Foundations of Quantum Physics

Charles E. Burkhardt · Jacob J. Leventhal

Foundations of QuantumPhysics

123

Charles E. BurkhardtSt. Louis Community CollegeDepartment of PhysicsSt. Louis, MOUSAcburkhardt@stlcc.edu

Jacob J. LeventhalDepartment of Physics and AstronomyUniversity of MissouriSt. Louis, MOUSAjake@umsl.edu

ISBN: 978-0-387-77651-4 e-ISBN: 978-0-387-77652-1DOI: 10.1007/978-0-387-77652-1

Library of Congress Control Number: 2008930210

c© 2008 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subject toproprietary rights.

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Helen, Charlie, Sarah, and MichelleBette, Andy, Bradley, Dan, and TinaIn loving memory of Steve Leventhal

Preface

This book is meant to be a text for a first course in quantum physics. It is assumedthat the student has had courses in Modern Physics and in mathematics throughdifferential equations. The book is otherwise self-contained and does not rely onoutside resources such as the internet to supplement the material. SI units are usedthroughout except for those topics for which atomic units are especially convenient.

It is our belief that for a physics major a quantum physics textbook should bemore than a one- or two-semester acquaintance. Consequently, this book containsmaterial that, while germane to the subject, the instructor might choose to omitbecause of time limitations. There are topics and examples included that are notnormally covered in introductory textbooks. These topics are not necessarily tooadvanced, they are simply not usually covered. We have not, however, presumed totell the instructor which topics must be included and which may be omitted. It isour intention that omitted subjects are available for future reference in a book thatis already familiar to its owner. In short, it is our hope that the student will use thebook as a reference after having completed the course.

We have included at the end of most chapters a “Retrospective” of the chapter.This is not meant to be merely a summary, but, rather, an overview of the importanceof the material and its place in the context of previous and forthcoming chapters. Forexample, the Retrospective in Chapter 3 we feel is particularly important because,in our experience, students spend so much time learning about eigenstates that theyget the impression that physical systems “live” in eigenstates.

We believe that students should, after a very brief review of salient experimentsand concepts that led to contemporary quantum physics (Chapter 1), begin solv-ing problems. That is, the formal aspects of quantum physics, operator formalism,should be introduced only after the student has seen quantum mechanics in action.This is certainly not a new approach, but we prefer it to the alternative of the for-mal mathematical introduction followed by problem solving. More importantly, webelieve that the students benefit from this approach. To this end we begin with aderivation (read: rationalization) of the Schrodinger equation in Chapter 2. Thischapter continues with a discussion of the nature of the solutions of the Schrodingerequation, particularly the wave function. We discuss at length both the utility of thewave function and its characteristics. It is our observation that the art of sketchingwave functions has been neglected. We are led to this conclusion from discussions

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with graduate students who have had the undergraduate course, but are unable tosketch wave functions for an arbitrarily drawn potential energy function. We thinkthat such a skill is crucial for understanding quantum mechanics at the introductorylevel and, thus, we spend a good deal of Chapter 2 discussing qualitative aspects ofthe wave function.

In Chapter 3 we solve the Schrodinger equation for two of the most important po-tential energy functions, the infinite square well and the harmonic oscillator. A pointof contrast between the these potentials is penetration of oscillator wave functionsinto the classically forbidden region. We discuss this penetration at length because,in our experience, students have a great deal of difficulty with this concept. We thenelaborate upon this concept by presenting the details of a problem not often seen inelementary texts, an infinite square well with a barrier in the middle. This affordsthe opportunity to see that, for energies less than the barrier height, the particlecan be found on either side of the classically impenetrable barrier, thus making theparticle’s presence inside the barrier undeniable. This problem also sets the stagefor solution of the more conventional barrier penetration problems in Chapter 5.

In Chapter 4 we discuss time-dependent states. We choose to do this at this pointto contrast these states with those studied in the previous chapter. While we discussthe free particle wave packet (as does virtually every other text), we also presentwave packets under the influence of a constant force and of a harmonic force. Thisdiscussion will, we believe, relate nicely to a later presentation of harmonic oscilla-tor coherent states (Chapter 7).

