ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS ANDWAVE MECHANICS

By

DAVID JOHN MASIELLO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2004

For Katie.

ACKNOWLEDGMENTS

Since August of 1999, I have had the privilege of conducting my Ph.D. research

in the group of Prof. Yngve Ohrn and Dr. Erik Deumens at the University of

Florida’s Quantum Theory Project. During my time in their group I learned a great

deal on the theory of dynamics, in particular, the Hamiltonian approach to dynam-

ics and its applications in electrodynamics and atomic and molecular collisions. I

also learned a new appreciation for scientific computing, of which I was previously

ignorant. Most importantly, Prof. Ohrn and Dr. Deumens taught me how to think

through a physical problem, sort out its underlying dynamical equations, and solve

them in a mathematically well-defined manner. I especially want to thank Dr. Erik

Deumens, with whom I worked most closely during my Ph.D. research. Erik had

a vision when I began my graduate studies and has promoted my work since then

to successfully realize it. Along the way, he challenged my creative, mathematical,

and physical intuitions and imparted on me a love for theoretical physics. Erik has

always taken time to listen to and carefully answer my questions and has always

respected my ideas. I thank him for being such an excellent mentor to me.

My understanding of physics has also been broadened by many others. Firstly, I

would like to thank Dr. Remigio Cabrera-Trujillo, who was a post doctoral associate

in the Ohrn-Deumens group, for his guidance especially during my first few years.

He has been a great source for advice on many topics from the details of quantum

scattering theory to simple computer problems like clearing printer jams. I have

joked on many occasions that he was my personal postdoc because he was always

so willing to help when I had questions. I would also like to thank the past and

present members of my research group, in particular, Dr. Anatol Blass, Dr. Maurıcio

iii

Coutinho Neto, Mr. Ben Killian, and Mr. Virg Fermo. In addition, I would like

to thank my officemates with whom I have spent almost five years. I thank Ms.

Ariana Beste, Mr. Igor Schweigert, and Mr. Tom Henderson for their friendship

and camaraderie. I have especially benefited from many conversations with Tom

Henderson on aspects of quantum mechanics, quantum field theory, and classical

electrodynamics.

Several other faculty and staff at the Quantum Theory Project, and the Depart-

ments of Chemistry, Physics, and Mathematics at the University of Florida have also

encouraged and promoted my Ph.D. research. At the Quantum Theory Project, I

thank Prof. Jeff Krause for taking sincere interest in my research and always finding

time to listen to me and provide guidance. I have taught with Jeff on a few occasions

and have known him to be a great teacher as well as mentor. I thank Prof. Henk

Monkhorst for his kindness and good humor. I will especially miss all of the LATEX

battles that we have fought over the past several years. In addition, I would like to

thank Dr. Ajith Perera for his friendship and patience. I thank the staff, especially

Ms. Judy Parker and Ms. Coralu Clements, for keeping all of the administrative

aspects of my graduate studies running smoothly. I would also like to thank the

custodians Sandra and Rhonda who have been so friendly to me and who keep the

Quantum Theory Project impeccably clean. In the Department of Chemistry, I

would like to thank the late Prof. Carl Stoufer, who was my undergraduate advisor

during my first year, for his friendship, wisdom, and advice. Throughout my entire

undergraduate career we would meet a few times per year to catch up over coffee and

donuts. It was due to Carl’s support that I was given the opportunity to study at

the Quantum Theory Project. In the Department of Physics, I would like to thank

Prof. Richard Woodard, from whom I learned quantum field theory. Richard is very

passionate about physics and is perhaps the best teacher that I have known. From

him I gained a deeper understanding of perturbation theory and its applications in

iv

quantum electrodynamics. In the Department of Mathematics, I would like to thank

Prof. Scott McCullough, who was effectively my undergraduate advisor. While I

was an undergraduate student of Scott’s, he imparted to me a deep appreciation for

mathematics and a particular interest in analysis. Scott was an excellent teacher

and mentor, and under his guidance, my undergraduate research was awarded by

the College of Liberal Arts and Sciences.

Outside of the University of Florida, many others have contributed to my scien-

tific career. At the University of Central Florida’s Center for Research and Education

in Optics and Lasers, I would like to thank Prof. Leonid Glebov, Prof. Kathleen

Richardson, and Prof. Boris Zel’dovich for first introducing me to the world of quan-

tum physics. In particular, Prof. Glebov and Prof. Richardson greatly stimulated

and encouraged my interests. With their recommendation, I received a fellowship

to study at the University of Bordeaux’s Department of Physics and Centre de

Physique Moleculaire Optique et Herzienne. While in Bordeaux, France, I had the

pleasure of working in the research group of Prof. Laurent Sarger. I wish to thank

Prof. Sarger as well as his colleagues for their hospitality during my time in France

and for introducing me to the field of atomic and molecular physics, which is the

setting for this dissertation.

Lastly, I would like to thank my family. My mother and father have always

provided unconditional love, support, and guidance to me. They have encouraged

my inquisitiveness of Nature and have promoted my education from kindergarten

to Ph.D. I thank my inlaws for their love and support and for providing a home

away from home while in graduate school. In conclusion, I would like to thank my

wonderful wife Katie for her encouragement, companionship, and unending love.

v

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Historical and Mathematical Background . . . . . . . . . . . . . . 2

1.2.1 Gauge Symmetry of Electrodynamics . . . . . . . . . . . . 31.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics 5

1.3 Approaches to the Solution of the Maxwell-Schrodinger Equations 71.4 Canonical Formulation of the Maxwell-Schrodinger Equations . . . 111.5 Format of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Notation and Units . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 THE DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 182.1.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Example: The Harmonic Oscillator in (qk, qk) . . . . . . . . 212.1.3 Geometry of TQ . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Example: The Harmonic Oscillator in (qa, pa) . . . . . . . . 252.2.2 Symplectic Structure and Poisson Brackets . . . . . . . . . 262.2.3 Geometry of T∗Q . . . . . . . . . . . . . . . . . . . . . . . 27

3 ELECTRODYNAMICS AND QUANTUM MECHANICS . . . . . . . . 29

3.1 Quantum Mechanics in the Presence of an Electromagnetic Field . 293.1.1 Time-Dependent Perturbation Theory . . . . . . . . . . . . 303.1.2 Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . . 333.1.3 Absorption of Electromagnetic Radiation by an Atom . . . 343.1.4 Quantum Electrodynamics in Brief . . . . . . . . . . . . . . 36

3.2 Classical Electrodynamics Specified by the Sources ρ and J . . . . 403.2.1 Electromagnetic Radiation from an Oscillating Source . . . 413.2.2 Electromagnetic Radiation from a Gaussian Wavepacket . . 47

vi

4 CANONICAL STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Lagrangian Electrodynamics . . . . . . . . . . . . . . . . . . . . . 564.1.1 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . . . . 564.1.2 The Lorenz and Coulomb Gauges . . . . . . . . . . . . . . 57

4.2 Hamiltonian Electrodynamics . . . . . . . . . . . . . . . . . . . . 594.2.1 Hamiltonian Formulation of the Lorenz Gauge . . . . . . . 614.2.2 Poisson Bracket for Electrodynamics . . . . . . . . . . . . . 66

4.3 Hamiltonian Electrodynamics and Wave Mechanics in ComplexPhase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Hamiltonian Electrodynamics and Wave Mechanics in Real PhaseSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 The Coulomb Reference by Canonical Transformation . . . . . . . 704.5.1 Symplectic Transformation to the Coulomb Reference . . . 714.5.2 The Coulomb Reference by Change of Variable . . . . . . . 78

4.6 Electron Spin in the Pauli Theory . . . . . . . . . . . . . . . . . . 794.7 Proton Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . 84

5.1 Maxwell-Schrodinger Theory in a Complex Basis . . . . . . . . . . 855.2 Maxwell-Schrodinger Theory in a Real Basis . . . . . . . . . . . . 88

5.2.1 Overview of Computer Program . . . . . . . . . . . . . . . 905.2.2 Stationary States: s- and p-Waves . . . . . . . . . . . . . . 935.2.3 Nonstationary State: Mixture of s- and p-Waves . . . . . . 935.2.4 Free Electrodynamics . . . . . . . . . . . . . . . . . . . . . 935.2.5 Analysis of Solutions in Numerical Basis . . . . . . . . . . 95

5.3 Symplectic Transformation to the Coulomb Reference . . . . . . . 995.3.1 Numerical Implementation . . . . . . . . . . . . . . . . . . 1015.3.2 Stationary States: s- and p-Waves . . . . . . . . . . . . . . 1025.3.3 Nonstationary State: Mixture of s- and p-Waves . . . . . . 1025.3.4 Free Electrodynamics . . . . . . . . . . . . . . . . . . . . . 1035.3.5 Analysis of Solutions in Coulomb Basis . . . . . . . . . . . 103

5.4 Asymptotic Radiation . . . . . . . . . . . . . . . . . . . . . . . . 1035.5 Proton Dynamics in a Real Basis . . . . . . . . . . . . . . . . . . 108

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

APPENDIX

A GAUGE TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . 113

A.1 Gauge Symmetry of Electrodynamics . . . . . . . . . . . . . . . . 113A.2 Gauge Symmetry of Quantum Mechanics . . . . . . . . . . . . . . 115

vii

B GREEN’S FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.1 The Dirac δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . 117B.2 The ∇2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.3 The ∂2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

C THE TRANSVERSE PROJECTION OF A(x, t) . . . . . . . . . . . . . 122

C.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.2 T ′kk(x′, t) Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C.2.1 Inside Step . . . . . . . . . . . . . . . . . . . . . . . . . . . 129C.2.2 Outside Step . . . . . . . . . . . . . . . . . . . . . . . . . . 131

C.3 Building AT (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

viii

LIST OF FIGURESFigure page

2–1 The configuration manifold Q = S2 is depicted together with the

tangent plane TqkQ at the point qk ∈ Q. . . . . . . . . . . . . . . . 22

3–1 The coefficient 1 of the unscattered plane wave exp(ik · x) is analo-gous to the 1 part of the S-matrix, while the scattering amplitudefk(Ω) which modulates the scattered spherical wave exp(ikr)/r isanalogous to the iT part. . . . . . . . . . . . . . . . . . . . . . . . 38

3–2 In the radiation zone, the observation point x is located far from thesource J. In this case the distance |x − x′| ≈ r − n · x′. . . . . . . . 44

3–3 The differential power dP/dΩ or radiation pattern corresponding toan oscillating electric dipole verifies that no radiation is emitted inthe direction of the dipole moment. . . . . . . . . . . . . . . . . . . 46

3–4 The norms of J and A are plotted with different velocities along thex-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3–5 The trajectory or world line r(t) of the charge is plotted. . . . . . . . 49

3–6 The bremsstrahlung radiation from a charged gaussian wavepacketmoves out on the smeared light cone with maximum at x = ct. . . . 50

3–7 The radiation pattern given by (3.63) shows the characteristic dipolepattern at lowest order. . . . . . . . . . . . . . . . . . . . . . . . . 53

4–1 A limited but relevant portion of the gauge story in the Lagrangianformalism is organized in this picture. . . . . . . . . . . . . . . . . 59

4–2 The Hamiltonian formulation of the gauge story is organized in thispicture with respect to the previous Lagrangian formulation. . . . . 65

4–3 Commutivity diagram representing the change of coordinates (q, p) to(p, q) at both the Lagrangian and equation of motion levels. . . . . 79

5–1 Schematic overview of ENRD computer program. . . . . . . . . . . . 92

5–2 Phase space contour for the coefficients of the vector potential A andits momentum Π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5–3 Phase space contour for the coefficients of the real-valued Schrodingerfield Q and its momentum P. . . . . . . . . . . . . . . . . . . . . . 94

ix

5–4 Phase space contour for the coefficients of the scalar potential Φ andits momentum Θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5–5 Real part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves. . . . . . . . . . . . . . . 97

5–6 Imaginary part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉,where η(t) is a superposition of s- and px-waves. . . . . . . . . . . . 97

5–7 Probability for the electron to be in a particular basis eigenstate. . . . 98

5–8 Phase space of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves. . . . . . . . . . . . . . . 98

5–9 Real part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves. . . . . . . . . . . . . . . 104

5–10 Imaginary part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉,where η(t) is a superposition of s- and px-waves. . . . . . . . . . . . 104

5–11 Probability for the electron to be in a particular basis eigenstate. . . . 105

5–12 Phase space of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves. . . . . . . . . . . . . . . 105

5–13 Schematic picture of the local and asymptotic basis proposed for thedescription of electromagnetic radiation and electron ionization. . . 107

B–1 The trajectory or world line r(t) of a massive particle moves from pastto future within the light cone. . . . . . . . . . . . . . . . . . . . . 120

C–1 Since A = hv, the transverse vector potential A⊥ = [v−k(k ·v)/k2]hand the longitudinal vector potential A‖ = [k(k · v)/k2]h, where his a scalar function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

x

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS ANDWAVE MECHANICS

By

David John Masiello

May 2004

Chair: Nils Yngve OhrnMajor Department: Chemistry

The interaction of electromagnetic radiation with atoms or molecules is often

understood when the timescale for the electromagnetic decay of an excited state is

separated by orders of magnitude from the timescale of the excited state’s dynamics.

In these cases, the two dynamics may be treated separately and a perturbative Fermi

golden rule analysis is appropriate. However, there do exist situations where the

dynamics of the electromagnetic field and the atomic or molecular system occurs

on the same timescale, e.g., photon-exciton dynamics in conjugated polymers and

atom-photon dynamics in cold atom collisions.

Nonperturbative methods for the solution of the coupled nonlinear Maxwell-

Schrodinger differential equations are developed in this dissertation which allow

for the atomic or molecular and electromagnetic dynamics to occur on the same

timescale. These equations have been derived within the Hamiltonian or canonical

formalism. The canonical approach to dynamics, which begins with the Maxwell

and Schrodinger Lagrangians together with a Lorenz gauge fixing term, yields a

set of first order Hamilton equations which form a well-posed initial value prob-

lem. That is, their solution is uniquely determined and known in principle once

xi

the initial values for each of the associated dynamical variables are specified. The

equations are also closed since the Schrodinger wavefunction is chosen to be the

source for the electromagnetic field and the electromagnetic field reacts back upon

the wavefunction.

In practice, the Maxwell-Schrodinger Lagrangian is represented in a basis of

gaussian functions with different widths and centers. Application of the calculus

of variations leads to a set of Euler-Lagrange equations that, for that choice of

basis, form and represent the coupled first order Maxwell-Schrodinger equations.

In the limit of a complete basis these equations are exact and for any finite choice

of basis they provide an approximate system of dynamical equations that can be

integrated in time and made systematically more accurate by enriching the basis.

These equations are numerically implemented for a basis of arbitrary finite rank.

The dynamics of the basis-represented Maxwell-Schrodinger system is investigated

for the spinless hydrogen atom interacting with the electromagnetic field.

xii

CHAPTER 1INTRODUCTION

Chemistry encompasses a broad range of Nature that varies over orders of mag-

nitude in energy from the ultracold nK Bose-Einstein condensation temperatures

[1, 2] to the keV collision energies that produce the Earth’s aurorae [3–5]. At the

most fundamental level, the study of chemistry is the study of electrons and nuclei.

