.) show that the resulting equationis just that for a classical rotating wave van der Pol oscillator.

8.3 From (8.4.34) and (8.4.35) solve for the relation between <p and /.

9Statistical Propertiesof a Laser

9.1 THE LASER MODEL [1-4]

In the prior chapter we discussed the semiclassical theory of a single modehomogeneously broadened laser in which we treated the electromagnetic fieldclassically. To discuss the statistical properties of the laser oscillator output,we must repeat the analysis and quantize the radition field. In addition wemust account for the damping of the radiation field as well as the quantumstatistical fluctuations of the reservoir associated with the damping as we didin Chapters 6 and 7. Finally, we must account for the fluctuations in theatomic system due to various damping mechanisms.

In Figure 9.1 we show a block diagram of the laser model. The reservoircoupled to a mode of the field is shown which accounts for the mode dampingand the quantum fluctuations which the damping introduces into the field.We have discussed such a reservoir in detail in Chapters 6 and 7 since thefield mode is equivalent to a damped harmonic oscillator. Below we show Nthree-level atoms per unit volume coupled to a pumping and dampingreservoir. Each atom is thought of as coupled to its own reservoir and theyare coupled to each other only through the atom-field interaction. Again wediscussed the damped atoms in detail in Chapters 6 and 7. Finally, the atomsand field are coupled by a dipole interaction as discussed in the prior chapter.However, we treat the field quantum mechanically.

The causal unperturbed hamiltonian of the atom and field is

Ho =2 le,(olfl l) i + fa»0atfl, (9.1.1)

where we are using the notation introduced in Section 3.10. Again we let

be the "number" operator for the number of atoms in level |/>.

(9.1-2)

469

9Statistical Propertiesof a Laser

9.1 THE LASER MODEL [1-4]

In the prior chapter we discussed the semiclassical theory of a single modehomogeneously broadened laser in which we treated the electromagnetic fieldclassically. To discuss the statistical properties of the laser oscillator output,we must repeat the analysis and quantize the radition field. In addition wemust account for the damping of the radiation field as well as the quantumstatistical fluctuations of the reservoir associated with the damping as we didin Chapters 6 and 7. Finally, we must account for the fluctuations in theatomic system due to various damping mechanisms.

In Figure 9.1 we show a block diagram of the laser model. The reservoircoupled to a mode of the field is shown which accounts for the mode dampingand the quantum fluctuations which the damping introduces into the field.We have discussed such a reservoir in detail in Chapters 6 and 7 since thefield mode is equivalent to a damped harmonic oscillator. Below we show Nthree-level atoms per unit volume coupled to a pumping and dampingreservoir. Each atom is thought of as coupled to its own reservoir and theyare coupled to each other only through the atom-field interaction. Again wediscussed the damped atoms in detail in Chapters 6 and 7. Finally, the atomsand field are coupled by a dipole interaction as discussed in the prior chapter.However, we treat the field quantum mechanically.

The causal unperturbed hamiltonian of the atom and field is

21=0

+ hotâ, (9.1.1)

where we are using the notation introduced in Section 3.10. Again we let

W,-£(aViDi (9-1.2)

be the "number" operator for the number of atoms in level |/>.469

9.2 THE FOKKER-PLANCK EQUATION FOR A LASER

(9.1.1)-(9.1.5)wehave

471

{where<$«(« - a) m <S(cc* - a

X <5( T2 - iVj) 6{JK - M) 3(« - a). (9.2.3)

We have already evaluated the last two terms in (9.2.2), since the atom andfield operators commute. The field term for normal ordering is given by(6.4.37) with v(() = 0 and the atomic term is given by (6.4.90) under theassumption that the number of atoms in levels 1 and 2 are small comparedwith those in the ground state, although we still may have a large number inthese upper levels. The Ho term has also been included in (6.4.37) and (6.4.90).We therefore need only to evaluate the Tr [S, W] term. We have

J = ± TTS [5(0, WA_Fin a) = \ TTS S(t)[6%* - «), WA_rl (9.2.4)ih

where we used the cyclic property of the trace. If we use (6.4.15), (9.2.3),and (9.1.3), we have

— (afM — ]

X 3(a) 8(a.*) d(Jf) diJ?) di^) 8(J^2). (9.2.5)

We must next use the commutation relations to put all terms into chosenorder. Since these techniques have already been developed, we leave thedetails as an exercise. The result is

(9.2.6)

When Jfi and JTZ are small compared with N, we may expand the expo-nential to second order as we have done in our previous work. This yields

(9.2.7)

STATISTICAL PROPERTIES OF A LASER

so that the interaction term reduces to

If we combine this with (6.4.37) and (6.4.90), we obtain the desired Fokker-

Planck equation

- (r12 +

I

1 JL- [Rt2 a^ra*

(9.2.9)

9.3 THE LASER ASSOCIATED LANGEVIN EQUATIONS 473

We have let co2I = wo above. The first derivative terms give the mean driftmotion while the second derivative terms give the diffusion.

9 3 THE LASER ASSOCIATED LANGEVIN EQUATIONS

We may immediately write down the Langevin c-number equations ofmotion by inspection from the Fokker-Planck equation (9.2.9). According tothe theory given in Chapter 7, we have

dx (v \j t

= ~\2 + ico°rdJldt

Mil.dt R2

dt= J ? , -

(9-3.1)

(9.3.2)

(9.3.3)

(9.3.4)

together with the adjoints of (9.3.1) and (9.3.2).The frequency <ue^ a>a and the laser will oscillate at a frequency close to

this, say co0. We may remove this high frequency from the equations by letting

a(0 = x'

Then (9.3.1>-(9.3.4) become

dJTdt

Mldt

dt i i i

where the new random forces are related to the old by

g. =

(9.3.5)

(9.3.6)

(9.3.7)

-, (9.3.8)

,. (9.3.9)

(9.3.10)

474 STATISTICAL PROPERTIES OF A LASER

If we neglect the random forces, these equations reduce to the semiclassicallaser equations of Chapter 8 as we shall show.

We have that

where / = <l|e*|2>. The polarization is

Vffi

(9.3.12)

since we removed the high frequency in (8.3.24) also. Furthermore, we makethe identification

(9.3.13)

With these identifications, (9.3.6) reduces to (8.2.20) and the others reduce to(8.3.21), (8.3.23), and (8.3.24) aside from the nonradiative transition termswî-yî and wlvA

rs which were neglected in the semiclassical approach. We

also assumed here that /t = ft* for simplicity.The moments of the Langevin forces may be read off by inspection from

the Fokker-Planck equation (9.2.9), since from the general theory of Chapter7 we have

<gi(0g,(u))R = 2(0«> d{t - u) (9.3.14)As an example

<g«*(0g»>« = - u)

[R2

(9.3.15)(9.3.16)

9.4 ADIABATIC ELIMINATION OF ATOMIC VARIABLES

In a typical gas laser the atomic decay rates (linewidths) may be of order108 sec"1 while the radiative decay rate y (cavity linewidth) may be of order10' sec"1. Under these conditions we may adiabatically eliminate the atomicvariables as we did in the semiclassical theory in Chapter 8. To simplifymodel for tutorial purposes without losing its essential physical features, wefirst assume that I \ is very large compared with other decay rates. This meansthat atoms in level 1 decay to the ground state so rapidly that we may set

JTX - 0, (9.4.1)

and neglect the Jfx equation 9.3.9 completely. Also the distribution functionwhich satisfies the Fokker-Planck equation (9.2.9) becomes independent of