Chapter 5 is an extension of Chapter 3 in that we solve the time-independentSchrodinger equation for several different one-dimensional potential energies. In-cluded is one of the most successful analytic potential energy functions for charac-terizing diatomic molecular vibrations, the Morse potential. The chapter concludeswith the WKB method for approximating solutions.

Chapter 6 presents the formalism of quantum physics, the mechanics of quan-tum mechanics, including a set of postulates. For completeness we also discuss theSchrodinger and Heisenberg pictures. Chapter 7 is devoted to the operator solu-tion of the Schrodinger equation for the harmonic oscillator with emphasis on theproperties of the ladder operators. Harmonic oscillator coherent states are also dis-cussed. Chapter 8 introduces three-dimensional problems and is devoted to angularmomentum. It is emphasized in this chapter that the concept of angular momen-tum in quantum mechanics transcends three-dimensional rotations (orbital angularmomentum).

Chapters 9 and 10 are devoted to solving the radial Schrodinger equation forseveral different central potentials. In addition to the common central potentials,Chapter 9 includes a thorough discussion of the isotropic harmonic oscillator usingthe shell model of the nucleus as an example. The isotropic oscillator also permitsintroduction the concept of accidental degeneracy. Because they are constituents ofoscillator eigenfunctions, an attempt is made to decrypt the different conventionsthat are used for Laguerre polynomials and associated Laguerre polynomials. Inour experience, this is a source of confusion to many students. Also contained inthis chapter is an elaboration on the Morse potential in which three-dimensional

Preface ix

molecular motion is considered through rotation–vibration coupling. The discus-sion of the hydrogen atom, the sole content of Chapter 10, is standard, but, as forthe isotropic oscillator, accidental degeneracy is stressed. Chapter 11 is included todemonstrate to the student that there are angular momenta in quantum mechanicsother than orbital and spin angular momenta. It includes the introduction of the Lenzvector, its consequences and ramifications. This subject is not usually covered at theintroductory level, but it is certainly not beyond the beginning student.

The material in the remaining four chapters depends heavily upon approxima-tion methods. Chapter 12 presents time-independent approximation methods, whileChapter 13 illustrates the use of these methods to solve problems of physical in-terest. One problem that is included in Chapter 13, albeit superficially, is the effectof fine structure on the shell model of the nucleus. Chapter 14 treats the Stark andZeeman effects. Particular attention is paid to the consequences of breaking thespherical symmetry of central potentials by application of an external field. Chapter15 presents time-dependent approximation methods, followed by a discussion ofatomic radiation including the Einstein coefficients.

There are more than two hundred problems. A detailed solutions manual is avail-able. There are a number of appendixes to the book, including the answers to allproblems for which one is required. Among the other appendixes is one listing theGreek alphabet with notations on common usage of these symbols in the book.There is also a short table of acronyms used in the book. The remaining appendixescontain material that is intended to be quick reference material and helpful withthe core material in the book. A list of (the inevitable) corrections can be found at:http://users.stlcc.edu/cburkhardt/ and http://www.umsl.edu/∼jjl/homepage/.

We are indebted to several people, without whose help this manuscript would nothave been completed. Helen and Charles Burkhardt, parents, read the manuscriptcritically. Discussions with Dr. J. D. Kelley were invaluable, as was his critical read-ing of the manuscript. Professor. S. T. Manson also read the manuscript and mademany useful suggestions. Discussions with Dr. M. J. Kernan were very helpful, aswere her suggestions. To all of these people we offer our sincere thanks.

Charles E. BurkhardtJacob J. Leventhal

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Early Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 The Franck–Hertz Experiment . . . . . . . . . . . . . . . . . 31.1.3 Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 Electron Diffraction Experiments . . . . . . . . . . . . . . . 71.1.5 The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Early Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 The Bohr Atom and the Correspondence Principle . 101.2.2 The de Broglie Wavelength . . . . . . . . . . . . . . . . . . . . 181.2.3 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 191.2.4 The Compton Wavelength Revisited . . . . . . . . . . . . 211.2.5 The Classical Radius of the Electron . . . . . . . . . . . . 23