The interaction of electrons and nuclei throughout this energy regime is mediated

by the photon which is the quantum of the electromagnetic field. The equations

which govern the dynamics of electrons, nuclei, and photons are therefore the same

equations which govern all of chemistry [6]. They are the Schrodinger equation [7, 8]

iΨ = HΨ (1.1)

and Maxwell’s equations [9]

∇·E = 4πρ ∇×B =4π

cJ+

E

c∇·B = 0 ∇×E+

B

c= 0. (1.2)

As they stand these equations are uncoupled. The solutions of the Schrodinger

equation (1.1) do not a priori influence the solutions of the Maxwell equations

(1.2) and vice versa. The development of analytic and numerical methods for the

solution of the coupled Maxwell-Schrodinger equations is the main purpose of this

dissertation. Before delving into the details of these methods a physical motivation

as well as a historical and mathematical background is provided.

1.1 Physical Motivation

Many situations of physical interest are described by the system of Maxwell-

Schrodinger equations. Often these situations involve electromagnetic processes that

occur on drastically different timescales from that of the matter. An example of such

1

2

a situation is the stimulated absorption or emission of electromagnetic radiation

by a molecule. The description of this process by (1.1) and (1.2) accounts for a

theoretical understanding of all of spectroscopy, which has provided an immense

body of chemical knowledge.

However, there do exist situations where the dynamics of the electromagnetic

field and the matter occur on the same timescale. For example, in solid state physics

certain electronic wavepackets exposed to strong magnetic fields in semiconductor

quantum wells are predicted to demonstrate rapid decoherence [10]. The dynamics of

the incident field, the electronic wavepacket, and the phonons that it emits is coupled

and occurs on the same femtosecond timescale. In atomic physics, the long timescale

for the dynamics of cold and ultracold collisions of atoms in electromagnetic traps

has been observed to exceed lifetimes of excited states, which are on the order of 10−8

s. This means that spontaneous emission can occur during the course of collision and

may significantly alter the atomic collision dynamics [11, 12]. Cold atom phenomena

are also being merged with cavity quantum electrodynamics to realize single atom

lasers [13–15]. The function of these novel devices is based on strong coupling of the

atom to a single mode of the resonant cavity. Lastly, in polymer chemistry, ultrafast

light emission has been detected in certain ladder polymer films following ultrafast

laser excitation [16]. A fundamental understanding of the waveguiding process that

occurs in these polymers is unknown. It is precisely these situations, where the

electromagnetic and matter dynamics occur on the same timescale and are strongly

coupled, that are the motivation for this dissertation.

1.2 Historical and Mathematical Background

The history of the Maxwell-Schrodinger equations dates back to the early twen-

tieth century when the founding fathers of quantum mechanics worked out the the-

oretical details of the interaction of electrodynamics with quantum mechanics [17].

It was realized early on that the electromagnetic coupling to matter was through

3

the potentials Φ and A, and not the fields E and B themselves [6, 18, 19]. The

potentials and fields are related by

E = −∇Φ − A/c B = ∇× A (1.3)

which can be confirmed by inspecting the homogeneous Maxwell equations in (1.2).

Unlike in classical theory where the potentials were introduced as a convenient math-

ematical tool, the quantum theory requires the potentials and not the fields. That

is, the potentials are fundamental dynamical variables of the quantum theory but

the fields are not. A concrete demonstration of this fact was presented in 1959 by

Aharonov and Bohm [20].

1.2.1 Gauge Symmetry of Electrodynamics

It was well known from the classical theory of electrodynamics [9] that working

with the potentials leads to a potential form of Maxwell’s equations that is more

flexible than that in terms of the fields alone (1.2). In potential form, Maxwell’s

equations become

∇2A − A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ (1.4a)

∇2Φ +∇ · Ac

= −4πρ. (1.4b)

The homogeneous Maxwell equations are identically satisfied. These potential equa-

tions enjoy a symmetry that is not present in the field equations (1.2). This sym-

metry is called the gauge symmetry and can be generated by the transformation

A → A′ = A + ∇F Φ → Φ′ = Φ − F /c, (1.5)

where F is a well-behaved but otherwise arbitrary function called the gauge gener-

ator. Applying this gauge transformation to the potentials in (1.4) leads to exactly

the same set of potential equations. In other words, these equations are invari-

ant under arbitrary gauge transformation or are gauge invariant. They possess the

4

full gauge symmetry. Notice also that the electric and magnetic fields are gauge

invariant. In fact, it turns out that all physical observables are gauge invariant.

That electrodynamics possesses gauge symmetry places it in a league of theories

known as gauge theories [21]. These theories include general relativity [22, 23] and

Yang-Mills theory [24–26]. Gauge theories all suffer from an indeterminateness due

to their gauge symmetry. In an effort to deal with this indeterminateness, it is

common to first eliminate the symmetry (usually up to the residual symmetry; see

Chapter 4) by gauge fixing and then work within that particular gauge. That is,

the flexibility implied by the gauge transformation (1.5) allows for the potentials to

satisfy certain constraints. These constraints imply a particular choice of gauge and

gauge generator. Gauge fixing is the act of constraining the potentials to satisfy

a certain constraint throughout space-time. For example, in electrodynamics the

potential equations (1.4)

∇2A− A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ

∇2Φ +∇ · Ac

= −4πρ

(1.4)

form an ill-posed initial value problem. However, they can be converted to a well-

defined initial value problem by adding an equation of constraint to them. For

example, adding the constraint Φ/c + ∇ · A = 0 leads to the well-defined Lorenz

gauge equations

∇2A − A

c2= −4π

cJ ∇2Φ − Φ

c2= −4πρ (1.6)

while adding ∇ ·A = 0 leads to the well-defined Coulomb gauge equations

∇2A − A

c2= −4π

cJT ∇2Φ = −4πρ, (1.7)

where JT is the transverse projection of the current J (see Appendix A). There

are many other choices of constraint, each leading to a different gauge. It is always

5

possible to find a gauge function that will transform an arbitrary set of potentials

to satisfy a particular gauge constraint. The subject of the gauge symmetry of

electrodynamics, which is a subtle but fundamental aspect of this dissertation, is

discussed in detail in Chapter 4. In particular, it will be argued that fixing a par-

ticular gauge, which in turn eliminates the gauge from the theory, is not necessarily

optimal. Rather, it is stressed that the gauge freedom is a fundamental variable of

the theory and has its own dynamics.

1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics

Since the gauge symmetry of electrodynamics was well known, it was noticed by

the founding fathers that if quantum mechanics is to be coupled to electrodynamics,

then the Schrodinger equation (1.1) needs to be gauge invariant as well. The most

simple way of achieving this is to require the Hamiltonian appearing in (1.1) to be

of the form

H =[P− qA/c]2

2m+ V + qΦ, (1.8)

where P is the quantum mechanical momentum, V is the potential energy, and m

is the mass of the charge q. This is in analogy with the Hamiltonian for a classical

charge in the presence of the electromagnetic field [27, 28]. The coupling scheme

embodied in (1.8) is known as minimal coupling, since it is the simplest possible

gauge invariant coupling imaginable. The gauge symmetry inherent in the combined

system of Schrodinger’s equation and Maxwell’s equations in potential form can be

generated by the transformation

A → A′ = A + ∇F Φ → Φ′ = Φ − F /c Ψ → Ψ′ = exp(iqF/c)Ψ.

(1.9)

The transformation on the wavefunction is called a local gauge transformation and

differs from the global gauge transformation exp(iθ), where θ is a constant. These

6

global gauge transformations are irrelevant in quantum mechanics where the wave-

function is indeterminate up to a global phase. Application of the gauge transfor-

mation (1.9) to the Schrodinger equation with Hamiltonian (1.8) and to Maxwell’s

equations in potential form leads to exactly the same equations after the transfor-

mation. Therefore, like the potential equations (1.4) by themselves, the system of

Maxwell-Schrodinger equations

iΨ =[P− qA/c]2Ψ

2m+ VΨ + qΦΨ (1.10)

∇2A − A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ (1.11a)

∇2Φ +∇ · Ac

= −4πρ (1.11b)

is invariant under the gauge transformation (1.9). There are several other symme-

tries that are enjoyed by this system of equations. For example, they are invariant

under spatial rotations, nonrelativistic (Galilei) boosts, and time reversal. As a

result, the Maxwell-Schrodinger equations enjoy charge, momentum, angular mo-

mentum, and energy conservation. That each continuous symmetry gives rise to

an associated conservation law was proven by Emmy Noether in 1918 (see Gold-

stein [27], Jose and Saletan [28], and Abraham and Marsden [29], and the references

therein). This issue is discussed in Chapter 2 in greater detail.

It is worthwhile mentioning that the Maxwell-Schrodinger equations are obtain-

able as the nonrelativistic limit of the Maxwell-Dirac equations

iΨD = βmc2ΨD + cα · [P − qA/c]ΨD + qΦΨD (1.12)

∇2A − A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ (1.13a)

∇2Φ +∇ · Ac

= −4πρ (1.13b)

7

which are the equations of quantum electrodynamics (QED) [19, 24, 30]. Here the

wavefunction ΨD is a 4-component spinor where the first two components represent

the electron and the second two components represent the positron, each with spin-

1/2. The matrices β and α are related to the Pauli spin matrices [7, 8] and c is the

velocity of light. This system of equations possesses each of the symmetries of the

Maxwell-Schrodinger equations and in addition is invariant under relativistic boosts.

1.3 Approaches to the Solution of the Maxwell-Schrodinger Equations

Solving the Maxwell-Schrodinger equations as a coupled and closed system em-

bodies the theory of radiation reaction [9, 26, 31], which is a main theme of this

dissertation. However, it should first be pointed out that (1.1) and (1.2) are com-

monly treated separately. In these cases, the effects of one system on the other are

handled in one of the following two ways:

• The arrangement of charge and current is specified and acts as a source for theelectromagnetic field according to (1.2).

• The dynamics of the electromagnetic field is specified and modifies the dynamicsof the matter according to (1.1).

It is not surprising that either of these approaches is valid in many physical situ-

ations. Most of the theory of electrodynamics, in which the external sources are

prescribed, fits into the first case, while all of classical and quantum mechanics in

the presence of specified external fields fits into the second.

As a further example of the first case, the dipole power radiated by oscillating

dipoles generated by charge transfer processes in the interaction region of p − H

collisions can be computed in a straightforward manner [32, 33]. It is assumed

that the dynamics of the oscillating dipole is known and is used to compute the

dipole radiation, but this radiation does not influence the p − H collision. As a

result energy, momentum, and angular momentum are not conserved between the

proton, hydrogen atom, and electromagnetic field system. As a further example of

the second case, the effects of stimulated absorption or emission of electromagnetic

8

radiation by a molecular target can be added to the molecular quantum mechanics as

a first order perturbative correction. The electrodynamics is specified and perturbs

the molecule but the molecule does not itself influence the electrodynamics. This

approach, which is known as Fermi’s golden rule (see Chapter 3 and Merzbacher [7],

Craig and Thirunamachandran [34], and Schatz and Ratner [35]) is straightforward

and barring certain restrictions can be applied to many physical systems.

The system of Maxwell-Schrodinger equations or its relativistic analog can be

closed and is coupled when the Schrodinger wavefunction Ψ, which is the solution

of (1.1), is chosen to be the source for the scalar potential Φ and vector potential

A in (1.4). In particular, the sources of charge ρ and current J, which produce

the electromagnetic potentials according to (1.4), involve the solutions Ψ of the

Schrodinger equation according to

ρ = qΨ∗Ψ J = qΨ∗[−i∇ − qA/c]Ψ + Ψ[i∇− qA/c]Ψ∗

/2m. (1.14)

On the other hand, the wavefunction Ψ is influenced by the potentials that appear

in the Hamiltonian H in (1.8).

The interpretation of the Schrodinger wavefunction as the source for the elec-

tromagnetic field was Schrodinger’s electromagnetic hypothesis, which dates back to

1926. The discovery of the quantum mechanical continuity equation and its sim-

ilarity to the classical continuity equation of electrodynamics only reinforced the

hypothesis. However, it implied the electron to be smeared out throughout the

atom and not located at a discrete point, which is in contradiction to the accepted

Born probabilistic or Copenhagen interpretation. Schrodinger’s wave mechanics

had some success, especially with the interaction of the electromagnetic field with

bound states, but failed to properly describe scattering states due to the probabilis-

tic nature of measurement of the wavefunction. In addition, certain properties of

electromagnetic radiation were found to be inconsistent with experiment.

9

Schrodinger’s electromagnetic hypothesis was extended by Fermi in 1927 and

later by Crisp and Jaynes in 1969 [36] to incorporate the unquantized electromag-

netic self-fields into the theory. That is, the classical electromagnetic fields pro-

duced by the atom were allowed to act back upon the atom. The solutions of this

extended semiclassical theory captured certain aspects of spontaneous emission as

well as frequency shifts like the Lamb shift. However, it was quickly noticed that

some deviations from QED existed [37]. For example, Fermi’s and Jaynes’s theo-

ries predicted a time-dependent form for spontaneous decay that is not exponential.

There are many properties that are correctly predicted by this semiclassical theory

and are also in agreement with QED. In the cases where the semiclassical theory

disagrees with QED [37], it has always been experimentally verified that QED is cor-

rect. Nevertheless, the semiclassical theory does not suffer from the mathematical

and logical difficulties that are present in QED. To this end, the semiclassical theory,

when it is correct, provides a useful alternative to the quantum field theory. It is

generally simpler and its solutions provide a more detailed dynamical description of

the interaction of an atom with the electromagnetic field.

Since 1969 many others have followed along the semiclassical path of Crisp and

Jaynes. Nesbet [38] computed the gauge invariant energy production rate from a

many particle system. Cook [39] used a density operator approach to account for

spontaneous emission without leaving the atomic Hilbert space. Barut and Van

Huele [40] and Barut and Dowling [41, 42] formulated a self-field quantum electro-

dynamics for Schrodinger, Pauli, Klein-Gordon, and Dirac matter theories. They

were able to eliminate all electromagnetic variables in favor of Green’s function inte-

grals over the sources and were able to recover the correct exponential spontaneous

decay from an excited state. Some pertinent critiques of this work are expressed

by Bialynicki-Birula in [43] and by Crisp in [44]. Bosanac [45–47] and Doslic and

10

Bosanac [48] argued that the instantaneous effects of the self interaction are un-

physical. As a result, they formulated a theory of radiation reaction based on the

retarded effects of the self-fields. Milonni, Ackerhalt, and Galbraith [49] predicted

chaotic dynamics in a collection of two-level atoms interacting with a single mode of

the classical electromagnetic field. Crisp himself has contributed some of the finest

work in semiclassical theory. He computed the radiation reaction associated with a

rotating charge distribution [50], the atomic radiative level shifts resulting from the

solution of the semiclassical nonlinear integro-differential equations [51], the interac-

tion of an atomic system with a single mode of the quantized electromagnetic field

[52, 53], and the extension of the semiclassical theory to include relativistic effects

[54].

Besides semiclassical theory, a vast amount of research has been conducted

in the quantum theory of electrodynamics and matter. QED [19, 24, 30, 55] (see

Chapter 3), which is the fully relativistic and quantum mechanical theory of elec-

trons and photons, has been found to agree with all associated experiments. The

coupled equations of QED can be solved nonperturbatively [56, 57], but are most

often solved by resorting to perturbative methods. As was previously mentioned,

there are some drawbacks to these methods that are not present in the semiclassi-

cal theory. In addition to pure QED in terms of electrons and photons, there has

also been an increasing interest in molecular quantum electrodynamics [34]. Power

and Thirunamachandran [58, 59], Salam and Thirunamachandran [60], and Salam

[61] have used perturbative methods within the minimal-coupling and multipolar

formalisms to study the quantized electromagnetic field surrounding a molecule. In

particular, they have clarified the relationship between the two formalisms and in

addition have calculated the Poynting vector and spontaneous emission rates for

magnetic dipole and electric quadrupole transitions in optically active molecules.