9.4 ADIABATIC ELIMINATION OF ATOMIC VARIABLES 475

Jfx and we may therefore neglect all derivatives with respect to Jfx\

0.(9.4.2)

Under this approximation the Langevin equations 9.3.6-9.3.8 reduce to

Hi = ~ [2 + Ko>e ~ "*]*'dt ~ l

where

In addition we assume

- J B H

L12 Z±L + ru*~L

(9.4.4)

(9.4.5)

(9.4.6)

(9.4.7)

is very large compared with y and ra so that we may neglect dJt'\dt com-pared with YXJM'. We may therefore solve (9.4.4) for the adiabatic value of

' + *"

If we use this and its conjugate, (9.4.3) and (9.4.5) become

4f = MTTJT2 -y + i\~^{co0 - «O - 2(cuc - o,0)l)a'dt 21 L I12 Ji

^ = Rz- (r8 + -n \^)jrat

1r1 2-/K-where we have let

, , • • ( . „ . -

IT =

^2 + K - <oaf

W)

(9.4.8)

(9.4.9)

(9.4.10)

(9.4.11)

(9.4.12)


In order to be self-consistent, as I \ -»• oo, F12 -* oo, we must also assumeft -*• oo in such a way that tt defined above remains finite to retain the essen-tial physical features of the model.

Equation (9.4.9) is a Langevin equation in which (fa(t))R — 0 and the firstcurly bracket term in (9.4.9) gives the entire drift motion. However, thesquare bracket term involving gjf and gM% in (9.4.10) has a finite reservoiraverage which we must evaluate and add to the drift motion before it is acorrect Langevin equation in which the reservoir average of the random forceis zero. Consider therefore the quantity

/=<«'*(0Sur(0>«. (9A13)

We may rewrite this as

(9.4.14)

where te = t — e and ultimately we let e -*• 0. Under the Markoff approxi-

mation<« f*to£ur(0>*-0 t>te. (9.4.15)

If we use this and (9.4.9), we obtain

" y "

ds + JY[«.K.) + ™f®B J«C\L r i 2 + i(eoo —

(9.4.16)

where we used (9.4.12). Again under the Markoff approximation since t > s,the first integral vanishes. Next since the forces are d-correlated we have

~ 0" 0,

so that / becomes on carrying out the integrals

(9.4.17)

. (9.4.18)

In a similar way we see that

(9.4.19)


Therefore, we see that the reservoir average of the g^ and gj^, terms in(9.4.10) become

- i(o>0 - (ott) T12 + i(co0 -

(9.4.20)

These coefficients may be read off directly from the Fokker-PIanck equation(9.2.9):

(9.4.21)

We used (9.4.7) and the fact that T12 » T2. If we therefore include the con-tribution (9.4.20) in the drift motion and define a new random force

- /* f| , (9.4.22)

where OL'C = a ' (O whose reservoir average vanishes we may replace (9.4.10)with

However,dt

IT

12

(9-4-23)

(9.4.24)

for a gas laser so that the entire correction to the drift motion of f"2 isnegligible and we have for the equation of motion for the population differ-ence

(9.4.25)at

w h e r e / ^ is given by (9.4.22). The reservoir average of the gjt andin (9.4.10) make negligible contribution to the drift motion.

If we take the reservoir average of both sides of (9.4.9), we obtain

- coa) - 2(coe -

In the steady-state, we see that the real and imaginary parts must vanish sothat we obtain the steady state frequency of the laser when

y(a.')R, (9.4.26a)

478

so that

STATISTICAL PROPERTIES OF A LASER

• (a>0 — (ft>e — tt>0), (9.4.26b)

which agrees with the classical result (8.4.19). Since F12 » y, we see that«>o = °>e- If w© tune the cavity so that «oe s* o0 , we may neglect detuningwithout serious error. The laser equation for the field and populations there-fore reduce to

at

dt|a'| V*

It =

where

• 1 2

(9.4.27)

(9.4.28)

(9.4.29)

(9.4.30)

(9.4.31)

We next need the two time correlation functions of these forces. From(9.2.9), we have by inspection (neglecting the nonradiative transition proba-bilities wlt s= w21 = 0)

- u)

- B])B

u)

u)

(9.4.32)

(9.4.33)

"))!, (9-4.34)

We have used (9.4.7) to simplify (9.4.34) where Vlt» T2 and let Jrl=Q

All others vanish. The B is given by (9.4.6).We next replace Jt' in the correlation functions above by its adia-

batic value. Therefore, we have on using (9.4.6), (9.4.8), (9.4.18), (9.4.19),

9.4 ADIABATIC ELIMINATION OF ATOMIC VARIABLES

and (9.4.21)

(B)R = *<|«'|» jrj + JL

479

Since 7r/ri2 « 1, (9.4.32) reduces to

(9.4.36)

>fl d(t - «). (9.4.37)

Equations (9.4.33) and (9.4.35) are unaltered, while (9.4.34) becomes

J ^(«)>i . (9-4.38)

We may next use (9.4.37), (9.4.38), (9.4.33), and (9.4.35) to evaluate thetwo time correlation functions of/, and/^-, given by (9.4.30) and (9.4.31). Wetherefore have that

2I\ 2• <gAt)gA«))n

• 1 2

Also, we have (9.4.39)

Similarly, we have

tr] ) N B ^ - « ) . (9.4,(9.4.40)

and (9.4.41)

(9.4.42)We may use these correlation functions to write down a new Fokker-Planckequation which describes the field and population. See Problem 9.3.


We next assume we may neglect dJTJdt compared with IVT 2 sinceP* 2> Y' From (9.4.28) we then obtain the adiabatic population difference

(9.4.43)

When we put this in (9.4.27), we obtain

d%' I f **2 1 / . r ,*AAA\

We leave it as an exercise for the reader to show that the reservoir averageof the last term in (9.4.44) is negligible compared with the drift term [5] (seeProblem 9.4). The Langevin equation for the field alone may therefore be

written as

where

(9.4.45)

(9.4.46)

The two-time correlation functions are found to be

12

/ _

\(T.

2ri2 Tr2|«f A+ *r|a'|") 2(r2 + ir|a'|2)2)/

(9.4.47)

B 2<Da»/> 3(< - ti).

In the limit in which 2F12 «« Tt -*• oo, we see that

(9.4.48)

\xt-u)

2<D«/> W - u). (9.4.49)


We may also obtain the Langevin equation for the photon number. Wehave by (9.4.45) and (9.4.46)

y a 'V = a'at a'V = a ' * « + <?„) + a'«* + <?a*)

t - y] l«f + cfflF. + «'^*. (9.4.50)^ t

The reservoir average of the 0 e and &£ terms are easily seen to be

Therefore, we rewrite (9.4.50) as

dt

where â«a is a new Langevin force with zero reservoir average. The ratio ofthe second to first term in irR2 is

At very low operating levels

V

l«'l2

At high operating levels

At intermediate levels

We therefore may always neglect the second term in R2 so that

dt

(9.4.53)


If we neglect fluctuations, we may set ^ B . B = 0 to obtain the steady-stateoperating intensity:

IT

(9.4.54)

We may use the drift vector (9.4.46) and the diffusion coefficients (9.4.48)and (9.4.49) to write the Fokker-Planck equation

+ £ D-Fp<- + 5 = « ^ » + s f e "*** <»-4-55>which describes the statistical properties of the radiation field of a singlemode homogeneously broadened laser when the atomic linewidths are largecompared with the cavity linewidth.