1.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Elementary Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 What is Doing the Waving? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 A Gedanken Experiment—Electron Diffraction Revisited . . . . . . 272.3 The Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Finding the Wave Function—the Schrodinger Equationo . . . . . . . 292.5 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Separation of the Schrodinger Equation—Eigenfunctions . . . . . . 332.7 The General Solution to the Schrodinger Equation . . . . . . . . . . . . 352.8 Stationary States and Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 382.9 Characteristics of the Eigenfunctions ψn (x) . . . . . . . . . . . . . . . . . 382.10 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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3 Quantum Mechanics in One Dimension—Bound States I . . . . . . . . . . . 473.1 Simple Solutions of the Schrodinger Equation . . . . . . . . . . . . . . . 47

3.1.1 The Infinite Square Well—the “Particle-in-a-Box” . 473.1.2 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Penetration of the Classically Forbidden Region . . . . . . . . . . . . . . 693.2.1 The Infinite Square Well with a Rectangular

Barrier Inside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Time-Dependent States in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 834.1 The Ehrenfest Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3 Quantum Representation of Particles—Wave Packets . . . . . . . . . 86

4.3.1 Momentum Representation of the Operator x . . . . . 904.3.2 The Dirac δ-function . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.3 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4 The Harmonic Oscillator Revisited—MomentumEigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.5 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1 Case I. The Free Packet/Particle . . . . . . . . . . . . . . . . 984.5.2 Case II. The Packet/Particle Subjected to a

Constant Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.5.3 Case III. The Packet/Particle Subjected to a

Harmonic Oscillator Potential . . . . . . . . . . . . . . . . . . 1044.6 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Stationary States in One Dimension II . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1 The Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 The Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3 The Finite Square Well—Bound States . . . . . . . . . . . . . . . . . . . . . 1235.4 The Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.5 The Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.6 The WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.6.1 The Nature of the Approximation . . . . . . . . . . . . . . . 1455.6.2 The Connection Formulas for Bound States . . . . . . 1485.6.3 A Bound State Example—the Linear Potential . . . . 1555.6.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.6.5 Comparison with a Rectangular Barrier . . . . . . . . . . 1625.6.6 A Tunneling Example—Predissociation . . . . . . . . . 163

5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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6 The Mechanics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.1 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.1.1 Matrix Representation of a Vector . . . . . . . . . . . . . . 1716.1.2 Dirac Notation for a Vector . . . . . . . . . . . . . . . . . . . . 1726.1.3 Operators in Quantum Mechanics . . . . . . . . . . . . . . . 173

6.2 The Eigenvalue Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.2.1 Properties of Hermitian Operators

and the Eigenvalue Equation . . . . . . . . . . . . . . . . . . 1806.2.2 Properties of Commutators . . . . . . . . . . . . . . . . . . . . 186

6.3 The Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 1896.3.1 Listing of the Postulates . . . . . . . . . . . . . . . . . . . . . . . 1896.3.2 Discussion of the Postulates . . . . . . . . . . . . . . . . . . . 1906.3.3 Further Consequences of the Postulates . . . . . . . . . . 198

6.4 Relation Between the State Vector and the Wave Function . . . . . 2006.5 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.6 Spreading of Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.6.1 Spreading in the Heisenberg Picture . . . . . . . . . . . . 2076.6.2 Spreading in the Schrodinger Picture . . . . . . . . . . . . 211

6.7 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7 Harmonic Oscillator Solution Using Operator Methods . . . . . . . . . . . . 2197.1 The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.1.1 The Schrodinger Picture . . . . . . . . . . . . . . . . . . . . . . 2197.1.2 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.1.3 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . 227

7.2 Coherent States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . 2297.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.4 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8 Quantum Mechanics in Three Dimensions—Angular Momentum . . . 2398.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.2 Angular Momentum Ladder Operators . . . . . . . . . . . . . . . . . . . . . . 241

8.2.1 Definitions and Commutation Relations . . . . . . . . . 2418.2.2 Angular Momentum Eigenvalues . . . . . . . . . . . . . . . 242

8.3 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2478.4 Orbital Angular Momentum Eigenfunctions—Spherical

Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.4.1 The Addition Theorem for Spherical Harmonics . . 2578.4.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2598.4.3 The Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8.5 Another Form of Angular Momentum—Spin . . . . . . . . . . . . . . . . 262