11

In both the semiclassical and quantum mechanical context the self-energy of the

electron has been studied [62–65]. The self-energy arises naturally in the minimal

coupling scheme as the qΦ term in the Hamiltonian (1.8). More specifically, the

electron’s self-energy in the nonrelativistic theory is defined as

U =∫

Vd3xqΦ(x, t)Ψ∗(x, t)Ψ(x, t) =

∫

Vd3x

∫

Vd3x′

ρ(x, t)ρ(x′, t)

|x − x′| . (1.15)

As a result of the qΦ term, the Schrodinger equation (1.10) is nonlinear in Ψ. It

resembles the nonlinear Schrodinger equation [66]

iu = −a(d2u/dx2) + b|u|2u (1.16)

which arises in the modeling of Bose-Einstein condensates with the Gross-Pitaevskii

equation and in the modeling of superconductivity with the Ginzburg-Landau equa-

tion.

In the relativistic theory, the electron is forced to have no structure due to

relativistic invariance. As a result, the corresponding self-energy is infinite. On the

other hand, the electron may have structure in the nonrelativistic theory. Conse-

quently, the self-energy is finite. The self-energy of the electron will be discussed in

Chapter 4 in more detail.

1.4 Canonical Formulation of the Maxwell-Schrodinger Equations

The work presented in this dissertation [67] continues the semiclassical story

originally formulated by Fermi, Crisp, and Jaynes. Unlike other semiclassical and

quantum mechanical theories of electrodynamics and matter where the gauge is fixed

at the beginning, it will be emphasized that the gauge is a fundamental degree of

freedom in the theory and should not be eliminated. As a result, the equations

of motion are naturally well-balanced and form a well-defined initial value problem

when the gauge freedom is retained. This philosophy was pursued early on by Dirac,

Fock, and Podolsky [68] (see Schwinger [19]) in the context of the Hamiltonian

12

formulation of QED. However, their approach was quickly forgotten in favor of

the more practical Lagrangian based perturbation theory that now dominates the

QED community. More recently, Kobe [69] studied the Hamiltonian approach in

semiclassical theory. Unfortunately, he did not recognize the dynamical equation

associated with the gauge and refers to it as a meaningless equation.

It is believed that the Hamiltonian formulation of dynamics offers a natural and

powerful theoretical approach to the interaction of electrodynamics and wave me-

chanics that has not yet been fully explored. To this end, the Hamiltonian or canoni-

cal formulation of the Maxwell-Schrodinger dynamics is constructed in this disserta-

tion. (Canonical means according to the canons, i.e. standard or conventional.) The

associated work involves nonperturbative analytic and numerical methods for the so-

lution of the coupled and closed nonlinear system of Maxwell-Schrodinger equations.

The flexibility inherent in these methods captures the nonlinear and nonadiabatic

effects of the coupled system and has the potential to describe situations where the

atomic and electromagnetic dynamics occur on the same timescale.

The canonical formulation is set up by applying the time-dependent variational

principle to the Schrodinger Lagrangian

LSch = iΨ∗Ψ − [i∇− qA/c]Ψ∗ · [−i∇− qA/c]Ψ

2m− VΨ∗Ψ − qΦΨ∗Ψ, (1.17)

and Maxwell Lagrangian together with a Lorenz gauge fixing term, i.e.,

LLMax = LMax −[Φ/c+ ∇ · A]2

8π

=[−A/c−∇Φ]2 − [∇× A]2

8π− [Φ/c+ ∇ · A]2

8π.

(1.18)

This yields a set of coupled nonlinear first order differential equations of the form

ωη = ∂H/∂η (1.19)

13

where ω is a symplectic form, η is a column vector of the dynamical variables, and

H is the Maxwell-Schrodinger Hamiltonian (see Chapter 4). These matrix equations

form a well-defined initial value problem. That is, the solution to these equations

is uniquely determined and known in principle once the initial values for each of

the dynamical variables η are specified. These equations are also closed since the

Schrodinger wavefunction acts as the source, which is nonlinear (see J in 1.14), for

the electromagnetic potentials and these potentials act back upon the wavefunction.

By representing each of the dynamical variables in a basis of gaussian functions GK,

i.e., η(x, t) =∑

KGK(x)ηK(t), where the time-dependent superposition coefficients

ηK(t) carry the dynamics, the time-dependent variational principle generates a hier-

archy of approximations to the coupled Maxwell-Schrodinger equations. In the limit

of a complete basis these equations recover the exact Maxwell-Schrodinger theory,

while in any finite basis they form a basis representation that can systematically be

made more accurate with a more robust basis.

The associated basis equations have been implemented in a Fortran 90 com-

puter program [70] that is flexible enough to handle arbitrarily many gaussian basis

functions, each with adjustable widths and centers. In addition, a novel numerical

convergence accelerator has been developed based on removing the large Coulombic

fields surrounding a charge (that can be computed analytically from Gauss’s law,

i.e., ∇ · E = −∇2Φ = 4πρ, once the initial conditions are provided) by applying a

certain canonical transformation to the dynamical equations. The canonical trans-

formation separates the dynamical radiation from the Coulombic portion of the field.

This in turn allows the basis to describe only the dynamics of the radiation fields

and not the large Coulombic effects. The canonical transformed equations, which

are of the form ω ˙η = ∂H/∂η, have been added to the existing computer program

and the convergence of the solution of the Maxwell-Schrodinger equations is studied.

14

The canonical approach to dynamics enjoys a deep mathematical foundation

and permits a general application of the theory to many physical problems. In par-

ticular, the dynamics of the hydrogen atom interacting with its electromagnetic field

has been investigated for both stationary and superpositions of stationary states.

Stationary state solutions of the combined hydrogen atom and electromagnetic field

system as well as nonstationary states that produce electromagnetic radiation have

been constructed. This radiation carries away energy, momentum, and angular mo-

mentum from the hydrogen atom such that the total energy, momentum, angular

momentum, and charge of the combined system are conserved. A series of plots are

presented to highlight this atom-field dynamics.

1.5 Format of Dissertation

A tour of the Lagrangian and Hamiltonian dynamics is presented in Chapter 2.

Hamilton’s principle is applied to the derivation of the Euler-Lagrange equations of

motion. Emphasis is placed on the Hamiltonian formulation of dynamics, which is

presented from the modern point of view which makes connection with symplectic

geometry. To this end, both configuration space and phase space geometries are

discussed.

In Chapter 3, the Schrodinger and Maxwell dynamics will be presented from the

point of view of perturbation theory. In the Schrodinger theory, the electromagnetic

field is treated as a perturbation on the stationary states of an atomic or molecular

system. In the long time limit, the Fermi golden rule accounts for stimulated transi-

tions between these states. As an example, the absorption cross section is calculated

for an atom in the presence of an external field. QED is discussed to emphasize the

success of perturbation theory. In the Maxwell theory, the electromagnetic fields

arising from specified sources of charge and current are presented. The first order

(electric dipole) multipolar contributions to the electromagnetic field are calculated.

Lastly, the bremsstrahlung from a gaussian charge distribution is analyzed.

15

Chapter 4 contains the main body of the dissertation, which is on the Hamilto-

nian or canonical approach to the Maxwell-Schrodinger dynamics. Nonperturbative

analytic methods are constructed for the solution of the associated coupled and

nonlinear equations. The gauge symmetry is discussed in detail and exploited to

cast the Maxwell-Schrodinger equations into a well-defined initial value problem.

The theory of canonical or symplectic transformations is used to construct a special

transformation to remove the Coulombic contribution to the dynamical variables.

The well-defined Maxwell-Schrodinger theory from Chapter 4 is numerically

implemented in Chapter 5. The associated equations of motion are expanded into

a basis of gaussian functions, which renders the partial differential equations as

ordinary differential equations. These equations are coded in Fortran 90. In ad-

dition, the (canonical transformed) equations associated with the Coulomb reference

are incorporated into the existing code. The dynamics of the spinless hydrogen atom

interacting with the electromagnetic field are presented in a series of plots.

A summary and conclusion of the dissertation are presented in Chapter 6.

1.6 Notation and Units

A brief statement should be made about notation. All work will be done in

the (1+3)-dimensional background of special relativity with diagonal metric tensor

gαβ = gαβ with elements g00 = g00 = 1 and g11 = g22 = g33 = −1. All 3-vectors

will be written in bold faced Roman while all 4-vectors will be written in italics. As

usual, Greek indices run over 0, 1, 2, 3 or ct, x, y, z and Roman indices run over 1,

2, 3 or x, y, z. The summation convention is employed over repeated indices. For

example, the 4-potential Aµ = (A0, Ak) = (Φ,A) and Aµ = gµνAν = (Φ,−A). The

D’Alembertian operator = ∇2 − ∂2/∂(ct)2 = −∂2 is used at times in favor of ∂2.

Fourier transforms will be denoted with tildes, e.g., F is the Fourier transform of F.

The representation independent Dirac notation |h〉 will be used in the discussion of

time-dependent perturbation theory, but for the most part functions h(x) = 〈x|h〉

16

or h(k) = 〈k|h〉 will be used. (It will be assumed that all of the functions of physics

are in C∞ and in L2 ∩ L1 over either the real or complex field.) Since it is the

radiation effects present on the atomic scale that are of interest, it is beneficial to

work in natural (gaussian atomic) units where ~ = −|e| = me = 1. In these units

the speed of light c ≈ 137 atomic units of velocity.

CHAPTER 2THE DYNAMICS

A dynamical system may be well-defined once its Lagrangian and associated

dynamical variables as well as their initial values are specified. This information

together with the calculus of variations [71] generates the equations which govern

the dynamics. Chapter 2 will detail the aspects associated with generating equations

of motion for dynamical systems.

Many different variational methods exist by which to generate dynamical equa-

tions, each having subtle differences [72]. However, all methods rely on the ma-

chinery inherent in variational calculus. Given a starting and ending point for the

dynamics, the calculus of variations determines the path connecting them. The dy-

namics is determined by extremizing (either minimizing or maximizing) a certain

function of these initial and final points.

In this chapter, the Lagrangian and Hamiltonian formalisms [27–29] are pre-

sented for discrete and continuous systems. The Lagrangian approach leads to sec-

ond order equations of motion in time, while the Hamiltonian or canonical approach

leads to first order equations of motion in time. The resulting dynamics are equiva-

lent in either case. However, the Hamiltonian approach enjoys a rich mathematical

foundation connecting differential geometry and dynamics [28, 29]. Much of the re-

mainder of this dissertation will be devoted to the canonical formulation of Maxwell

and Schrodinger theories.

The time-dependent variational principle [73], which has its origin in nuclear

physics [74], is the variational approach to the determination of the Schrodinger

equation. The Hamiltonian dynamics associated with the Schrodinger equation

evolves in a generalized phase space endowed with a Poisson bracket. With the

17

18

time-dependent variational principle, many-body dynamics may be consistently de-

scribed in terms of a few efficiently chosen dynamical variables (see Deumens et.

al. [75]). Additionally, the variational technology provides a means by which to

construct approximations to the resulting equations of motion in a systematic and

well-balanced way. As will be seen in Chapters 4 and 5, these approximations will be

of utmost importance in the numerical solutions of the coupled nonlinear Maxwell-

Schrodinger equations.

2.1 Lagrangian Formalism

Before delving into a detailed account of Lagrangian dynamics it is instructive

to say a few words about the Lagrangian itself. The Lagrangian is a scalar function

of the vectors qk and qk (k = 1, . . . , N) with dimensions of energy. However, it is

not the energy nor is it physically observable. The Lagrangian is a fundamental

ingredient in the determination of a dynamical system. That is, the dynamics of a

system may be known in principle once the system’s Lagrangian is known and the

dynamical variables are given at some time.

The Lagrangian may have a number of symmetries. In 1918, Emmy Noether

(see Goldstein [27] and the references therein) proved that to each continuous sym-

metry there is an associated conservation law. For example, since all observations

indicate that Nature is invariant under time and space translations as well as spatial

rotations, so should be the Lagrangian. If the Lagrangian possesses time transla-

tion invariance, then the energy of the system is conserved. If the Lagrangian is

invariant to space translations (rotations), then the linear (angular) momentum of

the system is conserved. One last symmetry of significance in this dissertation is the

gauge symmetry. Since Nature is invariant to the choice of gauge, the Lagrangian

should maintain this symmetry as well. If the gauge symmetry is preserved, then

the system enjoys conservation of charge. Depending on the particular system at

19

hand, other symmetries may be of importance and should also be respected by the

Lagrangian.

2.1.1 Hamilton’s Principle

Given a Lagrangian L(qk, qk, t) dependent upon the N position vectors qk, the

N velocity vectors qk, and also the time t, the action I is defined by the path integral

I(qk, qk, t) ≡∫ t2t1L(qk, qk, t)dt k = 1, . . . , N. (2.1)

That the variation of this integral between the fixed times t1 and t2 leads to a

stationary point is a statement of Hamilton’s Principle [27, 28]. Moreover, this

stationary point is the correct path for the motion. In mathematical symbols, the

motion is a solution of

δI = δ∫ t2t1Ldt = 0, (2.2)

where δI is the variation of the action I. Only those paths are varied for which

δqk(t1) = 0 = δqk(t2). A particular form of the variational path parametrized by the

infinitesimal parameter α is given by

qk(t, α) = qk(t, 0) + αηk(t), (2.3)

where qk(t) = qk(t, 0) is the correct path of the motion and the vectors ηk(t) are

well-behaved and vanish at the boundaries t1 and t2. By continuously deforming

qk(t, α) until it is extremized, the correct path can be found.

This parametrization of the path in turn parametrizes the action itself. Equa-

tion (2.2) may now be rewritten more precisely as

δI(α) =∂I(α)

∂α

∣∣∣α=0

dα = 0 (2.4)

20

which represents infinitesimal variations from the correct path. The calculus of

variations yields

∂I(α)

∂α=

∫ t2t1dt

∂L

∂qk∂qk

∂α+∂L

∂qk∂qk

∂α

=∂L

∂qk∂qk

∂α

∣∣∣

t2

t1+

∫ t2t1dt

∂L

∂qk− d

dt

∂L

∂qk

∂qk

∂α,

(2.5)

where a partial integration was performed in the second line. Since δqk(t1) = 0 =

δqk(t2), the surface term vanishes. The stationary point of the variation is therefore

determined by∫ t2t1dt

∂L

∂qk− d

dt

∂L

∂qk

∂qk

∂α

∣∣∣α=0

= 0. (2.6)

But since the vectors ∂qk/∂α are arbitrary (choose in particular ∂qk/∂α > 0 and

continuous on [t1, t2]), the integral is zero only when

∂L

∂qk− d

dt

∂L

∂qk= 0 (2.7)

by the fundamental lemma of the calculus of variations. Equation (2.7) defines

the system of N second order Euler-Lagrange differential equations in terms of the

local coordinates (qk, qk). Since these equations are valid on every coordinate chart,

the Euler-Lagrange equations are coordinate independent. It is demonstrated in

[28] that (2.7) can be written in a coordinate free or purely geometric form. If

these equations admit a solution, then the action has a stationary value. It is this

stationary value which determines the motion. The second order form of the Euler-

Lagrange equations can be seen be expanding the total time derivative to give

∂L

∂qk−

∂2L

∂ql∂qkql +

∂2L

∂ql∂qkql +

∂2L

∂t∂qk

= 0. (2.8)

It will always be assumed unless otherwise noted that the Hessian condition is sat-

isfied. That is det∂2L/∂ql∂qk 6= 0.