9.5 THE LASER AS A ROTATING WAVE VAN DER POLOSCILLATOR

Consider the Langevin equation for the field (9.4.45)

where

(9.5.1)

(9.5.2)

where we have letI=W\\ (9.5.3)

Let us linearize this equation about the operating intensity given in (9.4.54).That is, we write

r 2 + TT/ B T2 + TT/0 + TT(7 - 70) (9.5.4)and

If we let

- 1

(9.5.6)

9.5 THE LASER AS A ROTATING WAVE VAN DER POL OSCILLATOR 483

then

(9.5.7)

This expansion is very good in the range of operating intensities

which includes the threshold region and far above since

(9.5.8)

(9.5.9)

as we shall show. In this range we see from (9.4.48) that the diffusion coeffi-cients reduce to

( 9 5 1 0 )

Under this approximation, (9.5.1) is seen to be the equation for a rotatingwave van der Pol oscillator.

We may express the Langevin equation (9.5.1) in terms of dimensionless(scaled) variables as follows. Let

t — 7V a ' — £(2.

Then (9.5.1) becomes on using (9.5.7)

where

Also

dr 2

T,

l '

(9.5.11)

(9.5.12)

(9.5.13)

(9.5.14)

We may now choose | and T as follows. Define a pump parameter g as


Here we neglected rtl compared with Vt as well as w-compared with I \ . Atthreshold g — 0 while g > 0 above threshold and g < 0 below threshold.

We next require that the coefficient of |/J|2 be unity in (9.5.11) as one con-straint to determine £ and T. Thus

yTS

2so that (9.5.12) becomes

By (9.5.14) we have that

But since

dr

(9.5.16)

(9.5.17)

we have\<*\

<M

(9.5.19)

(9.5.20)

since / = Jr. We require this to be

as our second conintensity scaling is

while the time scaling is

At threshold, p = 0, so that

Accordingly, we have

4(5(^-1-,) (9.5.21)

as our second constraint. From these two constraints, we easily see that the:y scaling is

(9.5.22)

(9.5.23)

If we neglect n, we see that at threshold

which is the threshold intensity.

(9.5.24)

(9.5.25)

(9.5.26)

9.6 PHASE AND AMPLITUDE FLUCTUATIONS

Similarly, we see that at threshold

r,)or

We may therefore write g as

-

S

485

(9.5.27)

(9.5.28)

(9.5.29)

We therefore conclude that the laser is equivalent to a rotating wave van derPol oscillator over a wide range of operating intensities.

Under the approximation that I <,I <, TJv, the Fokker-Planck equation(9.4.55) becomes

dPdt'

When we use our scaled variables, this becomes(9.5.30)

% - - 1 to - KTOi- - ^ to - WVF + J ^ J : « . (».

which is the form we have already studied in great detail in Chapter 6.

9.6 PHASE AND AMPLITUDE FLUCTUATIONS, STEADY-STATE SOLUTION, LASER LINEWIDTH

Since we are interested in phase and amplitude fluctuations let us transformthe Fokker-Planck equation (9.4.45) into these variables. We let

a' - (9.6.1)

so that

= a'a'*; tp = - In — (9.6.2)


By the usual rules of differentiation, we see that

dV = \dl dtp

9 , 9 r j i 9— a = — I A =9a' 9/ 2 dtp92 _ 92 _ 9 92 1

9a'9a'* 9/2 9/ 99>24J

a8 _ a' . _ i a2 , * a , . a2

9a'2 9/* 49?>2 2 9? 9?>9/J

After minor algebra we see that (9.4.55) becomes

dP, 9 9* 9*

where we used (9.4.46), (9.4.48), and (9.4.49), and we have let

7_ J)

Also we have let

so thatP(a', a'*, 0 - 2Pr(/, y, r),

Jp(a', a'*, 0 d a' - jptf, tp, t) dl dtp = 1.

The corresponding Langevin equations are

(9.6.3)

(9.6.4)

(9.6.5)

(9.6.6)

(9.6.7)

(9.6.8)

(9.6.9)

9.6 PHASE AND AMPLITUDE FLUCTUATIONS

where the first moments are

6(s - u)

487

(9.6.10)= 0.

We should note that only at extremely high operating levels do we need retainthe diffusion correction to Ax in (9.6.8). This follows, since if ly> 1

^ / 1 _ - as

L r2 + 7r/Jsince F2 ^ TT. Therefore,

Ajs* f^2- - y)/ + yn I»1. (9.6.11)

Because Ar and the diffusion coefficients depend on / only, the Fokker-Planck equation (9.6.4) is separable, and we may look forsolutionsofthefonn

Px(J, ?, t) = e-ueil*Q(I), (9.6.12)where / is an integer or zero in order that Px be single-valued. Then (9.6.4)becomes

fj^jj (DIXQ) - AjQ^ + {X - PD^Q = 0,fj^jj ^ (9.6.13)which is an eigenvalue equation. The steady-state solutions correspond toX = 0. The steady-state solution which is independent of 1, the reader may easily show that

dDu

dl«

(9.6.14)

(9.6.15)

(9.6.16)

(9.6.17)

Therefore, if we let h = 0 for simplicity, we see after minor algebra that

di DU

il^ AL . «*/Jk ^ ! _ il) (t + si) (9.6.18)di D R\R rj\ r/


We used (9.5.24). In the vicinity of threshold, we may neglect the quadraticterm in w//r2 so that the steady-state solution becomes

(9.6.20)which is a gaussian for

0 < Rt < 2Rith.

As a next application let us obtain the laser linewidth due to phase fluctua-tions above threshold where the amplitude is stabilized sufficiently that wemay adiabatically eliminate the intensity.

From (9.6.5) and (9.6.10) we have that

41

where the mean population difference is

(9.6.21)

(9.6.22)

We used (9.4.43).In the region for above threshold (a factor of 10 in photon number), the

fluctuations in / become relatively small. This stabilization of the oscillatorpermits us to replace / in (9.6.21) by its operating value given by

j o = * 2 _ I ? (9.6.23)y tr

(compare 9.4.54). Since/is stable, the Fokker-Planck equation (9.6.4) reduces

which is a diffusion equation which has the Green's function solution (as wehave seen earlier)

(9.6.25)

The steady-state distribution is

_1_2ir'

(9.6.26)

PROBLEMS 489

The laser spectrum when amplitude fluctuations are suppressed is given bythe Fourier transform of

(a\t)a(0)) =But by (9.6.25) and (9.6.26) we have

(9.6.27)

, t/<p0, 0) dtp

2TT 2n —** dq>

The spectrum is

r i n

which is Lorentzian whose full width at half power is

4/n

(9.6.28)

(9.6.29)

(9.6.30)

and is due to phase fluctuations at this operating level.Lax and Zwanziger [6] have recently applied these techniques to study

laser photon statistics near threshold. Stephens [7] has also used thesetechniques to study the noise properties of Josephson junction oscillators.

REFERENCES

[1] M. Lax, Phys. Rev., 145,110 (1966).[2] M. Lax and W. H. Louisell, IEEEJ. Quantum Electron., QE-3,47 (1967).[3] M. Lax, Phys. Rev., 172, 350 (1968).[4] M. Lax, Phys. Rev., 172, 362 (1968).[5] M. Lax and W. H. Louisell, Phys. Rev., 185, 568 (1969).[6] M. Lax and M. Zwanziger, presented at VII International Quantum Electronics Con-

ference, Montreal, 1972, and to be published.[7] M. J. Stephens, Phys. Rev., 182, 531 (1969); M. O. Scully and P. A. Lee, Ann. N.Y.