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8.5.1 Matrix Representation of the Spin Operators andEigenkets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

8.5.2 The Stern–Gerlach Experiment . . . . . . . . . . . . . . . . . 2708.6 Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

8.6.1 Examples of Angular Momentum Coupling . . . . . . 2778.6.2 Spin and Identical Particles . . . . . . . . . . . . . . . . . . . . 285

8.7 The Vector Model of Angular Momentum . . . . . . . . . . . . . . . . . . 2928.8 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2948.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

9 Central Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979.1 Separation of the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . 298

9.1.1 The Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . 3009.1.2 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3029.1.3 Behavior of the Wave

Function for Small and LargeValues of r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

9.2 The Free Particle in Three Dimensions . . . . . . . . . . . . . . . . . . . . . 3059.3 The Infinite Spherical Square Well . . . . . . . . . . . . . . . . . . . . . . . . . 3089.4 The Finite Spherical Square Well . . . . . . . . . . . . . . . . . . . . . . . . . . 3099.5 The Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 316

9.5.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 3179.5.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 319

9.6 The Morse Potential in Three Dimensions . . . . . . . . . . . . . . . . . . . 3399.7 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3439.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

10 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34710.1 The Radial Equation—Energy Eigenvalues . . . . . . . . . . . . . . . . . . 34710.2 Degeneracy of the Energy Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 35210.3 The Radial Equation—Energy Eigenfunctions . . . . . . . . . . . . . . . 35410.4 The Complete Energy Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 36110.5 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36210.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

11 Angular Momentum—Encore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36511.1 The Classical Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36511.2 The Quantum Mechanical Kepler Problem . . . . . . . . . . . . . . . . . . 36711.3 The Action of A+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37111.4 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37211.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

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12 Time-Independent Approximation Methods . . . . . . . . . . . . . . . . . . . . . . 37512.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

12.1.1 Nondegenerate Perturbation Theory . . . . . . . . . . . . . 37512.1.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . 382

12.2 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

13 Applications of Time-Independent Approximation Methods . . . . . . . . 39713.1 Hydrogen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

13.1.1 Breaking the Degeneracy—Fine Structure . . . . . . . . 39713.2 Spin–Orbit Coupling and the Shell Model of the Nucleus . . . . . . 40913.3 Helium Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

13.3.1 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41113.3.2 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

13.4 Multielectron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42213.5 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

14 Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43114.1 Hydrogen Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . 431

14.1.1 Electric Fields—the Stark Effect . . . . . . . . . . . . . . . . 43114.1.2 Magnetic Fields—The Zeeman Effect . . . . . . . . . . . 436

14.2 Multielectron Atoms in External Magnetic Fields . . . . . . . . . . . . 44214.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44614.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

15 Time-Dependent Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44915.1 Time Dependence of the State Vector . . . . . . . . . . . . . . . . . . . . . . . 44915.2 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

15.2.1 Harmonic Perturbation—Rotating WaveApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

15.2.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 45515.3 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 45715.4 Two-state Systems Using Perturbation Theory . . . . . . . . . . . . . . . 459

15.4.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 45915.4.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 462

15.5 Extension to Multistate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 46415.5.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 46415.5.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 46515.5.3 Transitions to a Continuum of States—The

Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46515.6 Interactions of Atoms with Radiation . . . . . . . . . . . . . . . . . . . . . . . 468

15.6.1 The Nature of Electromagnetic Transitions . . . . . . . 469

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15.6.2 The Transition Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 47015.6.3 The Einstein Coefficients—Spontaneous Emission 47315.6.4 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47615.6.5 Transition Rates and Lifetimes . . . . . . . . . . . . . . . . . 480

15.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

A Answers to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

B Useful Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

C Energy Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

D Useful Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

E Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

F Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

G �-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507G.1 Integral �-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507G.2 Half-Integral �-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

H Useful Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

I Useful Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511I.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511I.2 Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511I.3 Gauss’ Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Contents xvii

J Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

K Commutator Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519K.1 General Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519K.2 Quantum Mechanical Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

L Miscellaneous Operator Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521L.1 Baker–Campbell–Hausdorff (BCH) Formula . . . . . . . . . . . . . . . . 521L.2 Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525