Lastly, notice that the Lagrangian is arbitrary up to the addition of a total time

derivative. That is, if L → L′ = L + (d/dt)K for K a well-behaved function of the

21

dynamical variables, then the action

δI → δ∫ t2t1L+ (d/dt)Kdt = δK(t2) − δK(t1) +

∫ t2t1δL dt =

∫ t2t1δL dt = δI (2.9)

since δK(t2) = 0 = δK(t1). Thus the same Euler-Lagrange equations (2.7) are

generated for L′ as for L. In other words, there are many Lagrangians that lead

to the same equations of motion. There is no unique Lagrangian for a particular

dynamical system. All Lagrangians differing by only a time derivative will lead to

the same dynamics. More generally, in the dynamics of continuous systems two

equivalent Lagrangians may differ by a purely surface term in time and space.

2.1.2 Example: The Harmonic Oscillator in (qk, qk)

It is now useful to present a brief illustrative example. In two freedoms, the dy-

namics of a scalar mass subjected to the force of a harmonic potential with frequency

ωk is determined by the Lagrangian (no summation)

L(qk, qk) =1

2mqkqk − 1

2mω2

kqkqk k = 1, 2 (2.10)

which is a function of the real-valued vectors qk and qk. Application of the calculus

of variations to the associated action functional leads to (2.7) with ∂L/∂qk = mqk

and ∂L/∂qk = −mω2kqk. The second order Euler-Lagrange equations of motion are

1

2mqk +

1

2mω2

kqk = 0 k = 1, 2 (2.11)

with initial value solution qk(t) = qk(t0) cos(ωkt) + qk(t0) sin(ωkt)/ωk. It is said that

qk is an integral curve of the dynamical equation (2.11). Once the initial values

qk(t0) and qk(t0) are provided, the dynamics of the harmonic oscillator is known.

This dynamics occurs in a space whose coordinates are not just the qk, but both the

qk and qk. Some geometric aspects of this space will now be presented.

22

PSfrag

replacem

ents

Q

TqkQqk

Figure 2–1: The configuration manifold Q = S2 is depicted together with the tangent

plane TqkQ at the point qk ∈ Q.

2.1.3 Geometry of TQ

In the Lagrangian formalism, the dynamics unfolds in a velocity phase space

whose points are of the form (qk, qk). The position vectors qk lie in a differential

manifold called the configuration manifold Q, while the velocity vectors lie in the

manifold of vectors tangent to Q. The space formed by attaching the space spanned

by all vectors tangent to the point qk ∈ Q is called the tangent fiber above qk or

the tangent plane at qk and is denoted by TqkQ. The union of the configuration

manifold Q and the collection of all fibers TqkQ for each point qk ∈ Q (together

with local charts on TqkQ) is called the velocity phase space, tangent bundle, or

tangent manifold of Q and is denoted by TQ. It is that manifold that carries the

Lagrangian dynamics, not the configuration manifold Q. A picture is presented in

Figure 2–1 corresponding to the case where Q is the two-dimensional surface S2 of

the unit ball in R3. The tangent plane at the point qk reaches out of S

2 and into R3.

This larger manifold is where the associated Lagrangian dynamics occurs.

23

The integral curves of a dynamical system are vector fields and are called the

dynamics or the dynamical vector fields. The velocity phase space dynamics is a

vector field on TQ denoted by ∆L ≡ qk(∂/∂qk) + qk(∂/∂qk), where qk and qk are

the components of ∆L and ∂/∂qk and ∂/∂qk form a local basis for ∆L. The time

dependence of a dynamical variable F (qk, qk), which is an implicitly time-dependent

function on TQ, is determined by its variation along the dynamics. That is

F (qk, qk) ≡ ∆L(F ) =∂F

∂qkqk +

∂F

∂qkqk. (2.12)

The accelerations qk can be substituted directly from the dynamical equations.

Thus, the time dependence of a dynamical variable is determined by the equations

of motion themselves without even the knowledge of their solution.

Beyond functions and vector fields on TQ, there is another important geo-

metrical quantity called the one-form that is worth defining. One-forms on TQ

are linear functionals that map vector fields to functions. That is, if the one-form

α = A1adq

a +A2adq

a is applied to the vector field X = Xb1(∂/∂q

b) +Xb2(∂/∂q

b), then

their inner product results in

〈α|X〉 = A1aX

b1dq

a(∂/∂qb) + A1aX

b2dq

a(∂/∂qb) + A2aX

b1dq

a(∂/∂qb) + A2aX

b2dq

a(∂/∂qb)

= A1aX

b1δab + A2

aXb2δab

= A1aX

a1 + A2

aXa2 ,

(2.13)

where dqa(∂/∂qb) = dqa(∂/∂qb) = δab and dqa(∂/∂qb) = dqa(∂/∂qb) = 0, and where

Aa and Xb are the local components, which are functions, of the one-form α and

vector field X. It is common to write 〈α|X〉 ≡ α(X). It should be pointed out that

the differential of a function is a one-form. That is

dF =∂F

∂qkdqk +

∂F

∂qkdqk (2.14)

24

is a one-form and may be applied to the dynamical vector field ∆L to give

dF (∆L) ≡ 〈dF |∆L〉 =∂F

∂qkqk +

∂F

∂qkqk = F . (2.15)

The one-forms are also called covariant vectors or covectors and are dual to the

vector fields which are sometimes called contravariant vectors.

2.2 Hamiltonian Formalism

The Lagrangian formalism set up N second order dynamical equations which

required 2N initial values to fix the dynamics. Alternatively, and equivalently, the

dynamics may be described in terms of 2N first order equations of motion with 2N

initial values. This so called Hamiltonian dynamics evolves in a different tangent

manifold or phase space with generalized coordinates qa and pa, which are governed

by the dynamical equations

qa =∂H

∂paand − pa =

∂H

∂qa, (2.16)

where the function H is called the Hamiltonian (see (2.18) below). It is itself a

dynamical variable and for many physical systems it is the energy. Since (2.16) are

of first order, the associated trajectories are separated on the new phase space. The

change of variables from (qa, qa) to (qa, pa) is accomplished by a Legendre transfor-

mation [27, 28]. The momentum conjugate to the vector qa is defined in terms of

the Lagrangian L by

pa ≡∂L(qa, qa)

∂qa. (2.17)

Notice that this conjugate momentum is not a vector as is the velocity qa and does

not lie in the tangent manifold TQ. Rather the momentum pa is dual to the position

vector qa. It is a one-form and lies in the cotangent manifold T∗Q. This difference will

soon be elaborated on. With the momentum pa and the Lagrangian, the Hamiltonian

25

function is constructed according to

H(qa, pa) = paqa(qa, pa) − L(qa, pa). (2.18)

Here it is assumed that the relation (2.17) can be inverted to solve for the velocity

qa. Hamilton’s canonical equations of motion (2.16), which are first order differential

equations in time, can now be obtained from an argument similar to that presented

in Section 2.1.1 on Hamilton’s Principle. That is, if the Lagrangian in the action

integral (2.1) is replaced by L = paqa −H from (2.18), i.e.,

I(qa, pa) ≡∫ t2t1[paq

a(qa, pa) −H(qa, pa)]dt, (2.19)

then Hamilton’s equations follow in a straightforward manner.

2.2.1 Example: The Harmonic Oscillator in (qa, pa)

It is now useful to compare the Lagrangian and Hamiltonian dynamics for a

simple dynamical system. Recall the Lagrangian for the two freedom harmonic

oscillator in (2.10). That is

L(qa, qa) =1

2mqaqa − 1

2mω2

aqaqa a = 1, 2. (2.10)

The momentum conjugate to qa is pa ≡ ∂L/∂qa = mqa and with it the Hamiltonian

becomes

H(qa, pa) = paqa −

papa2m

− 1

2mω2

aqaqa

=papa2m

+mω2

aqaqa

2. (2.20)

With this Hamiltonian the equations of motion are:

qa =∂H

∂pa= pa/m

−pa =∂H

∂qa= mω2

aqa

(2.21)

and have the initial value solutions qa(t) = qa(t0) cos(ωat) + qa(t0) sin(ωat)/ωa and

pa(t) = pa(t0) cos(ωat)−mωaqa(t0) sin(ωat). These are the integral curves of the first

26

order differential equations (2.21) and may be compared to those in the Lagrangian

formulation.

2.2.2 Symplectic Structure and Poisson Brackets

One of the many powerful aspects of the Hamiltonian or canonical approach

to dynamics is the flexibility and ability to treat positions and momenta similarly.

This similarity among the coordinates is made explicit by the following notation:

ξa = qa a = 1, . . . , N

ξa = pa−N a = N + 1, . . . , 2N.(2.22)

Similarly, the forces become ∂H/∂pa ≡ ∂H/∂ξa+N and ∂H/∂qa ≡ ∂H/∂ξa so that

the equations of motion are:

ξa =∂H

∂ξa+Na = 1, . . . , N

−ξa =∂H

∂ξa−Na = N + 1, . . . , 2N.

(2.23)

These Hamilton equations may be written more compactly as

ωabξb =

∂H

∂ξa, (2.24)

where ωab are the matrix elements of the symplectic form ω. The symplectic form is

an antisymmetric 2N × 2N -dimensional matrix of the form

ω =

(0N − 1N1N 0N

)

, (2.25)

where 0N and 1N are the N×N -dimensional zero and identity matrices respectively.

The matrix (2.25) is also referred to as the canonical symplectic form because it

satisfies the properties:

ω2 = −1 and ωT = −ω, (2.26)

or equivalently ωabωbc = δac and ωab = −ωba. The matrix element ωab with both

indices up is the inverse of ωab.

27

In (2.12), the time derivative or variation of a (implicitly time-dependent) dy-

namical variable F on TQ was demonstrated. In a similar fashion, F can be viewed

in the momentum phase space T∗Q, which will be discussed shortly. It is

F (qb, pb) =∂F

∂ξbξb =

∂F

∂ξbωba

∂H

∂ξa, (2.27)

where the equation of motion (2.24) was inverted and substituted for ξb. The right

hand side of this equation is called the Poisson bracket of F with H. In general, it

may be written for any two functions in T∗Q as

F,G ≡ ∂F

∂ξbωba

∂G

∂ξa=∂F

∂qa∂G

∂pa− ∂F

∂pa

∂G

∂qa. (2.28)

In particular, an alternative form of Hamilton’s equations is derived when the Pois-

son bracket is applied to the coordinate ξ. That is

ξa = ξa, H. (2.29)

Since the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity

f, gh = gf, h + f, gh, the set of functions on T∗Q forms a Lie algebra under

Poisson bracket ·, ·. In fact, the Hamiltonian dynamics can naturally be studied

from this point of view [29, 73].

2.2.3 Geometry of T∗Q

As was previously mentioned, the dynamics associated with Hamilton’s equa-

tions of motion (2.24) do not unfold in the same velocity phase space TQ that was

defined in Section 2.1.3. These equations of motion define a vector field ξ on a dif-

ferent phase space whose components are the functions ωba(∂H/∂ξa). The integral

curves of this vector field are the dynamics.

Recall that the points of TQ are made up of qk and qk. The velocities qk are the

local components of the vector field qk(∂/∂qk). However, the momenta are the local

components of the one-form padqa ≡ (∂L/∂qa)dqa, which are not the components

28

of a vector field. Since one-forms are dual to vector fields, padqa lies in the dual

space of TqaQ. This space is the cotangent space at qa and is denoted by T∗qaQ. In

analogy with TQ, the cotangent bundle or cotangent manifold T∗Q is made up of

Q together with its cotangent spaces T∗qaQ. Consequently, the carrier manifold for

the Hamiltonian dynamics is not TQ, but rather it is the phase space T∗Q. The

dynamical vector field on T∗Q is given by

∆H ≡ ξb∂

∂ξb= qa

∂

∂qa+ pa

∂

∂pa=∂H

∂pa

∂

∂qa− ∂H

∂qa∂

∂pa, (2.30)

where Hamilton’s equations of motion (2.16) were substituted for the qa and pa.

There is one last geometric quantity that needs to be defined. The symplectic

form ω is a two-form on T∗Q. Two-forms are bilinear, antisymmetric forms that map

pairs of vector fields to functions. That is, if X = Xa(∂/∂ξa) and Y = Y b(∂/∂ξb)

are vector fields on T∗Q, then

ω(X, Y ) = XaY bω(∂/∂ξa, ∂/∂ξb) = XaωabYb = XaYa − Y aXa. (2.31)

The matrix elements ωab = −ωba are identical to those presented earlier. Since ω is

nonsingular and the differential dω = 0, i.e., ω is closed, the two-form ω is called

a symplectic form. In general, phase space is naturally endowed with a symplectic

form or structure. For this reason T∗Q is also a symplectic manifold [29]. Lastly, it

should be mentioned that ω(X, Y ) is a measure of the area between the vectors X

and Y. In fact, there is a powerful theorem attributed to Liouville [27–29] that states

that the phase space volume must be invariant under canonical transformations in

phase space. Canonical transformations are those transformations that maintain the

symplectic structure of Hamilton’s dynamical equations

ωabξb =

∂H

∂ξa. (2.24)

More will be said on canonical transformations in Chapter 4.

CHAPTER 3ELECTRODYNAMICS AND QUANTUM MECHANICS

The coupling of electrodynamics to charged matter is a complicated problem.

This complexity is compounded by the fact that the fields produced by charges in

motion react back upon the charges, thus causing a modification of their trajectory.

As mentioned in the introduction, the corresponding physics is often analyzed in

one of two ways. Either:

• The electromagnetic field is taken as an influence on the dynamics of the charges.

• The sources of charge and current are used to calculate the dynamics of theelectromagnetic field.

Chapter 3 will discuss both of these cases in detail. The first portion of this chapter

will set up the time-dependent perturbation theory which will be used to make

calculations in quantum mechanics under the influence of an electromagnetic field.

The second portion of this chapter will explore the electrodynamics resulting from

a given ρ and J. In particular, the multipole expansion will be introduced and used

to calculate the power radiated from an oscillating electric dipole. Additionally, the

electromagnetic fields corresponding to a gaussian wavepacket will be presented. In

the narrow width limit of the gaussian, the resulting physics reduces to the expected

textbook results for a point source.

3.1 Quantum Mechanics in the Presence of an Electromagnetic Field

The dynamics of charges in an external electromagnetic field may be studied

at varying levels of sophistication from a purely classical description of both charge

and field to a fully quantum treatment. Various semiclassical or mixed quantum-

matter/classical-field descriptions are available as well as fully quantum and rela-

tivistically invariant treatments such as quantum electrodynamics.

29

30

Time-dependent perturbation theory [7] is a systematic method by which to

calculate (among other things) properties of the dynamics of charges in an external

electromagnetic field. In this section, the time-dependent perturbation theory is in-

troduced for a general perturbation in the context of quantum mechanics. Emphasis

is then placed on the classical electromagnetic field as a particular time-dependent

perturbation V. Within this framework the perturbation is seen as causing tran-

sitions between two stationary states |Ψk〉 and |Ψm〉 of an atomic system, and is

symbolized to lowest order by the matrix element Vkm = 〈Ψk|V |Ψm〉. Experimen-

tal observables such as the rate of transition or absorption cross section may be

calculated from Vkm. Additionally, time-dependent perturbation theory gives a pre-

scription for calculating successively higher order corrections to Vkm, which may

in turn provide better and better agreement with experiment. This section con-

cludes with a discussion of quantum electrodynamics, in which both matter and

fields are quantized and the description is relativistically invariant. Here again the

time-dependent perturbation theory (often in the form of Feynman diagrams) is the

essential machinery used in calculations.