Acad. Sci., 168, 387 (1970).

PROBLEMS

9.1 Derive (9.2.6).9.2 Give a physical interpretation to each term in (9.4.27) and (9.4.28).9.3 From the two-time correlation functions (9.4.39M9.4.42) and the drift

vectors given by (9.4.27) and (9.4.28) write out the new Fokker-Planck equa-tion in the variables a', a'*, and f*2. The new distribution function depends


only on these variables. How is it related to the old distribution function of(9.2.29)?

9.4 Evaluate the reservoir average of the last term in (9.4.44) and show that it isnegligible compared with the drift term.

9.5 Derive (9.4.51).9.6 Calculate the mean number of photons in steady state using (9.6.19).9.7 Justify (9.6.27).9.8 Calculate the laser spectrum far below threshold.

Appendix A

Method of Characteristics

(A.1)

Consider the two equationsdx dy dzP ~ Q~ R'

where P, Q, and R are functions of a;, y, z. Assume we can find a solution ofthese equations of the form

u(x, y, z) = a, (A.2)

where a is a constant. Then, we have that

0 = du = —• dx + — dy + — dzdx dy dz

_ fdu ,dudydt± dzl ,Ldx dy dx dz dx\Vdu duQ duRl ,

dx dy dzj P

where we used (A.1). From (A.3) it therefore follows that a satisfies thepartial differential equation

dx dy dz(A.4)

as well as the system (A.I).From (A.2) we also have when we assume x and y are independent variables

du du dzdx dz dxdu dudz .dy dz dy

(A.5)

491

492

When we

APPENDIX A

solve for du\dx and du/dy and substitute into (A.4), we have

or

dy

dx oy(A.7)

That is, u(x, y,z) — a also solves (A.7). Therefore, to obtain solutions for(A.4) or (A.7), we need only find a solution of (A.1). If /, m, and n are anyfunctions of x, y, z, we also easily see that

I dx + m dy + n dz = [/ + m &• + n —"1 dxL dx dx}

dxmQ + nR)—, (A.8)

(A.9)

where we used (A.1). Therefore, we see that

dx dy dz _ Idx + mdy + ndzP~Q~R~ IP + mQ + nR

If we can solve the original set, we may be able to find functions /, m, and nwhich we may solve. In fact if we can find l,m,n such that

Idx + mdy + ndz*=0, (A.10)

then this is an exact differential and may be integrated so that

du — 0 = / dx + m dy + n dz,

and it follows from (A.8) thatIP + mQ + nR = 0

also. But this is just (A.4), sincedu , du du .. . , .— = J; — = m; r - = « (A.13)dx dy dz

if Idx + mdy + ndz = du is exact.It is straightforward to generalize these results to more variables: solutions

of the set.*&..;._& (A.14)

(A.11)

(A.12)

APPENDIX A

also satisfy

<=i oxt

493

(A.15)

Equation (A.1) are the so-called characteristic equations associated with(A.4) or (A.7). We next show that if u(x, y,z) = a and v(x, y,z) = b are twoindependent solutions of the characteristic equations [which also satisfy (A.4)or (A.7)], then any arbitrary function

<f>(u, v) = 0or equivalently,

u = g(v)

also satisfies (A.4) or (A.7). This follows, since we have

?2 _ ivffhi I §£ d£\ . (hpfdv , dv dzl _ Q9a; 9«L9a; dz dxj dvLdx dzdxj

9y ~ duldy dz dy\ dvldx dzdyj ~

(A. 16)

(A.17)

(A.19)

However, since u and v both satisfy (A.5), both relations are automaticallysatisfied identically.

Appendix J3

Hamiltonian for Radiation Fieldin Plane-Wave Representation

We derive the hamiltonian (4.3.45) for the energy in the radiation field in acubical cavity in the plane-wave representation.

We substitute for E and H from (4.3.44) into (4.3.45) and obtain, afterminor algebra,

X f dr [<*„(*) exp (ik, • r) - au,\t) exp ( - ik, • r)]J cavity

x [(hrA*) « p (*i ' • r) - ««'/(*) exp ( - ik r • r)]. (B.I)

Since, by (4.3.36),

we see that

- I dr exp [±i(k, + kr) • r] = dr._,

- f dr exp [± »(k, - k,.) • r] = «,,.,.T J cavity

(B,2)

(B.3)

If we substitute these into (B.I) and carry out the sum over /', we have

(since «i s «_,)

H - 2 —».«,»' 4

494

APPENDIX B

By the ordinary rules of vector analysis,

(gto x k\)• (Sto- x fe«) = [8to x ( £ , x ! „ )

495

(B.5)since €,„ • k", = 0 and £ , • £ , = 1. We also have by the same argument

(€„ x fc,) • («_„. x ft.,) = (gto • t^ykt • £_, = - ( # „ • «_,„.), (B.6)since fc_, = —fc,. From (B.6), the last term in (B.4) vanishes identically,while by (B.5) the first term reduces to 2daa. and so

2,1.9 2

in agreement with (4.3.45).

Appendix

Momentum of Fieldin Cavity

In this appendix we evaluate the classical momentum associated with theradiation field in a cavity.

If we substitute E and H from (4.3.44) into (4.3.47) we obtain

G - - f 2 2 V^v Pi. * (•?•• * Ml2c \.«vc

x - I dT[ai9(t) exp (ik, • r) - a^t) exp ( - »k, • r)]T ./cavity

x [a,v(0 exp (ik,- • r) - aVa.\t) exp ( - ik,. • r)]. (C.I)The integral is identical with the one evaluated in Appendix B for the field

energy. If we use this result, we see that G becomes

= 2 ' 2Cx fc,)]

. x k_,)]}. (C.2)

We have omitted the explicit time dependence for simplicity.By a well-known vector identity, we see that

( C ' 3 )

since S,oand fc_, = —fe,. If we use these, (C.2) reduces to

APPENDIX C 497

since k, = cofcjc. We now show that the last sums over /, a, and a' vanishidentically. The ala and a_lar commute classically as well as quantum-mechanically. Since k, = — k_t and since a' and a are dummy indices, wemay write for the first sum over /, a, and <f

A similar argument shows that the afar term also vanishes. Therefore, (C.4)reduces to the value given in (4.3.48) of the text.

Appendix D

Properties of TransverseDelta Function*

The transverse b function is defined by

V(P) - i 1 Va ~ &)<&),] exp (Ik, • p) (D.I)

or

where k = fck, k* = |k|2, and dk = dkx dkvdkx.We shall now derive some useful properties of dfy

T(p).

1° <5«T(P) = V ( p ) -This property follows by simple inspection of (D.2) since 8if = 6H anckfc.

2° a</(p) - 8itT(-9) (D.4)

If in (D.2) we let k -»- —k, then dk -*• —dk and we have from (D.2)

(D.3)

J.00

00

3-

Q.E.D.

(D.5)

* These properties of the transverse S function were presented in a course at StanfordUniversity by Dr. D Walecka.

498

APPENDIX D 499

From (D.2), when we differentiate with respect to xj and sum over/,

However,

since

f dx{

This follows from (D.3) and (D.5) directly.

5° di(T(p) - 8

where

However,

±± fcoêxp(fk.p)_ r<ofc,fciexP(ik.p)dXtdXjJ-ao k2 J-oo fc*

Therefore (D.10) may be written

p)B

Q.E.D

(D.6)

(D.7)

(D.8)W = ' ^ JJJrfk exp (ik • r) = J(») d(y) d(z)—00

is the ordinary three-dimensional Dirac d function since

«$(*) = i - f"rf*,e** (D.9)2 T T J— oo

To prove (D.7), we have from (D.2) and (D.8)

,T7, f" * IT e*P (*' P>-(2ff) •'—oo K

The last integral may be indirectly evaluated from electrostatic theory.The potential of a charge e is

A-nr(D.13)

APPENDIX D500

This satisfies the Poisson equation

V*K = -e d(t). (D.14)

We therefore find that

V * ( - ) = - 4 i r 8 ( t ) = - — 9 \ d k e x p ( i k • r ) , (D.15)

where we used (D.8). It is easily verified that

J L . _l_rêxp0k.r)Airr (2ir)*J k% V '

that is, if we operate with V2 on both sides of (D.16), we see that (D.15) issatisfied. If we then use (D.16) in (D.12), (D.7) follows:

dxk dxl

= r-r'.p = r r .This property follows very easily by differentiating (D.2) alternately with

respect to xk and x'k.

Appendix

Commutation Relationsfor D and B

In this appendix we derive the commutation relations (4.6.17) and (4.6.18) ofthe text.

Since B = curl A, we see that in the Schrodinger picture

From (4.6.5) of the text and the symmetry properties (D.3) and (D.4) ofAppendix D, we may write (E.I) as

) , B^r')] = ih -?-, <513r(p) - ifi

oy(E.2)

where p = r — r'. From (D.2) of Appendix D,

= + (2TT) 3 J-CO fc2exT> ( ik • p)

(E.3)

If we substitute these equations into (E.2), we see that [D^r), 5j(r')] = 0 inthe Schrodinger picture so that (4.6.17) of the text follows in the Heisenbergpicture. (A similar proof follows for [D2, Bz] and [D3, B3].)

501

502 APPENDIX E

We next consider the commutator for two perpendicular components,say Dx and Bz. We therefore have

M Btf)] =

(E.4)

J

where we used the commutation relations (4.6.5) of the text.Again from (D.2) and (D.8) of Appendix D, we see that

(E.5)

If we put these into (E.4), we have

(E.6)

since p = r — r'. Similar proofs follow for [D2, B3] and [D3, Bx] so that(4.6.18a) is proved. The proof of (4.6.18b) is similar.

Appendix

Heisenberg Equations ofMotion for D and B

We derive the Heisenberg equations of motion given in (4.6.20) of the text.The hamiltonian is given by (4.6.19). Since by (4.6.13a) all components of Dcommute, we see by (4.6.19) that for the x component of D

ih i ^ d i = J _ rdT> [Di(r> 0> B^f 0]>dt 2pQ J

where B2 = Bf + B2* + B32. From (4.6.17), [P l t Bf\ - 0 so that (F.I)

reduces to

ih j ^ = -*- fdr' {[Dx(r, t), B,V, 01 + [U** 0, **(*'> 0]}- (F.2)dt 2/t^J

From Problem 1.8(/), we see that

[Di, Bi2] = [Ox. KM + B&Dlt B& (F.3)

where we use the notation B' = B(r', t). From (4.6.18), Eq. F.3 is given by

(F.4)

since (9/3z)<5(p) is a c number and commutes withSimilarly,

>dyIf we substitute (F.4) and (F.5) into (F.2), we have

(F.5)

t'- ° H - (R6)503

504

But from the definition of the 6 function, we know that

>dr'/(r ')3(r-r ')=/(r),—00

so that (F.6) reduces to

.»<*Pi(r,Q MrdBJt, t) dB2(r,t)lunL dy oz J

f°J—

APPENDIX F

(F.7)

(F.8)dt ft9L dy

But B = fi0H, and the expression in brackets is just the x component of thecurl B.

Similar proofs follow for the y and z components of D, and we have proved(4.6.20a) of the text. We leave as an exercise the proof of (4.6.20b).

Appendix IJT

Evaluation of FieldCommutation Relations

We show here that

. t),j * (A(r\ 0)2 rfr ] = ihAffy, t).

Since r' is a variable of integration, we may write the left side of (G.I) as

2 J jdr' [Ak(t), Atftftf)] = ih 2 jdr' Atf) dk?(r - n (G.2)where we used the identity

[A, BC] = [A, B]C + B[A, C],and the commutation relations (4.9.2).

To proceed, we need another property of the transverse d function.We consider the expansion of an arbitrary vector

(G.3)

B(r) = 4 A , exp (-i%. r), (G.4)

where €,„ (a = 1,2,3) are three mutually perpendicular unit vectors and Biaare expansion coefficients.

We may write B as the sum of its component parallel to k, (longitudinal)and its two components perpendicular to kt (transverse). We have

B(r) = B r + BL. (G.5)If we let 6W = kj/|k,|, then from (G.4)

(G.6)i2 iLi I <r=l

= -k 15l3gI3 exp (-ik, • r),Ju I

where div B r = 0 and curl Bx = 0.

505

506

We consider now the integral

APPENDIX G

- , (G.7)

We shall show that/-V(O, (G.8)

that is, the transverse d function projects out the transverse component of thevector.

If we put (G.4) and (D.I) into (G.7), we have

i 2 2i^L i - 1 I ff-1

If we note that

recall that

X 2 ^ exp [ikr • (r -

l C31dT exp [l'*r ~ kl)'rl =

-k r i k>,) . (G.9)

(G.10)

(G.ll)

and carry out the sum over /' in (G.9), then

T7* *— *->,Lt I «—•

It is easily shown that

s

)ii—

i-X= 0

if a =1,2

if a = 3.

(G.13)

When we use this, (G.12) becomes

I - exp ( - ik , • r'

From (G.6), this reduces to (G.8), the desired result. If we recall that duT =

6itT, we see that (G.I) follows from (G.2) when (G.7) and (G.8) are used.

Appendix Xx

Evaluation of Sums inEquation (5.10.17.)*

In this appendix we evaluate the sum in (5.10.17) of the text.We consider the sum over the polarizations a. Figure H. 1 shows fc„ 0,, q>o x,

and x'. For simplicity, we have assumed that x and x' have only x components.The propagation vector has components

kt — (sin 0j cos <plt sin 0, sin <pu cos 0,). (H.I)

The unit orthogonal polarization vectors 6n and ia may be taken as

&a = (—cos 0, cos <pt, —cos 6t sin <plt sing,2 = (sin 9?,, - cos <plt 0),

and the atom dipole moments have components

x = |x| [1,0,0]

(H.2)

(H.3)

With this choice of polarization vectors, the sum over a in (5.10.17) becomes

We may change the sum over I in (5.10.17) by means of (4.5.4) and (4.5.10)to an integral

2 - TTTs f V da>i fsin *« d6i r V » (H"5)

I (2irc) Jo Jo Jo* E. Fermi, Rev. Mod. Phys., 4, 87 (1932).