3.1.1 Time-Dependent Perturbation Theory

An important class of solutions to the Schrodinger equation (1.1) are those

which are eigenfunctions of the Hamiltonian operatorH. These solutions |Ψm〉 satisfy

the time-independent Schrodinger equation

H|Ψm〉 = Em|Ψm〉 (3.1)

and are called stationary states. A general solution |Ψ(t)〉 of the Schrodinger equa-

tion (1.1) may be constructed from these stationary states according to

|Ψ(t)〉 = e−iH(t−t0)|Ψ(t0)〉 =∑

me−iEm(t−t0)|Ψm〉〈Ψm|Ψ(t0)〉, (3.2)

31

where |Ψ(t0)〉 is an initial state vector and where the sum over m may imply inte-

gration if the energy spectrum is continuous.

Equation (3.2) is only applicable when the Hamiltonian is time-independent.

For if H ≡ H(t), then the energy of the system is not conserved and H admits no

strictly stationary states. However, it may be possible to split a time-dependent

Hamiltonian into the sum of two terms:

H = H0 + V (t), (3.3)

where H0 is time-independent and describes the unperturbed system while V (t)

accounts for the time-dependent perturbation. To fix ideas, consider for example

the electronic transition induced by a passing electromagnetic disturbance that is

localized in both space and time. In other words, the system is initially unperturbed

for some long time and is in an eigenstate of H0. While in the interaction region

the system is perturbed by V after which it settles down into another unperturbed

eigenstate of H0 for a sufficiently long time.

Time-dependent perturbation theory seeks to connect the stationary states of

the unperturbed system, i.e., those states satisfying

H0|Ψm〉 = Em|Ψm〉, (3.4)

with the time-dependent perturbation V (t). These calculations are most clearly

demonstrated in the interaction picture. In the interaction picture the perturbation

is singled out by applying the unitary operator U0 = exp(iH0t) to |Ψ(t)〉. That is

|ΨI(t)〉 = eiH0t|Ψ(t)〉 (3.5)

and the time-dependent Schrodinger equation (1.1) becomes

i(d/dt)|ΨI(t)〉 = VI(t)|ΨI(t)〉, (3.6)

32

where VI(t) = U0V (t)U †0 . In other words, the interaction picture separates the

physics that depends upon the perturbation from the physics that depends upon

only the unperturbed system. The state vector at time t is obtained from that at

time t0 via

|ΨI(t)〉 = UI(t, t0)|ΨI(t0)〉, (3.7)

where UI is the time evolution operator which satisfies

UI(t, t0) = 1 − i∫ t

t0dt′ VI(t

′)UI(t′, t0). (3.8)

The time evolution operator connects the (orthonormal) stationary states |Ψk〉 and

|Ψm〉 according to

〈Ψk|UI(t, t0)|Ψm〉 = δkm − i∫ t

t0dt′〈Ψk|VI(t′)UI(t′, t0)|Ψm〉

= δkm − i∑

n

∫ t

t0dt′〈Ψk|VI(t′)|Ψn〉〈Ψn|UI(t′, t0)|Ψm〉

= δkm − i∑

n

∫ t

t0dt′eiωknt

′〈Ψk|V (t′)|Ψn〉〈Ψn|UI(t′, t0)|Ψm〉,

(3.9)

where∑

n |Ψn〉〈Ψn| = 1 and ωkn = Ek − En were used.

The time-dependent perturbation theory is now set up by iterating on (3.8). If

the perturbation V is small then the time evolution operator becomes a power series

in V. That is

UI(t, t0) = 1 − i∫ t

t0dt′ VI(t

′) + (−i)2∫ t

t0dt′ VI(t

′)∫ t′

t0dt′′ VI(t

′′) + · · · . (3.10)

And so at first order the transition amplitude between two distinct states of energy

Ek and Em (with k 6= m) is

〈Ψk|UI(t, t0)|Ψm〉 = −i∫ t

t0dt′eiωkmt

′〈Ψk|V (t′)|Ψm〉. (3.11)

33

Assuming that the perturbation is sufficiently small, the probability of finding the

system in the state |Ψk〉 is given by

Pk←m(t) =∣∣∣〈Ψk|UI(t, t0)|Ψm〉

∣∣∣

2

=∣∣∣ − i

∫ t

t0dt′eiωkmt

′〈Ψk|V (t′)|Ψm〉∣∣∣

2

. (3.12)

If the perturbation is localized in time then t0 and t may be naively extended to

infinity to yield the transition probability

Pk←m(+∞) =∣∣∣〈Ψk|UI(+∞,−∞)|Ψm〉

∣∣∣

2

=∣∣∣− i

∫∞

−∞dt eiωkmt〈Ψk|V (t)|Ψm〉

∣∣∣

2

(3.13)

which involves a Fourier integral of the matrix element Vkm = 〈Ψk|V (t)|Ψm〉.

3.1.2 Fermi Golden Rule

The formalism set up thus far is also applicable for time-independent perturba-

tions V 6= V (t). In this case the transition probability can be obtained from (3.12)

as

Pk←m(t) = 2|〈Ψk|V |Ψm〉|21 − cos(ωkmt)

(Ek − Em)2(3.14)

which is proportional to t2 if Ek ≈ Em.

Now consider the situation in which there is a near continuum of final states

available having energies in the interval (Em − ∆E/2, Em + ∆E/2). If the density

of the near continuum states is denoted by ρF (E), then the transition probability

to all of these states is given by

∑

k∈FPk←m(t) =∫ Em+∆E/2

Em−∆E/22|〈Ψk|V |Ψm〉|2

1 − cos(ωkmt)

(Ek − Em)2ρF (Ek)dEk, (3.15)

where the sum runs over all states |Ψk〉 belonging to the near continuum of final

states. The quotient [1−cos(ωkmt)]/(Ek−Em)2 is sharply peaked at Ek = Em which

confirms that the dominant transitions are those that conserve the unperturbed

energy. Since both |〈Ψk|V |Ψm〉|2 and ρF (Ek) are approximately constant around Em

and t is such that ∆E 2π/t (i.e., long time behavior), the transition probability

34

becomes

∑

k∈FPk←m(t) ≈ 2|〈Ψk|V |Ψm〉|2ρF (Ek)∫∞

−∞

1 − cos(ωkmt)

ω2km

dωkm

= 2πt|〈Ψk|V |Ψm〉|2ρF (Ek)

(3.16)

which increases linearly with time.

The total transition probability per unit time or transition rate Γ is given by

Γ = (d/dt)∑

k∈FPk←m(t) = 2π|〈Ψk|V |Ψm〉|2ρF (Ek) (3.17)

and is constant. Fermi’s golden rule of time-dependent perturbation theory [7, 34,

35] embodies the tendency for the perturbed system to make energy conserving tran-

sitions for which the probability increase as t2 or to make nearly energy conserving

transitions which oscillate in time. Either way the transition rate Γ is constant.

Fermi’s golden rule may be extended to include perturbations that vary har-

monically in frequency ω. An electromagnetic disturbance of a charge would be an

example. In this case the golden rule generalizes to

Γ = 2π|〈Ψk|V |Ψm〉|2ρF (Em + ω). (3.18)

3.1.3 Absorption of Electromagnetic Radiation by an Atom

Recall the electromagnetic field coupling to quantum mechanics is given by the

minimal coupling prescription

i(∂/∂t) → i(∂/∂t) − qΦ

−i∇ → −i∇− qA/c,

(3.19)

where Aα = (Φ,A) are the dynamical variables of the electromagnetic field. Apply-

ing this transformation to the Schrodinger equation iΨ = P2Ψ/2m+ V0Ψ results in

the Schrodinger equation coupled to the electromagnetic field

iΨ =[P − qA/c]2

2mΨ + V0Ψ + qΦΨ (3.20)

35

with Hamiltonian

H =[P− qA/c]2

2m+ V0 + qΦ. (3.21)

In the Coulomb gauge (see Appendix A for details) this Hamiltonian becomes

H =P2

2m+ V0 −

q

mcA · P +

q2

2mc2A2. (3.22)

The external free electromagnetic field evolves according to ∇2A − A/c2 = 0 with

∇·A = 0 and Φ = 0 since it is assumed that the charges do not themselves contribute

to the field. By ignoring the quadratic term in A, the Hamiltonian H separates into

an unperturbed portion

H0 =P2

2m+ V0 (3.23)

plus the perturbation

V = − q

mcA · P. (3.24)

It should be pointed out here that substantial confusion has existed in the litera-

ture over the A · P appearing in the perturbation V. This confusion was due the

widespread use of E · r and its higher order approximations [59, 60] instead of A ·P.

The relationship between these two approaches have been thoroughly investigated

in [76–79].

The cross section for stimulated absorption (or emission) of radiation by an

atom may be calculated via Fermi’s golden rule. If the external field varies harmon-

ically in frequency as a plane wave, then the perturbation becomes

V (x, t) = − q

mc

A0e

i(k·x−ωt) + A∗0e−i(k·x−ωt)

ε · P (3.25)

where ε is the field’s polarization. The rate of energy absorption by the atom is

Γω =q2ω

m2c2|A0|2

∣∣∣〈Ψk|eik·xε ·P|Ψm〉

∣∣∣

2

ρF (Em + ω). (3.26)

36

If the density of the near continuum states is narrow then

ρF (Em + ω) = δ(Ek − Em + ω) = δ(ωkm + ω) (3.27)

and the absorption cross section σabs(ω) = Γ/I0 becomes

σabs(ω) =q2|A0|2

∣∣∣〈Ψk|eik·xε ·P|Ψm〉

∣∣∣

2

δ(ωkm + ω)/m2c2

ω|A0|2/2πc

=4π2q2

ωm2c|A0|2

∣∣∣〈Ψk|eik·xε ·P|Ψm〉

∣∣∣

2

δ(ωkm + ω),

(3.28)

where I0 = ω|A0|2/2πc is the incident flux of photons of frequency ω. Similarly the

emission cross section is

σem(ω) =4π2q2

ωm2c|A0|2

∣∣∣〈Ψk|e−ik·xε ·P|Ψm〉

∣∣∣

2

δ(ωkm − ω). (3.29)

Notice that the time-dependent perturbation theory gives properties of the so-

lution but not the solution. That is, the cross section is easily accessible but the

wavefunction and 4-potential are not. The cross section is a property of the solution

and can be calculated from knowledge of the solution. Of course, the wavefunction

and 4-potential constitute the actual solution.

3.1.4 Quantum Electrodynamics in Brief

The quantum theory of electrodynamics [19, 24, 30], also known as QED, is the

interacting quantum field theory of electron and photon fields. The relativistically

invariant QED is one of the most successful physical theories to date, in that there

is no evidence for any discrepancy between experiment and prediction. However,

QED is beset by many mathematical and logical difficulties. These difficulties are

in some cases avoided by physical arguments or simply concealed from view as in

the renormalization of mass and charge.

Putting aside its inconsistencies, QED is a prime example of the success of

time-dependent perturbation theory. A combination of the free Dirac theory and

the free Maxwell theory provide the unperturbed states on which the interaction

37

Lint = −JµAµ/c operates. The free QED Lagrangian density

Lfree QED = ψ[iγµ∂µ −mc]ψ − 1

16πFµνF

µν (3.30)

gives the equations of motion for the free electron

[iγµ∂µ −mc]ψ = 0 (3.31)

and the equations of motion for the free electromagnetic field

∂µFµν = ∂2Aν − ∂ν(∂ · A) = 0ν, (3.32)

where the Dirac γ-matrices are related to the Pauli spin matrices (4.66), ψ = ψ†γ0

is the Dirac adjoint of the four component spinor ψ, and Fµν = ∂µAν − ∂νAµ is

the electromagnetic field tensor. This noninteracting theory sets up the free un-

perturbed in-state |p1 · · ·pn〉in and out-state |k1 · · ·km〉out, which will be connected

by UI(+∞,−∞) = Texp[i∫d4xLint], where T is the time-ordering operator. The

resulting matrix elements will yield some properties of the dynamics.

Working in the interaction picture, the machinery of time-dependent perturba-

tion theory is used to construct the scattering matrix or S-matrix

out〈k1 · · ·km|p1 · · ·pn〉in = in〈Ω|a(k1) · · ·a(km)UI(+∞,−∞)a†(p1) · · ·a†(pn)|Ω〉in

(3.33)

where S = 1 + iT and |Ω〉in is the in-vacuum. The respective fermion creation

and annihilation operators a†(k) and a(k), create and annihilate single fermions of

momentum k according to a†(k)|Ω〉 = |k〉 and a(k)|Ω〉 = 0 where the spin has been

neglected for simplicity. The situation in which the particles do not interact at all

(the 1 part) as well as the interesting interactions (the iT part) are both included in

the S-matrix. The interacting components are commonly collected and are referred

38

to as the T-matrix. Together 1+ iT is used to define the invariant amplitude M as

out〈k1 · · ·km|UI(+∞,−∞)|p1 · · ·pn〉in

= (2π)4δ(p1 + · · · + pn − k1 − · · · − km) · iM(p1, · · · ,pn → k1, · · · ,km).

(3.34)

This invariant amplitude is analogous to the scattered wavefunction of quantum

mechanics, i.e.,

Ψk(x) ∼ N[

eik·x + fk(Ω)eikr

r

]

, (3.35)

where the unscattered field exp(ik ·x) and the spherically scattered field exp(ikr)/r

are indicated schematically in Figure 3–1. In fact all of quantum mechanics is

PSfrag

replacem

ents

Scattered field

Incident field

Figure 3–1: The coefficient 1 of the unscattered plane wave exp(ik · x) is analogousto the 1 part of the S-matrix, while the scattering amplitude fk(Ω) which modulatesthe scattered spherical wave exp(ikr)/r is analogous to the iT part.

just the nonrelativistic limit of QED. Unfortunately, while QED is suitable for the

scattering of single particle states to single particle states, it requires great effort to

deal with bound states.

The probability of finding |k1 · · ·km〉out in |p1 · · ·pn〉in is given by

P (+∞) ∼∣∣∣out〈k1 · · ·km|p1 · · ·pn〉in

∣∣∣

2 d3k1

(2π)3· · · d

3km(2π)3

=∣∣∣in〈Ω|a(k1) · · ·a(km)UI(+∞,−∞)a†(p1) · · ·a†(pn)|Ω〉in

∣∣∣

2 d3k1

(2π)3· · · d

3km(2π)3

(3.36)

39

which is analogous to (3.13). A similar connection can be made in the cross section.

If n = 2 in the in-state, then the differential scattering cross section dσ becomes

dσ ∼∣∣∣iM(p1,p2 → k1, · · · ,km)

∣∣∣

2 d3k1

(2π)3· · · d

3km(2π)3

(2π)4δ(p1 + p2 − k1 − · · · − km)

(3.37)

which is analogous to (3.28). As in quantum mechanics, time-dependent pertur-

bation theory in QED gives a prescription by which to calculate properties of the

solution which rely on scattering amplitudes, e.g., cross sections, decay rates, and

probabilities. It is considerably more difficult to compute the actual solution, which

in this case would be the states on which the field operators

Aµ(x, t) =∫

V

d3k

(2π)3

∑

λ

εµ(k, λ)c(k, λ)e−ik·x√

2k+ ε∗µ(k, λ)c†(k, λ)

eik·x√2k

ψi(x, t) =∫

V

d3k

(2π)3

∑

s

ui(k, s)a(k, s)e−ik·x√

2ω+ vi(k, s)b

†(k, s)eik·x√

2ω

(3.38)

act. In (3.38), εµ and ε∗µ are helicity eigenstates of Aµ, and c, c† are photon creation

and annihilation operators. Similarly ui and vi are eigenspinors of ψi, and a, a†

and b, b† are electron and positron [80] creation and annihilation operators.