507

508 APPENDIX H

Figure H.1 Geometry for propagation vector and atomic dipole moments,


0- e*\x\\x'\<o<o' f" *td*1Ll-e«m~r*\* * ' 2^0(27rc)8i Jo [i(o> - ft),) + y^/2](a), - ft)')

X J'sin 0, deS'dvt (sin8 p, + cos8 0, cos8 p,)e**lr cos "•

The integrals over <pi and 0, are easily evaluated:

f t /*«»

sin8 9>i'<If>t = I cos8 <Pid(pt — 7r. (H.7)Jo

(H.6)

If we let u = cos 0,, we have

[sin fc,r cos fc,r sin fc,r (H.8)

We make the assumption that the atoms are located many wavelengths apartso that ktr » 1. In this case, we need retain only the first term in (H.8) so that(H.6) reduces to

(H.9)

eg -1- o- n - - 1 e*|x| |x>l mm> [°° sin (ft)t/c)r[1

r 47rVc 8 i Jo [f(o> - <o{) +

since a>t = ckt.

APPENDIX H 509

It has been assumed that \\yA, the half life of atom A, is very short so thatthe emission may be considered as part of a continuous spectrum. Since theintegral in (H.9) is very strongly peaked at a>, = a>', we may remove the factor[i(o> — eo,) + y/2]"1 from under the integral sign and replace a>, by a/.For the same reason, with little error, we may let the lower limit on et>, go to— oo. In this case (H.9) reduces to

1 e8 |x[ | x ' | coa>'fi0

J^00 <jq>t sin (ft)t/c)r[l —

- 0 0 ft), — ft)

To evaluate the integral, we let

2TJ-£ = O , — <w',

and the integral may be written as

'»]. (H.10)

(H.11)

f ^ s i n ( 2

. (o'r^di= sm — —

C J-oo 4cos

•oo £ c

— i cos — I — sinc J-oo £ c

(1 — cos 2TT*£)

sin 2ttt I

+ cos2irrl

sin (1 — cos 2irtt)c

(o'rC^dt— — si

C J-oo £

. . <o'rf«d£ . 2w£— J sin — I — sm 2trtl cos 3 .

C J-co C C(H.12)

The first two integrals vanish since the integrands are odd functions of £. Thelast two are easily evaluated by means of the following integrals:

(1)I*00 sin rx

J-oo Xdx = (H.13)

This may be verified by taking the Laplace transform with respect to T of bothsides.

(2) fa sin px cos qxdx = (ir p>qJ-oo x [0 p < q

This is proved by writing the integrand as

sin/«; cos qx = ^[sin (p + q)x + sin (p — q)x],

(H.14)

510APPENDIX H

and using (H.13). When we use (H.13) and (H.14), (H.12) reduces to

r0

crc

Therefore, (H.10) becomes

f°<l;2;0;0=1

,+Ut'r/e

"'r 4nHi[Ko) - a/) + yJ2]

This is the result we wished to prove.

(H.15)

t<r-cr

t>~.c

(H.16)

Appendix 1

Wiener-Khintchine Relations*

Let v(t) be a random variable. Define

(v(t) for -T<t<T

[0 otherwise.Let V(<o) be the Fourier transform of v^t):

V(a>) = 7- f° vÔe-^' dt'. (12)2.7T J—to "

If we multiply by ei<o* and integrate over dm from — oo to +00, we obtain

where we used the result

If v is real, it follows that2TT J—

(1.3)

•>. (1-4)

(1.5)

(1.6)

where »<w(r) is the value of v(t) for the Arth system of the ensemble and Nis thenumber of systems in the ensemble (JV is very large).

The time average of v(t) for a given system of the ensemble is defined by

V*(co) = F(-<o).The ensemble average of v(t) is defined by

where Tis very long.

* See Ref. 6, Chap. 4.

IT J-T a.7)

511

512 APPENDIX I

An ensemble is stationary with respect to v if there is no preferred origin intime so that the ensembles {vlk)(t)} and {vm(t + t')} have the same statisticalproperties. If in the course of time vw(t) for each system passes through allvalues accessible to it, the ensemble is ergodic. For an ensemble which isergodic, we have that

^ ( 0 = K0, (1-8)

which means that the time average will be the same for all systems in theensemble. Moreover if the ensemble is stationary

(1-9)

will be independent of t. Similarly, for a stationary ensemble,

»>• (1.10)

Therefore, the time and ensemble averages are equal for stationary ergodicensembles.

The correlation function of v(t) for a stationary ensemble is

(t + s)). (1.11)

(1.12)

It is independent of t since v is stationary. Note that

If we take the Fourier transform of K(s), we have

2irJ-

2iT J—oo

e~imK(s) ds

>ds,

which is defined as the spectral density of v(/). If we multiply both sides ofei<at and integrate over all co and use (1.4), we obtain

K(s) = f"J(co)e+ia" dm.J-IO

(1.14)

Equations (1.13) and (1.14) are the Wiener-Khintchine relations.We next relate these to V(co). If »(/) is ergodic and stationary, we have

K(s) = <t*0Ms)> =

(1.15)

APPENDIX I

If we use (1.1), we have (on using (1.3) twice)

XT f-Jdo.

i r00^ ,f°°= 2T J— J W" T J-oo J rft°' K ^ F (= - \ d c o V{co)V{~(»)eia"

1 J—CO

" I dt' ei(a+a'H'

1 •'—

where we used (1.5). Comparison with (1.14) shows that

513

which is the spectral density of v(t). Note also that

= <»*> = £ fVcoOprfcu.i J—00

(1-16)

(1-17)

Appendix J

Atom-Field Hamiltonian underDipole Approximation*

In this appendix we show that under the dipole approximation, we mayapproximate the hamiltonian for an atom interacting with an electromag-netic field

by

H = — [p - eA(r,2m

2mwhere jx. = ei is the atomic dipole moment and £ is the electric field.

If we make the unitary transformation

- wp [i j - A(r, o] W0> » U\x(t)),the Schrodinger equation

otbecomes

ihUot

^ \X) = HU\X),ot

(J.I)

(J.2)

(J.3)

(J.4)

(J-5)

where U is defined by (J.3) and H is given by (J.I). Since

we have on multiplying (J.5) from the left by IP

ot(J.6a)

* Paul I. Richards, Phys. Rev., 73, 254 (1948); sec E. A. Power and S. Zineau, Phil.Trans. A., 251,427 (19S9) and references contained therein.

514

APPENDIX J

where

KsV'HU-iHU' —dt

From (J.3) we see first that

In the Coulomb gauge

515

(J.6b)

Tra'Tt- wdiv A = O, (j.8)

and when we neglect the field source [p = 0 = J], we have [cf. Section 4.8]

(J.9)B(r, 0 = curl A.

If we use (J.7)-(J.9), the transformed hamiltonian (J.6b) becomes

K = -a • E(r, /) + WHU. (J. 10)

We next proceed to evaluate U*HU. We have

H' m U*HU = — J 7 V - <A • p + P • A)} 1/ + — A2 + eV{t), (J.ll)2m 2m

where we used (J.I), (J.3), and the fact that {/commutes through V(r) andA*(r,0-Also

by (J.8) so that p • A = A • p in the Coulomb gauge. Therefore, we have

H' = t/f i - U - - A • (U'pU) + — Aa + eV.2m m 2m

Our next task is to commute U through p2 and p. We have

. . . . hdU Sfr.A^

(J.13)

(J.14)Therefore,

m

since IPU *= 1

- -A-U'lUp + eC7grad(r-A)]

--A.p--A-[grad(r.A)],m m (J.15)

516

Next, since

we have

A[B,C]+[A,C\B,

APPENDIX

(J.16)

2[Pi

By (J.14), we have

(J.17)

e«_a.rl7a(LLA)-ii dxX dxt J

(X18)

Thus (J.17) becomes when we use (J.14)

[p2, I/] = uifr VV • A) + e'ferad (r • A)]2 + 2e grad (r . A) • pj. ( J-19)

If we use (J.15) and (J.19), (J.13) becomes

+

2m 2im2m

m

F. (120)m

Now

(J.21)

In the last term we used (J.8).When we use these, (J.20) reduces to

eh

2m i.i.ic

+ ex • E, (J.22)

where we used (J.10).