Lastly it should be pointed out that a beautiful representation of the time-

dependent perturbation theory was introduced by Feynman [55]. These so called

Feynman diagrams provide a pictorial version of the invariant amplitude

iM = C · out〈k1 · · ·km|UI(+∞,−∞)|p1 · · ·pn〉in

= C · out〈k1 · · ·km|T

ei∫d4xLint

|p1 · · ·pn〉in (3.39)

=[out〈k1 · · ·km|

(1 + i

∫d4xT

Lint

+ · · ·

)|p1 · · ·pn〉in

]

connected.

40

For example, the invariant amplitude for Bhabha scattering, i.e., e+e− → e+e−, is

iMBhabha =[out〈k1k2|p1p2〉in + out〈k1k2|i

∫d4xT

Lint

|p1p2〉in +O(L2

int)]

connected

=

p1

p2

k1

k2

+

p1

p2

k1

k2

+ O(L4int),

(3.40)

where each of the above diagrams corresponds to a term (or portion thereof) in

the perturbative expansion of iMBhabha. These tree order diagrams are the lowest

order nonvanishing diagrams that contribute to and are the largest part of the

Bhabha scattering invariant amplitude. Higher order perturbative corrections to the

amplitude also have pictorial representations and may be systematically constructed

using Feynman’s prescription.

In this manner the time-dependent perturbation theory may be diagrammati-

cally written to any order, translated into mathematical expressions, and computed.

While this is by no means an easy task, the invariant amplitude may in principle be

calculated to any order. Notice again that this machinery produces the amplitude

iM, which is a property of the solution but not the actual solution.

3.2 Classical Electrodynamics Specified by the Sources ρ and J

If the sources of charge and current are known, then the dynamics of the result-

ing electromagnetic field can be calculated from Maxwell’s equations at each point

in space-time. These fields may behave quite differently depending on the motion

of their source. For example, a static source gives rise to a purely electrostatic field,

while a uniformly moving source creates both an electric field and a magnetic field.

More importantly, if the source is accelerated then electromagnetic radiation is pro-

duced. Electromagnetic radiation is a unique kind of electromagnetic field in that

41

it carries away energy, momentum, and angular momentum from its source. The

radiation field is not bound to the charge as are the static fields.

In this section the electromagnetic fields produced by an oscillating electric

dipole are calculated to lowest order via the multipole expansion. The correspond-

ing power and radiation pattern are also presented. Then, the dynamics of the

bremsstrahlung produced by a wavepacket source is analyzed. It is shown that the

wavepacket’s fields reduce in the narrow width limit to the usual point source results.

The consistent coupling of electrodynamics and quantum mechanics is needed

because the sources of charge and current produce electromagnetic fields and these

fields act back upon the sources. The understanding of this process requires the

inclusion of recoil effects on the charges due to the electromagnetic field. These

effects, known as radiation reaction effects, are a main aspect of this dissertation

and will be discussed in detail in Chapter 4.

3.2.1 Electromagnetic Radiation from an Oscillating Source

In this section the Lorenz gauge (see Appendix A for details) is used to inves-

tigate the electromagnetic radiation produced by a localized system of charge and

current [9] which vary sinusoidally in time according to (the real part of):

ρ(x, t) = ρ(x)e−iωt

J(x, t) = J(x)e−iωt.

(3.41)

It is assumed that the electomagnetic potentials and fields also have the same time-

dependence. The general solutions to the wave equations of (A.5) are given by

Φ(x, t) =∫

Vd3x′

∫∞

−∞dt′ ρ(x

′, t′)

|x − x′|δ(

t′ −[

t+|x − x′|

c

])

A(x, t) =1

c

∫

Vd3x′

∫∞

−∞dt′

J(x′, t′)

|x − x′|δ(

t′ −[

t+|x − x′|

c

])

,

(3.42)

where G(+)(x, t;x′, t′) = δ(t′ − [t + |x − x′|/c])/4π|x − x′| is the retarded Green’s

function for the wave operator ∂2 = − = ∂2/∂(ct)2 − ∇2 (see Appendix B). It is

42

assumed that there are no boundary surfaces present. With the oscillating sources

from (3.41), it will be seen that all of the dynamics of the electromagnetic field for

which ω 6= 0 can be described in terms of the A alone. The component of the

electromagnetic field for which ω = 0 is just the static electric monopole field

Φmonopole(x, t) =q

|x| . (3.43)

The vector potential for all other frequencies is

A(x) =1

c

∫

Vd3x′

eik|x−x′|

|x − x′|J(x′), (3.44)

where the wavevector k = ω/c and it is understood that A(x, t) = A(x)e−iωt.

For a given charge density J, (3.44) could in principle be computed. With the

resulting vector potential the electromagnetic field may be calculated from Ampere’s

law. That is

B = ∇× A (3.45a)

E =i

k∇× B (3.45b)

in a region outside the source. Instead of evaluating (3.44) exactly, general properties

of its solution may be determined whenever the dimensions of J are much smaller

than a wavelength. That is, if the dimensions of the charge density are of order d

and the wavelength λ = 2π/k, then d λ. From these distances, the following three

spatial regions may be constructed:

The near or static zone: d r λ

The intermediate zone: d r ∼ λ

The far or radiation zone: d λ r

In each region the electromagnetic field behaves quite differently. For example in

the near zone, the fields behave as if they were static fields which show strong

43

dependence on their source. On the other hand in the far zone, the fields display

properties of radiation fields which are transverse and fall off as r−1.

The static near zone fields may be obtained from (3.44) by noting that kr 1

since r λ. In this case exp(ik|x − x′|) ∼ 1 and the vector potential becomes

Anear(x) =1

c

∫

Vd3x′

J(x′)

|x − x′|

=1

c

∫

Vd3x′J(x′)

∑

lm

4π

2l + 1

r′l

rl+1Ylm(Ω)Y ∗lm(Ω′)

= − ikpr

+ · · · ,

(3.46)

where |x − x′|−1 has been expanded into the spherical harmonics Ylm and an inte-

gration by parts was performed with all surface terms vanishing. The equation of

continuity iωρ+∇·J = 0 was also used in the computation as well as the definition

of the dipole moment p =∫

Vd3xxρ(x). From (3.45), the resulting magnetic and

electric fields are:

Bnear =ik

r2n × p + · · · (3.47a)

Enear =3n(n · p) − p

r3+ · · · , (3.47b)

where n is the unit vector in the direction of the observation point x. Notice that

Enear is independent of the frequency ω and is thus purely static. As expected Bnear

is zero in the static limit ω → 0. A multipole expansion of the near zone vector

potential can now be made and successively better results may be obtained by going

to higher orders in (l, m).

44

At the other extreme, the far zone fields for which kr 1 may be obtained

from (3.44) by noticing that

|x − x′| =√

(x − x′) · (x − x′)

= |x|√

1 − 2x · x′|x|2 +

|x′|2|x|2

≈ |x|(

1 − x · x′|x|2

)

= r − n · x′

(3.48)

since |x′| |x| = r|n| = r. A picture of the corresponding situation is shown in

Figure 3–2, where the x′-integration runs over the domain of the source J. With the

PSfrag

replacem

ents

dO

n r = |x|

x

Figure 3–2: In the radiation zone, the observation point x is located far from thesource J. In this case the distance |x − x′| ≈ r − n · x′.

approximation (3.48), the far zone vector potential becomes

Afar(x) =eikr

r

1

c

∫

Vd3x′J(x′)e−ikn·x

′

=eikr

r

∞∑

m=0

(−ik)mm!

1

c

∫

Vd3x′J(x′)(n · x′)m

= −ikpeikr

r+ · · · ,

(3.49)

where |x − x′|−1 ≈ r−1 if only the leading term in kr is kept. It can now be seen

that the vector potential is an outgoing spherical wave with mth-angular coefficient∫

Vd3x′J(x′)(−ikn · x′)m/cm!. From (3.45), the corresponding fields are:

Bfar = k2 eikr

r(n × p)

[

1 − 1

ikr

]

+ · · · (3.50a)

Efar = k2 eikr

r(n × p) × n +

[ 1

r3− ik

r2

]

eikr[3n(n · p) − p] + · · · . (3.50b)

45

The magnetic field is transverse to the radius vector x = rn while the electric

field has components longitudinal and transverse to x. Both fields fall off like r−1

at leading order. The r−1-fields are the true radiation fields which carry energy,

momentum, and angular momentum to infinity. This can be seen from the time-

averaged differential power radiated per unit solid angle

dP

dΩ=

1

2Re

[

r2n · c

4πE × B∗

]

=ck4

8π|n × p|2

=ck4

8π|p|2 sin2 θ

(3.51)

which in this case is a measure of the energy radiated per unit time per unit solid

angle by an oscillating electric dipole p. Integrating this expression over Ω = (θ, φ)

gives the total power radiated, i.e.,

P =∫dΩ

dP

dΩ=ck4

3|p|2. (3.52)

The corresponding radiation pattern is shown in Figure 3–3. In general, the power

radiated by an l-pole goes like k2(l+1). Notice that it is the r−1-fields whose power

makes it to infinity in three dimensions. This is because E×B ∼ r−2 which exactly

cancels the r2 in the measure factor d3x = r2drdΩ. In two dimensions, it is the

r−1/2-fields whose power makes it to infinity since d2x = rdrdθ. As before, a more

accurate description of the radiation field is obtained by including higher order terms

in the sum (3.49). The lowest order (nonvanishing) multipole contributes the most

to the field.

In the intermediate zone, neither of the previous approximations are valid. In

fact all terms in the previous series expansions would have to be kept. The under-

standing of the behavior of the fields in this zone requires the more sophisticated

machinery of vector multipole fields. The interested reader is referred to [9] for a

detailed discussion of multipole fields of arbitrary order (l, m).

46

θ

p

x

Figure 3–3: The differential power dP/dΩ or radiation pattern corresponding toan oscillating electric dipole verifies that no radiation is emitted in the directionof the dipole moment. Rather the dipole radiation is a maximum in the directiontransverse to p.

Outside of the physics literature there is also a large amount of engineering

literature in the field of computational electrodynamics. In this area, Maxwell’s

field equations are often solved numerically by finite element methods (see Jiao and

Jin [81] and references therein). Many applications of this work lie in electromagnetic

scattering, waveguiding, and antenna design. The inverse source problem [82] is also

another area of interest in engineering. Here, the goal is to determine the sources

of charge and current with only the knowledge of the electromagnetic fields outside

of the source’s region of support. This problem has benefited from the work of

Goedecke [83], Devaney and Wolf [84], Marengo and Ziolkowski [85], and Hoenders

and Ferwerda [86], who have demonstrated the decomposition of the electromagnetic

field into nonradiating and purely radiating components.

47

3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket

Consider the gaussian wavepacket with initial position r moving with constant

velocity v

Ψ(x, t) =[2`2

π

]3/4

e−`2[x−(r+vt)]2eimv·x, (3.53)

where b = 1/√

2` is the wavepacket width. The corresponding probability current

is given by

J(x, t) = vρ(x, t) =q

2m[Ψ∗(−i∇Ψ) + Ψ(i∇Ψ∗)]. (3.54)

In Fourier space this current becomes J(k, t) = qv exp(−ik · [r + vt] − k2/8`2) and

the vector potential is obtained by integrating against the Green’s function D(+)k for

the wave operator (see Appendix B). The vector potential becomes

A(x, t) =∫

V

d3k

(2π)3eik·x

∫∞

−∞dt′D

(+)k (t, t′)4πcJ(k, t′)

= 4πcqv∫

V

d3k

(2π)3eik·(x−r)−k2/8`2

∫∞

−∞dt′

Θ(t− t′) sin ck(t− t′)

cke−ik·vt

′

= 4πcqv∫

V

d3k

(2π)3

eik·[x−(r+vt)]−k2/8`2

c2k2 − (k · v)2

(3.55)

which is difficult to perform analytically due to the complicated angular dependence

of the integrand. For nonrelativistic velocities, A can be approximated by

A(x, t) = 4πcqv∫

V

d3k

(2π)3

eik·[x−(r+vt)]−k2/8`2

c2k2[1 − (v/c)2 cos2 θ]

≈ qv

2π2c

∫

V

d3k

k2eik·[x−(r+vt)]−k2/8`2

=qv

c

erf(√

2`|x − (r + vt)|)

|x − (r + vt)| ,

(3.56)

where Gradshteyn and Ryzhik [87] was used. The norms of this vector potential and

its associated current density J are plotted along the x-axis in Figure 3–4 for two

different velocities. The charge q is taken to be negative. Notice that A follows the

charge distribution and that A will generate an electromagnetic field. For v/c 1

this result is equivalent to a Galilei boost of the fields from the rest frame of the

48

x20151050-5

1

0.5

0

Figure 3–4: The norms of J and A are plotted with different velocities along thex-axis.

source. Only the electrostatic field remains by going to the rest frame. And so,

there is little difference between uniform motion and no motion.

As stated previously, the more interesting field dynamics is created whenever

the source is accelerated. To this end, consider the vector potential arising from a

moving charge whose current has the simple time dependence

J(k, t) = qv(t)e−ik·(r+vt)−k2/8`2

= qve−ik·(r+vt)−k2/8`2Θ(t− t0)Θ(t1 − t),

(3.57)

where v is constant. This time dependence corresponds to a situation in which a

source is suddenly accelerated from a standstill to a uniform movement with velocity

v and is then instantaneously decelerated again to a standstill (see Figure 3–5). In

each of the three temporal regions of the current, the vector potential has a different

49

PSfrag replacements

r(t)

ct

x

past

elsewhere

future

t1

t2

Figure 3–5: The trajectory or world line r(t) of the charge is plotted. Electromag-netic radiation is produced at t1 and t2 and moves out on the light cone.

behavior. Obviously for t < t0, A(x, t) = 0. For t0 ≤ t < t1,

A(x, t) ≈ qv

c

erf(√

2`|x − (r + vt)|)

|x− (r + vt)|

+1

2

erf(√

2`[c(t− t0) − |x − (r + vt0)|])

|x − (r + vt0)|

− 1

2

erf(√

2`[c(t− t0) + |x − (r + vt0)|])

|x − (r + vt0)|

(3.58)

and for t ≥ t1,

A(x, t) ≈ qv

2c

erf(√

2`[c(t− t0) − |x − (r + vt0)|])

|x − (r + vt0)|

− erf(√

2`[c(t− t0) + |x − (r + vt0)|])

|x − (r + vt0)|

+erf

(√2`[c(t− t1) + |x − (r + vt1)|]

)

|x − (r + vt1)|

− erf(√

2`[c(t− t1) − |x − (r + vt1)|])

|x − (r + vt1)|

.

(3.59)

Again nonrelativistic velocities are assumed. A space-time plot of the norm of this

piecewise vector potential is shown in Figure 3–6. Note that the charge was at rest

until the time t0, where it was instantaneously accelerated to a velocity of magnitude

50

PSfrag

replacem

ents

10

10

-10

0

0

0

5

5

-5

2

-2

t

x

Figure 3–6: The bremsstrahlung radiation from a charged gaussian wavepacketmoves out on the smeared light cone with maximum at x = ct.

v. Then the charge moved uniformly with v until the time t1, when it was instan-

taneously decelerated to rest again. Since electromagnetic radiation is produced

whenever the velocity changes in time, electromagnetic ripples are produced at t0

and t1. The ripples move out as radiation at the velocity c of light. Figure 3–6 shows

the light cone, which is smeared out due to the nonpointlike structure of the charge.

The vector potential presented so far has both longitudinal and transverse com-

ponents. For the time being, the tranversality of the A is not important. It turns

out that the only fields which contribute to the Poynting vector or to the power are

the transverse fields. And so it does no harm to keep the full vector potential. For

the interested reader, the transverse vector potential AT associated with (3.58) and

(3.59) is calculated in Appendix C by analogy to the quadrupole moment tensor.

51

The electric and magnetic fields corresponding to (3.59) are

E ≈ −A

c

= −qv`c

√

2

π

[e−2`2[c(t−t0)−R(t0)]2 − e−2`2[c(t−t0)+R(t0)]2

R(t0)

+e−2`2[c(t−t1)+R(t1)]2 − e−2`2[c(t−t1)−R(t1)]2

R(t1)

]

= −qv`c

√

2

π

[g−0 (t) + g−1 (t)

]

︸ ︷︷ ︸

h−(t)

(3.60)

neglecting the purely longitudinal −∇Φ, and

B = ∇× A

= −v × q`

c

√

2

π

[

u(t1)e−2`2[c(t−t1)+R(t1)]2 + e−2`2[c(t−t1)−R(t1)]2

R(t1)

− u(t0)e−2`2[c(t−t0)−R(t0)]2 + e−2`2[c(t−t0)+R(t0)]2

R(t0)

]

= −q`c

√

2

πv ×

[

u(t1)g+1 (t) − u(t0)g

+0 (t)

]

,

(3.61)

where R(t) ≡ |R(t)| = |x− (r+vt)| and where the unit vectors u(t0) = R(t0)/R(t0)

and u(t1) = R(t1)/R(t1). With (3.60) and (3.61), the Poynting vector is

S =c

4πE × B

=q2`2

2π2ch−(t) v ×

v ×[

u(t1)g+1 (t) − u(t0)g

+0 (t)

]

.

(3.62)

The differential power radiated into the solid angle dΩ at time t becomes

dP (x, t)

dΩ= R(t)2 n · S(t)

=q2`2

2π2ch−(t)R(t)2

[(v · u(t1)

)g+1 (t) −

(v · u(t0)

)g+0 (t)

]

n · v

− v2[(

n · u(t1))g+1 (t) −

(n · u(t0)

)g+0 (t)

]

=q2`2

2π2ch−(t)R(t)2

g+0 (t)

[(v × n

)·(v × u(t0)

)]

− g+1 (t)

[(v × n

)·(v × u(t1)

)]

(3.63)

52

where the unit vector n = R(t)/R(t) is normal to the surface of the ball that

emanates from the radiation source. The vectors v, n, u(t0), and u(t1) are all

constant in time. By choosing the z-axis along the velocity v, the angles between

v and the unit vectors u(t0) and u(t1) are δ0 and δ1 respectively. With a little

geometry, it can be verified that

sin δ0 ≈sin θ

√

1 + (v/ct)(t1 − t0) cos θand sin δ1 ≈

sin θ√

1 − (v/ct)(t1 − t0) cos θ(3.64)

by suppressing terms of quadratic order and higher in c−1, where θ is the angle

between v and n. In terms of the angles δ0, δ1, and θ, the differential power becomes

dP

dΩ=q2`2v2

2π2ch−(t)R(t)2

[

g+0 (t) sin δ0 sin θ − g+

1 (t) sin δ1 sin θ]

(3.65)

which is independent of the polar angle φ. The corresponding radiation pattern

is shown in Figure 3–7 and shows that power is radiated in all directions except

along the direction of motion. Notice that the “dipole-like” pattern is modified by

contributions arising from the expansion of the square roots in the angles δ0 and δ1.

That is

sin2 θ√

1 ± (v/ct)(t1 − t0) cos θ= sin2 θ

[

1∓ v

2ct(t1− t0) cos θ+O

((v/c)2 cos2 θ

)]

. (3.66)

These contributions are more significant at higher velocities. The quadrupole pat-

tern in Figure 3–7 is obviously overemphasized. By integration over the unit sphere,

the total power is found to be

P (x, t) =q2`2v2

πch−(t)R(t)2

∫ π

0dθ

g+0 (t) sin δ0 sin2 θ − g+

1 (t) sin δ1 sin2 θ

(3.67)

which is equivalent to dE/dt where E is the total field energy. Both of the integrals

in (3.67) can be done analytically. Since both h− and g+ are proportional to 1/R,

the power does not decay with the radius x.

53

PSfragreplacements

x

v

Figure 3–7: The radiation pattern given by (3.63) shows the characteristic dipolepattern at lowest order. Keeping O(v/c) terms reveals the quadrupole pattern.Higher order multipole patterns are generated by O(v2/c2) and higher terms.

For an electron whose charge distribution has a width corresponding to the

Bohr radius a0 and has a velocity of ve = 1 a.u. between the times t0 = 0 a.u. and

t1 = 1 a.u., the instantaneous power is P ≈ 2 × 10−3 a.u. ≈ 3 × 10−4 J/s at the

maximum of the peak from t0. The power from the t1 peak is the same.

In order to put the previous results into perspective it is useful to make a

comparison with the Larmor result. The Larmor power

P (t) =2q2

3c3v(t)2 (3.68)

is the instantaneous power radiated by an accelerated point charge that is observed

in a reference frame where the velocity of the charge is significantly less than that

of light. The angular behavior of the emitted radiation may be determined by

examining the differential power

dP (t)

dΩ=

q2

4πc3v(t)2 sin2 θ (3.69)

which is the dipole radiation pattern. If the result of (3.65) is correct, then it should

reduce to the Larmor formula in the limit of the wavepacket width b going to zero

54

(point charge). Making use of the identity

δ(x) = lim`→+∞

`√πe−`

2x2

, (3.70)

where ` = 1/√

2b, the differential power in (3.65) becomes

dP

dΩ=

q2

4πc3sin2 θ

[

vδ(t− t0) − δ(t− t1)]2

︸ ︷︷ ︸

a2

. (3.71)

Again v/c 1 was assumed. The term in square brackets has the dimensions of

acceleration. And so, (3.71) reduces to the Larmor result (3.69) for the stepwise

velocity v(t) = vΘ(t− t0)Θ(t1 − t). These results are presented in [88].

CHAPTER 4CANONICAL STRUCTURE

The governing equation of quantum mechanics is the Schrodinger equation [7, 8].

In the minimal coupling prescription it is

iΨ =[−i∇− qA/c]2Ψ

2m+ VΨ + qΦΨ. (1.1)

The dynamics of the scalar potential Φ and vector potential A are not described

by this linear equation. Specification of these potentials as well as the initial values

for the wavefunction Ψ casts the Schrodinger equation into a well-defined boundary

value problem that is also a well-defined initial value problem.

The governing equations of electrodynamics are Maxwell’s equations [9]:

∇·E = 4πρ ∇×B =4π

cJ+

E

c∇·B = 0 ∇×E+

B

c= 0. (1.2)

The dynamics of the charge density ρ and current density J are not described by

these linear equations. Specification of the external sources as well as the initial

values for the electric and magnetic fields E and B satisfying ∇ · E = 4πρ and

∇ · B = 0 casts the Maxwell equations into a well-defined boundary value problem

that is also a well-defined initial value problem.

Each of these theories are significant in and of themselves. Given a particular

arrangement of sources throughout space-time and the initial values for E and B,

the Maxwell equations govern the dynamics of the resulting electromagnetic field.

Likewise, given a particular external field throughout space-time and the initial value

for Ψ, the Schrodinger equation governs the dynamics of the sources. However, notice

that the Maxwell equations do not say anything about the dynamics of the sources

and the Schrodinger equation does not say anything about the electrodynamics.

55

56

It is possible to couple the linear Maxwell and Schrodinger equations. The

resulting nonlinear Maxwell-Schrodinger theory accounts for the dynamics of the

charges and the electromagnetic field as well as their mutual interaction. For exam-

ple, given an initial source and its corresponding Coulomb field, a wavefunction and

electromagnetic field are generated. The electromagnetic field has its own dynamics

and acts back upon the wavefunction. This in turn causes different fields to be gen-

erated. It will be demonstrated that these coupled nonlinear Maxwell-Schrodinger

equations can be cast into a well-defined initial value problem and solved in an

efficient numerical manner.

4.1 Lagrangian Electrodynamics

Consider the Maxwell Lagrangian density

LMax =[−A/c−∇Φ]2 − [∇× A]2

8π− ρΦ +

J · Ac

(4.1)

with external sources ρ and J. Variation of this Lagrangian leads to the governing

equations of electrodynamics, i.e.,

∇2A− A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ

∇2Φ +∇ · Ac

= −4πρ

(1.4)

These Maxwell equations (in terms of the potentials) do not form a well-defined

initial value problem. But, by choosing a particular gauge they can be turned into

one. In other words, these equations are ill-posed as they stand. However, they do

enjoy both Lorentz and gauge invariance as does the Lagrangian (4.1).

4.1.1 Choosing a Gauge

Working in a particular gauge can be organized into the following hierarchy:

1. At the solution level, a gauge generator F can be chosen so that a gauge trans-formation of the solutions, i.e., Φ → Φ′ = Φ− F /c and A → A′ = A+∇F, mapsthem to new solutions that satisfy the gauge condition.

57

2. At the equation level, the set consisting of (1.4) together with a gauge constrainthas only solutions that satisfy the gauge condition.

3. At the Lagrangian level, a gauge fixing term can be added to (4.1) so that theresulting Euler-Lagrange equations automatically include the gauge constraint.

4.1.2 The Lorenz and Coulomb Gauges

The first two tiers can be elaborated on as follows. With a gauge function F

satisfying ∇2F − F /c2 = −[Φ/c + ∇ · A] a solution Aα = (Φ,A) of the potential

equations (1.4) can be mapped to the Lorenz gauge solution AαLorenz according to

the gauge transformation:

Φ → ΦLorenz = Φ − F /c A → ALorenz = A + ∇F. (4.2)

Alternatively, adding the gauge constraint Φ/c + ∇ · A = 0 to (1.4) leads to the

Lorenz gauge equations of motion:

∇2A − A

c2= −4π

cJ ∇2Φ − Φ

c2= −4πρ. (4.3)

With ρ and J specified throughout space-time, the Lorenz gauge equations of mo-

tion are well-defined once the initial values for A, A, Φ, and Φ are known. There

is some symmetry left in the solutions to these equations. Namely, the residual

gauge freedom left in the homogeneous equation ∇2F − F /c2 = 0 allows for gauge

transformations on the solutions such that the new solutions do not leave the Lorenz

gauge. However, these gauge transformed solutions do correspond to different initial

conditions. Note that the Lorenz gauge enjoys relativistic or Lorentz invariance. It

will be shown, that the Lorenz gauge is the most appropriate gauge for dynamics.

With another gauge function G satisfying ∇2G = −∇·A a solution Aα = (Φ,A)

of the potential equations (1.4) can be mapped to the Coulomb gauge solution

AαCoulomb according to:

Φ → ΦCoulomb = Φ − G/c A → ACoulomb = A + ∇G. (4.4)

58

Alternatively, adding the gauge constraint ∇ ·A = 0 to (1.4) leads to the Coulomb

gauge equations of motion:

∇2A − A

c2= −4π

cJ +

∇Φ

c∇2Φ = −4πρ. (4.5)

Again with ρ and J specified throughout space-time, the Coulomb gauge equations

of motion are well-defined once the initial values for A, A, Φ, and Φ are known. As

before, there remains a symmetry or residual gauge freedom from the homogeneous

equation ∇2G = 0. Note that in the Coulomb gauge Gauss’s law reduces to ∇2Φ =

−4πρ. Inverting this equation specifies Φ in terms of ρ. That is Φ = (1/∇2)[−4πρ].

The scalar potential can now be totally removed from the theory by substitution

of this Green’s function integral. This may be done at the expense of Lorentz

invariance. In practice, where the equations are to be expanded in a basis of s-

gaussians, either transverse basis functions would have to be used or the transverse

fields would have to be generated from a standard basis. The former case would

require a major revision of most existing integral codes, which are in direct space,

while the latter would require the instantaneous transverse projection P abT = δab −

∂a∂b/∇2 (see Appendix B) This operation, which is over all space, is difficult to

describe in terms of a local set of basis functions.

Lastly, for the third tier, consider the Lagrangian density (4.1) together with a

gauge fixing term for the Lorenz gauge, i.e.,

LLMax = LMax −[Φ/c+ ∇ · A]2

8π

=[−A/c−∇Φ]2 − [∇× A]2

8π− ρΦ +

J · Ac

− [Φ/c+ ∇ · A]2

8π.

(1.18)

The resulting Euler-Lagrange equations obtained from LLMax are identical to the

Lorenz gauge wave equations in (4.3) which are equivalent to the general potential

equations (1.4) together with the constraint Φ/c+ ∇ · A = 0.

59

PSfrag

replacem

ents

LMax LLMaxLCMax

gauge invariant

ddt∂LMax

∂ξ− ∂LMax

∂ξ= 0d

dt∂LMax

∂ξ− ∂LMax

∂ξ= 0 d

dt∂LMax

∂ξ− ∂LMax

∂ξ= 0

Maxwell’s equations

well-posed IVP

well-posed IVP

well-posed IVP

uniqueuniquesolutionsolution solutions

many

∇ · A = 0 ∇ ·A + Φ/c = 0ill-posed IVP addadd

gaugegaugetransformation transformation

constraintconstraint

Figure 4–1: A limited but relevant portion of the gauge story in the Lagrangianformalism is organized in this picture. The middle column (i.e., the column belowLMax) enjoys full gauge freedom. The far left (Coulomb gauge) and far right (Lorenzgauge) columns have limited gauge freedom. That is, there are a limited class ofgauge transformations that can be made on the solutions such that they remain inthe same gauge. This symmetry is due to the residual gauge freedom. Note thatthese solutions correspond to different initial conditions within the gauge. Also notethat the Euler-Lagrange equations together with a particular gauge constraint areequivalent to the Euler-Lagrange equations derived from that particular gauge fixedLagrangian.

There are many other known gauges, the choice of which is arbitrary. All

choices of gauge lead to the same physically observable electromagnetic fields E and

B. Together with the definitions E = −A/c−∇Φ and B = ∇×A, the Lorenz and

Coulomb gauge equations of motion as well as the general potential equations (1.4)

imply Maxwell’s equations (1.2). A diagram of this gauge story in the Lagrange

formulation is presented in Figure 4–1.

4.2 Hamiltonian Electrodynamics

In the Hamiltonian prescription, the momentum conjugate to A with respect

to the Maxwell Lagrangian (4.1) is

Π ≡ ∂LMax

∂A=

1

4πc[A/c+ ∇Φ]. (4.6)

60

The momentum conjugate to Φ is identically zero, i.e.,

Θ ≡ ∂LMax

∂Φ= 0. (4.7)

A Hamiltonian density can still be defined as the time-time component of the

Maxwell stress-energy tensor T αβMax = ∂LMax/∂(∂αξ)∂βξ − gαβLMax. It is

HMax ≡ T 00Max = Π · A+ΘΦ−LMax =

[−4πcΠ]2 + [∇× A]2

8π− c∇Φ ·Π+ρΦ− J ·A

c

(4.8)

and the resulting equations of motion are:

A ≡ ∂HMax

∂Π= 4πc2Π − c∇Φ −Π ≡ ∂HMax

∂A=

∇[∇ · A] −∇2A

4π− J

c+ c∇Θ

(4.9)

Φ ≡ ∂HMax

∂Θ= 0 −Θ ≡ ∂HMax

∂Φ= ρ+ c∇ · Π.

Since the momentum Θ defined in (4.7) is identically zero, so is its time derivative

Θ and gradient ∇Θ. Notice that these Hamilton equations form a well-posed initial

value problem. The machinery inherent in the Hamiltonian formalism automatically

adds a momentum and automatically adds the additional equation of constraint

Φ = 0. It turns out that this extra equation fixes a particular gauge where Φ = 0.