APPENDIX J

As an example, consider a plane wave

A = A0e-<<oWk",where

Then (J.22) becomes

»K = -2- + eK(r) - iwer-A(r, r) + » - ( r2m m p)

517

(J.23)

(J.24)

where we used (J.9).In doing perturbation theory, we take the unperturbed hamiltonian as

*o = 7 - + eV(r), (J.26)Zm

and we would expect that the first order terms would be those linear in e.However, we see that these terms are in the ratio

k-p hk* . , hk*(o:—- : — = ck:kv: —

m 2m 2m

2m :

c 2mcwhere we have let v — pint, the electron velocity and used (J.24). Since v « cwe may neglect the k • p term in (J.25). Also at optical frequencies hk\2mc « 1and we may neglect the ks term linear in e in (J.25). Thus for plane wavesunder the dipole approximation the hamiltonian is

Ks*— + eV(r) - er - E(r, t), (J.27)2m

so that it is legitimate to treat — er • E as the perturbing energy rather than

Appendix

Properties ofFokker-Planck Equations

Let %, a 2 , . . . , a, = a be a set of c-number independent variables. LetP(a, 0 is a probability distribution function which obeys an equation of theform

at{

where the repeated indices are summed over from 1 to / and the s/t and Qit

are functions of the a's. This equation is called a Fokker-Planck equation.The s/{ are called the components of a drift vector and the 3ti are called thecomponents of a diffusion matrix

To obtain insight into the meaning of the drift vector, let us obtain theequation for the mean value of afc defined by

J (K.2)Jwhere da s d<tlt da2,... ,d*f If we take the time derivative of both sides of

(K.2) and use (K.I), we obtain

(K.3)

If we integrate the first term by parts once and the second by parts twice, note

that

(K.4)

APPENDIX K 519

and assume the integrated parts vanish since P -*• 0 as a, -*• ± oo, we obtain

(K.5)

Therefore, the mean equation of motion for at is determined by the kthcomponent of the drift vector.

Consider next the equation of motion

d(oca.) = I dtL

dt

f- §- (s/{P) + - £ - (<^P)i (K.6)If we again integrate the first term by parts once and the second twice, weobtain

T <a*ai> » [dziat J

since

As a simple example consider the Fokker-Planck equation

(K.7)

(K.8)

f - W - » g ] . (K.9)where ^4', J5', and ^ are constants. By (K.5), we have

since

Jt<.*)=-A.'-.B'{z) (K.10)

(K.11)From (K.10) it follows that

«0> = e~B\x(P)) + p (1 - e~st)-^ £ . (K.12)

Thus <*(0) approaches a steady-state value when B' > 0. The mean motionis unaffected by the diffusion constant S>.

By (K.7) and (K.9) we see that

j(x(t)) = 2{z(A'-B'x)) + 2®at

(K.13)

520 APPENDIX K

If we use (K.12) and integrate, we find that

(At)) - (*(0>2 = | (1 - e-2B'O + [<*2(0)> - <*(0)>V2*'', (K.14)D

which gives the variance or the mean square deviation. The initial fluctuationsdie out and in the steady-state

- ,(K.15)

which shows that residual fluctuations are due to the presence of the diffusionterm. This result could have been obtained more directly from (K.10), since

j <*>* = 2{x) ^ = 2A\x) - 2B\x)\ (K.16)at at

If we subtract this from (K.13), we obtain

d_dt

whose solution is given by (K.14).We next return to the general equation (K.1). To solve this equation we

begin by separating out the time dependence. We let

{<**> ~ <*>*} - 22 - 2B'{(x*) - <*>*}, (K.17)

P(a, r) = e-A'*P,(a).

If we put this in (K.1) P, satisfies the eigenvalue equation

where L is the operator

r __L—L. — - •ooc,-

(K.18)

(K.19)

(K.20)9a, i ,

The steady-state solution is the eigenfunction associated with the eigenvalueA, = 0.

We next look for a transformation of the type

P,(S) = e*MQt(.x),

which will remove the first derivative terms of P, with respect to a in (K.19).If we can do this by some choice of % the equation for Qx is called self-adjoint.

If we note that

APPENDIX K

and

521

dxidxj dxt Ldxj dxtl

we see that

where

m -L - j -

(K.23)

R, (K.24)

and

\dxidoLi

J l -

xi dxt dx, dx{

(K.25)

dxf oxjloxi

We therefore see that if we can choose % so that R is zero, then all first deri-vative terms of Qt with respect to a will vanish. We therefore require that

(K.27)VW.J VXj

If the determinant of the matrix &u is not zero, its inverse exists such that

^kTû = V (K.28)If we multiply both sides of (K.27) by 3>ki~^ and sum on i, we obtain theequations for %

(K.29)

(K.30)

To integrate these equations, it is sufficient that

dUk dUt

3a, dxj,.Then % is independent of the path of integration and is given by

(K.31)

The relations (K.30) are called the integrability conditions.

522APPENDIX K

If we assume the s/{ and 3>ti are such that (K.30) is satisfied then R = 0and when we put (K.21) into (K.19) we have

LezQl - A,e*fi, (K.32a)or

e-*Le*Q, m EQt = A, Q%, (K.32b)where

(K.33)

j

A possible solution corresponds to Jt = 0 or

If we multiply by Sb^r1 an^ s u m o v e r '» w e n a v e

VU1.J —j

Before proceeding to discuss (K.32), let us consider the steady state solutionof (K.I) for which dPjdt = 0. In this case we have

i-2£. (K.34)doCf

(K.35)

a 2Uk (K.36)

by (K.29). Again the relations (K.30) are satisfied if In P is given by the lineintegral

In P. = 2 f ~Uk dxk a 2*(a) (K.37)Ja

and is independent of the path. Therefore, the steady-state distribution isP. - el*l\ (K.38)

which is the eigenfunction for A : = 0.Let us return to (K.32). We have

\^9«~ + m\Qte> - -AtQ,(a). (K.39)

To show the eigenfunctions are orthogonal we write this for Am and have

mQ»(s). (K.40)

Then we multiply (K.39) by Qm from the left, (K.40) by Qt from the left andsubtract. This gives on integrating

&> j-QQ~ ® -f a ) * = ( A , - A J f QtQm dS .J

(K.41)

APPENDIX K523

If we integrate the first terms by parts twice and note that the integrated partsvanish, we see it cancels with the second term on the left (i and/' are summedover so we may interchange them). Therefore, if A, ?* Am , we have

J&(«)a»(«) <*« = <$,„, (K.42)where we normalize / = m to unity. We could not have carried through thisproof if we had not eliminated the first derivative terms and made L' self-adjoint.

It can be proved that if the eigenvalues have a lower bound, then the Qtform a complete set:

I Q^Q.fe') = *«-« ' ) . (K-43)In terms of the P,'s, we have

(K.44)

where Pt is the steady-state weighting function.To illustrate these techniques, consider (K.9). For the steady-state, we have

or

dP, {A' - B'x)S3 P

dx Si "(K.45)

P.

where we have normalized so that

In this case (K.29) reduces to

E l exp - •?- (x - 4fF 20\ BV'

(K47)

so that(K.48)

B (K.49)in agreement with the general relation between P, and exp 2X.

524

If we use (K.48) and (K.25), (K.32b) becomes

APPENDIX K

(K.50)

or

If we let

this reduces to

da?

The eigenvalues of this are

A,and

' • L20 ® *<** '

4' - B'aQ

«0 ,1 ,2 , . . .

(K.52)

(K.53)

(K.54)

(K.55)

where the #,(a) are hermite polynomials.Finally, the Green's function solution may be written as

(Ffc '/2o. 0) = "A20

since as / -> 0, by (K.44) this solution approaches the required product of d-

functions:

d(cc - a0) = I I d(a, - a,°). (K.57)

The reader should try to solve (K.9) with a function of the form exp

where

K0.which reduces at t = 0 to

7 P(.x,tJxo,O).

(K.58)

(K.59)

Index

Absorption, 353Absorption of radiation, 271Adiabatic approximation, 460,474Angular momentum, eigenvectors, 113

matrix elements, 114orbital, 110

Anticommutator, 100Antinormal order, 139Anti-Stokes line, 305Arbitrary operator, ordering of, 190Associated distribution function, equation

of motion for, 370Associated functions, antinormal order, 141

arbitrary operators, 190fourier transform for, 192normal order, 140

B

Baker-Hausdorff theorem, 137Basis vectors, 27Bloch equations, 357Boltzmann distribution, 335Boson operators, 150

ordered, 138traces of, 161

Bosons, 98Bra vectors, 6

matrix representation, 25

Characteristic function, 168,195damped oscillator, 399Poisson distribution, 179

Chosen order, 191Classical theory of radiation, 238Coherent state, 104Collision broadening, 323Commutation relations, 35

for fields, 251Commutator, 11Completeness relations, 19,21,24Contraction, 187Correlation functions, 426Correspondence principle, 2, 218,249

D

Damped oscillator, 372, 388driven, 344, 366equation of motion for, 418

Damping, quantum theory of, 331ffde Broglie wave length, 51Density of modes, 250Density operator, methods, 331

in Heisenberg picture, 76in interaction picture, 78,79in Schrodinger picture, 76radiation field, 264spin-1/2 particles, 220

Diagonalization, 30simultaneous, 33

Diffusion coefficients, damped atoms, 389,435

damped oscillator, 425Dirac 5-function, 19Distribution function, 195

harmonic oscillator, 211homogeneously broadened atoms, 377two-level atom, 207

Dopper effect, 309Dynamical state, 3

525

S26

Dynamical variable, 13Dyson time ordering operator, 57,64

Ehrenfest's principle, 226,322Eigenvalues, 14Einstein coefficients, 282Einstein relation, 430,435Election in electromagnetic field, classical,

129Electron spin resonance, 318Emission, 271,277,353Energy of radiation field, 239,248Entropy, 215Equation of motion, for associated distribu-

tion function, 370Heisenberg, 55for system operator, 385

Equipartition theorem, 218Euler Angles, 307Exclusion principle, 127Exponential distribution, 180

Fermi's golden rule, 64Fermions, 98Fluctuation-dissipation theorem, 425Fokker-Planck equation, 372,390,518

for homogeneously broadened atoms, 385,401

for laser, 470Free particle, 71

INDEX

Generalized Wick's Theorem for bosons, 182Green's function for damped oscillator, 399

H

Hamiltonian, 34atom in radiation field, 270

Harmonic oscillator, eigenfunctions,102,213

eigenvalues, 94eigenvectors, 97Heisenberg picture for, 90

Heisenberg operator, equations of motion, 55

perturbation theory, 68Heisenberg picture, 54Hermitian adjoint, 13Hermitian operators, 13

matrix representation, 26Hilbert space, 10Homogeneously broadened atoms, 375

Induced emission, 277Interaction picture, 57

Ket vector, 5matrix representation, 25

Kramers-Heisenberg cross section, 296Kronecker 6,18

Lamb shift, 258,285,346,354Langevin equation, 431,433

Laser, 473multilevel atom, 435N-atoms, 438photon number, 428

Langevin force, 423,429Langevin noise sources, 432,441Laser, Helium-Neon, 445

linewidth, 485quantum theory of, 469ffsemiclassical theory of, 444ffstatistical properties, 469as Van der Pol oscillator, 482

Light propagation in vacuum, 314Linear operators, 10Linewidth, 285,347,367,420,485

M

Magnetic dipole moment, 318Mass renormalization, 291Master equation, 343Markoff approximation, in Heisenberg

picture, 360in Schrodinger picture, 336

Matrices, 25Hermitian, 26

INDEX

inverse, 26traces, 25unitary, 26

Maxwell-Boltzmann distribution, 219Maxwell equations, with sources, 259

without sources, 238Method of characteristics, 401,491Minimum uncertainty wave function, 50Minimum uncertainty wave pocket time

development, 71,73Mixed state, 77,219Modes in spherical resonator, 447Momentum of field, 239,249Momentum representation, 3

N

Normal modes, 240Normal order, 139, 203

Observable, 19function of, 25measurement of, 43

One-time averages, 368Operator algebra, 132ffOrdering for arbitrary operator, 190Orthogonality, 10,18,23

Particle in central field, 116Partition function, 217Pauli equations, 347Pauli exclusion principle, 99Pauli spin matrices, 125,220

operators in Heisenberg picture, 127Perturbation theory, 57

for Heisenberg operator, 68using density operator, 74ff

Planck distribution, function, 282law, 181

Poisson bracket, 57Poisson distribution, 105,176, 233

characteristic function for, 179Projection operator, 44Pure state, 77

527

Quantization, 34of electromagnetic field, 230ofLC-circuit,231of transmission line, 235

Quantum theory of damping, densityoperator approach, 33 Iff

Langevin approach, 418ff

Raman scattering, 304Raman tensor, 305Rayleigh'slaw.302Rayleigh scattering, 301Reduced density operator, 81

in interaction picture, 83,338in Schrodinger picture, 337

Representation, change of, 30Representatives, 28Reservoir model, 332Resonance fluorescence, 308Rotating wave approximation, 324,336Rotation matrix, 305

Scalur product, 6Schrodinger equation, 2,51,58, 71Schrodinger picture, 53Schrodinger-Markoff picture, 343Schwary inequality, 47Self adjoint, 14Similarity transformation, 32,55Spectra, 19,404

for damped oscillator, 424for Van der Pol oscillator, 415

Spectral densities, 342,345Spin, 122Spin-lattice relaxation, 360Spin-spin relaxation, 360Spontaneous emission, 276Stokes' line, 305Superposition principle, 6

Thermal averages, 182

528

Thomson scattering, 303Three level atoms, 389Transformation functions, 30,32,39,42,

102,108,119Translation operators, 36, 38Transverse 5-function, 253,498Two time averages, 404

U

Uncertainty principle, 45Unitary operator, 52Unitary transformation, 32

INDEX

Van der Pol oscillator, 408

Wave mechanics, 70Wick's theorem for bosons, 185Wiener-Khintchine relations, 511Wigner-Weisskopf theory of natural line-

width, 285,420Wigner dbtribution function, 168

Zero point energy, 235Zero point field fluctuations, 256

Documents

Louisell Text