This gauge can always be fixed by a gauge transformation whose generator satisfies

F /c = Φ. The residual gauge freedom left in the homogeneous equation F = 0

does allow for a gauge transformation on the solutions to (4.9). These new gauge

transformed solutions do not leave the Φ = 0 gauge, but do correspond to a different

initial value problem within this gauge. In other words, they are solutions to (4.9)

with different initial values. Pay careful attention to the fact that these Hamilton

equations of motion form a well-posed initial value problem even though a gauge

fixed Lagrangian was not knowingly used. The Hamiltonian formalism automatically

added the extra equation Φ = 0.

61

4.2.1 Hamiltonian Formulation of the Lorenz Gauge

Rather than fixing the Coulomb gauge at the equation level it may be bene-

ficial to work in a more general theory where a gauge is chosen at the Lagrangian

level and retains all of the 4-potential, is Lorentz invariant, and does not require

any instantaneous or nonlocal operations. To this end, consider the Lorenz gauge

Lagrangian density from (1.18), i.e.,

LLMax =[−A/c−∇Φ]2 − [∇× A]2

8π− ρΦ +

J · Ac

− [Φ/c+ ∇ · A]2

8π. (1.18)

It will be shown that the equations of motion derived from LLMax are well-defined

because of the addition of the last term in this expression. It turns out that this

term is known in the literature [24, 68] and is a gauge fixing term for the Lorenz

gauge. From (1.18), the momentum conjugate to A is

Π ≡ ∂LLMax

∂A=

1

4πc[A/c+ ∇Φ] (4.10)

and the momentum conjugate to Φ is

Θ ≡ ∂LLMax

∂Φ= − 1

4πc[Φ/c+ ∇ · A]. (4.11)

With these momenta and coordinates, electrodynamics is given a symplectic struc-

ture. The Hamiltonian density is

HLMax =

[−4πcΠ]2 + [∇× A]2 − [4πcΘ]2

8π− c∇Φ ·Π− cΘ∇·A+ ρΦ− J · A

c(4.12)

and the resulting equations of motion are:

A = 4πc2Π − c∇Φ −Π =∇[∇ · A] −∇2A

4π− J

c+ c∇Θ

(4.13)

Φ = −4πc2Θ − c∇ · A −Θ = ρ+ c∇ ·Π.

62

These equations, which are a generalization of (4.3), together with the initial values

for A, Π, Φ, and Θ form a well-posed initial value problem. The residual gauge

freedom resulting from the homogeneous equation F = 0 does allow for a gauge

transformation on the solutions to (4.13). These new gauge transformed solutions

do not leave the Lorenz gauge, but do correspond to a different initial value problem

within the Lorenz gauge. In other words, they are solutions to (4.13) with different

initial values.

Notice that a relationship exists between the momentum Θ and the gauge func-

tion F leading to the Lorenz gauge. That is, from Θ = −[Φ/c + ∇ · A]/4πc and

F /c2 − ∇2F = Φ/c + ∇ · A notice that Θ ≡ F/4πc. So the D’Alembertian of

the gauge function F acts a generalized coordinate in this phase space. It is the

momentum conjugate to the scalar potential Φ.

In matrix form, the dynamical equations in (4.13) are

0 0 −1 0

0 0 0 −1

1 0 0 0

0 1 0 0

A

Φ

Π

Θ

=

∇× [∇× A]/4π − J/c+ c∇Θ

ρ + c∇ · Π

4πc2Π − c∇Φ

−4πc2Θ − c∇ · A

, (4.14)

where 1 is the 3× 3 identity matrix. Notice that (4.14) is of the Hamiltonian form

ωη = ∂H/∂η. (4.15)

More specifically ωabηb = ∂H/∂ηa, where ηb is a column matrix of the generalized

positions and momenta, i.e.,

ηb(x, t) =

Ak

Φ

Πk

Θ

, (4.16)

63

where k = 1, 2, 3. The antisymmetric matrix ωab is the (canonical) symplectic form

associated with the phase space of electrodynamics in the Lorenz gauge. By substi-

tution, these first order Hamiltonian equations of motion can be shown to be equiv-

alent to the second order Lorenz gauge equations Φ = −4πρ and A = −4πJ/c.

Together with the definition of the electric and magnetic fields, (4.13) imply

∇ ·E = 4πρ+ 4πΘ ∇× B =4π

cJ +

E

c− 4πc∇Θ

(4.17)

∇ ·B = 0 ∇× E +B

c= 0.

These equations are not equivalent to Maxwell’s equations unless Θ(x, t) remains

constant in space-time throughout the dynamics. In order to analyze this question,

the dynamics of the sources must be considered. It should be noticed that the

inhomogeneous equations in (4.17) imply

Θ ≡ ∇2Θ − Θ

c2=

1

c2[ρ + ∇ · J]. (4.18)

If the matter theory is such that the equation of continuity ρ = −∇ · J is satisfied,

then Θ = 0. So if Θ(t = 0) = Θ(t = 0) = 0, then Θ(t) = 0 at all times t. In other

words, if the sources of charge and current satisfy the equation of continuity, then

the dynamical theory arising from the Lagrangian (1.18) is the Maxwell theory of

electrodynamics.

Note that while (4.9) and (4.13) do not enjoy the full gauge symmetry as do the

general potential equations (1.4), this does not mean that the observables resulting

from (4.9) or (4.13) are not gauge invariant. Any observable that is calculated will

be invariant to the choice of gauge generator. Moreover, once the solutions to these

well-defined equations are constructed, these solutions belong to the many solutions

of (1.4). This family of solutions is the most general solutions of the potential form

of Maxwell’s equations. In fact, gauge transformations can even be made from one

64

particular gauge to another [89]. A diagram depicting the relevant gauge story in

the Hamiltonian formulation is presented in Figure 4–2. Notice that there is no

Hamiltonian theory that enjoys the full gauge symmetry of (1.4). The Hamiltonian

HMax in the far right column is obtained by a Legendre transformation of the gauge

invariant Lagrangian LMax in (4.1). However, the Hamiltonian dynamics stemming

from the gauge invariant LMax is not gauge invariant, but rather occurs in the gauge

where Φ = 0.

65

PSfrag

replacem

ents

LMax LLMaxLCMax

gauge invariant

ddt∂LMax

∂ξ− ∂LMax

∂ξ= 0 d

dt∂LMax

∂ξ− ∂LMax

∂ξ= 0d

dt∂LMax

∂ξ− ∂LMax

∂ξ= 0

Maxwell’s equations

well-posed IVP

well-posed IVPwell-posed IVPwell-posed IVP

well-posed IVP

well-posed IVP

uniqueunique uniquesolutionsolution solutionsolutions

many

∇ · A = 0 ∇ · A + Φ/c = 0ill-posed IVPadd add

constraintconstraint

gaugegauge gaugetransformationtransformationtransformation

ωη = ∂HMax

∂ηωη =∂HL

Max

∂ηωη =

∂HCMax

∂η

HMax = ∂LMax

∂ξξ − LMaxHL

Max =∂LL

Max

∂ξξ − LLMaxHC

Max =∂LC

Max

∂ξξ − LCMax

Figure 4–2: The Hamiltonian formulation of the gauge story is organized in this picture with respect to the previous Lagrangianformulation. Figure 4–1 is depicted in the box with dotted borders. It can now be seen how the Coulomb and Lorenz gaugesconnect in both formalisms.

66

4.2.2 Poisson Bracket for Electrodynamics

The phase space that carries the associated dynamics is naturally endowed

with a Poisson bracket ·, · (recall Chapter 2). This may be seen by considering

the variation of ξ along the dynamics ∆H ≡ (∂/∂η)η. That is

∆H(ξ) ≡ (d/dt)ξ = (∂ξ/∂ηb)ηb = (∂ξ/∂ηb)ω−1ab (∂H/∂ηa) ≡ ξ,H, (4.19)

where η are the generalized coordinates. In general, the Poisson bracket of the

dynamical variable F with the dynamical variable G is

F,G =

∂F/∂A

∂F/∂Φ

∂F/∂Π

∂F/∂Θ

T

0 0 −1 0

0 0 0 −1

1 0 0 0

0 1 0 0

−1

∂G/∂A

∂G/∂Φ

∂G/∂Π

∂G/∂Θ

. (4.20)

Since the symplectic form ω is canonical its inverse is trivial, i.e., ω−1 = ωT = −ω.

Also notice that ω2 = −1, ωTω = 1, and det ω = 1.

4.3 Hamiltonian Electrodynamics and Wave Mechanics in Complex

Phase Space

Consider the matter theory associated with the Schrodinger Lagrangian (~ = 1)

LSch = iΨ∗Ψ − [i∇− qA/c]Ψ∗ · [−i∇− qA/c]Ψ

2m− VΨ∗Ψ − qΦΨ∗Ψ (1.17)

where Ψ is the wavefunction for a single electron, V = qq/|x| is the static Coulomb

potential energy of a proton, and (Φ,A) are the electron’s scalar and vector poten-

tials. Notice that this Lagrangian is already written in phase space. The momentum

conjugate to the wavefunction Ψ is iΨ∗. Together with the previous Maxwell La-

grangian, the coupled nonlinear dynmical theory arising from the Lagrangians

LMax =1

2[Π · A− Π · A] −

[−4πcΠ]2 + [∇× A]2

8π− c∇Φ ·Π

(4.21)

67

LSch =i

2[Ψ∗Ψ − Ψ∗Ψ] −

[i∇− qA/c]Ψ∗ · [−i∇− qA/c]Ψ

2m+ VΨ∗Ψ + qΦΨ∗Ψ

(4.22)

Lgauge =1

2[ΘΦ − ΘΦ] − −2πc2Θ2 − cΘ∇ · A (4.23)

yields the following equations of motion:

A = 4πc2Π− c∇Φ −Π =∇[∇ · A] −∇2A

4π− J

c+ c∇Θ

Φ = −4πc2Θ − c∇ · A −Θ = ρ + c∇ · Π (4.24)

iΨ =[−i∇− qA/c]2Ψ

2m+ VΨ + qΦΨ −iΨ∗ =

[i∇− qA/c]2Ψ∗

2m+ VΨ∗ + qΦΨ∗.

Surface terms of the form (d/dt)pq/2 have been added in the above Lagrangians

in order to symmetrize them, i.e., L = pq − H − (d/dt)pq/2 becomes L = [pq −

pq]/2 −H. This can always be done since the action I =∫Ldt =

∫[L + (d/dt)g]dt

is invariant to the addition of a pure surface term to the Lagrangian. Note that

the Schrodinger wavefunctions Ψ and Ψ∗ are complex-valued while the remaining

electromagnetic variables are all real-valued. These dynamical equations may be

put into matrix form as

i 0 0 0 0 0

0 −i 0 0 0 0

0 0 0 0 −1 0

0 0 0 0 0 −1

0 0 1 0 0 0

0 0 0 1 0 0

Ψ

Ψ∗

A

Φ

Π

Θ

=

[−i∇− qA/c]2Ψ/2m+ VΨ + qΦΨ

[i∇− qA/c]2Ψ∗/2m+ VΨ∗ + qΦΨ∗

∇× [∇× A]/4π − J/c+ c∇Θ

ρ+ c∇ · Π

4πc2Π − c∇Φ

−4πc2Θ − c∇ ·A

,

(4.25)

where the symplectic form is canonical. The electromagnetic sector of it is identi-

cal to (4.14). These dynamical equations define the coupled Maxwell-Schrodinger

68

theory. This theory is well-defined and closed. In other words, the dynamics of

the charges, currents, and fields are all specified as well as their mutual interaction.

Given initial values for Ψ, Ψ∗, A, Π, Φ, and Θ determines their coupled dynamics

throughout space-time.

With the dynamics of the charges defined, the problem in (4.17) can now be

addressed. The Schrodinger equation in (4.24) implies the continuity equation

(d/dt)qΨ∗Ψ = −∇ · qΨ∗[−i∇− qA/c]Ψ + Ψ[i∇− qA/c]Ψ∗

/2m (4.26)

which may be written more compactly as ρ = −∇ · J. From the definition of the

momentum Θ in (4.11) and the wave equations Φ = −4πρ and A = −4πJ/c,

notice that

Θ =−1

4πc[(d/dt)Φ/c+∇·A] =

1

4πc[(d/dt)4πρ/c+∇·4πJ/c] =

1

c2[ρ+∇·J] = 0

(4.27)

by appealing to (4.26). So if Θ(t = 0) = Θ(t = 0) = 0, then the electrodynamics

stays in the Lorenz gauge for all time since the only solution of Θ = 0 with

Θ(t = 0) = Θ(t = 0) = 0 is Θ(t) = 0.

It is worth mentioning that if Θ(t = 0) = 0 for all time, then the electron-

electron self interaction makes no contribution to the Schrodinger energy. This

is true since the self interaction term qΦΨ∗Ψ in the above Schrodinger Lagrangian

cancels exactly with −c∇Φ·Π in the Maxwell Lagrangian. The cancellation requires

a partial integration of −c∇Φ · Π to cΦ∇ · Π followed by a substitution of 0 =

ρ + c∇ · Π from Θ(t = 0) = 0 in (4.24). However, there is still a contribution from

the self-energy arising in the Maxwell energy of the Coulombic field.

69

4.4 Hamiltonian Electrodynamics and Wave Mechanics in Real Phase

Space

The dynamical equations (5.16) are mixed, real and complex. For consistency

these equations are put into real form with the Lagrangian densities:

LMax =1

2[Π · A− Π · A] −

[−4πcΠ]2 + [∇× A]2

8π− c∇Φ ·Π

(4.28)

LSch =1

2[PQ− PQ] −

[∇Q+ qAP/c]2 + [−∇P + qAQ/c]2/4m

+V [Q2 + P 2]/2 + qΦ[Q2 + P 2]/2

(4.29)

Lgauge =1

2[ΘΦ − ΘΦ] − −2πc2Θ2 − cΘ∇ · A (4.30)

The functions P and Q are related to the real and imaginary parts of Ψ and Ψ∗

according to Ψ = [Q + iP ]/√

2 and Ψ∗ = [Q − iP ]/√

2. The equations of motion

that are associated with these Lagrangians are:

A = 4πc2Π − c∇Φ −Π =∇[∇ · A] −∇2A

4π− J

c+ c∇Θ

(4.31a)

Φ = −4πc2Θ − c∇ · A −Θ = ρ+ c∇ ·Π

Q =−∇2P + q∇ · (AQ)/c+ qA · ∇Q/c+ q2A2P/c2

2m+ V P + qΦP

(4.31b)

−P = −∇2Q + q∇ · (AP )/c+ q∇P · A/c− q2A2Q/c2

2m+ V Q + qΦQ.

70

These dynamical equations may be put into matrix form as

0 0 0 −1 0 0

0 0 0 0 −1 0

0 0 0 0 0 −1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

A

Φ

Q

Π

Θ

P

=

∇× [∇× A]/4π − J/c+ c∇Θ

ρ+ c∇ · Π

−[∇2Q + q∇ · (AP )/c+ q∇P · A/c− q2A2Q/c2]/2m+ V Q + qΦQ

4πc2Π − c∇Φ

−4πc2Θ − c∇ · A

[−∇2P + q∇ · (AQ)/c+ qA · ∇Q/c+ q2A2P/c2]/2m+ V P + qΦP

,

(4.32)

where the symplectic form is again canonical. Note that the equation of continuity

ρ = −∇ · J still holds with the real charge and current densities

ρ = q[Q2 + P 2]/2 J =q

2mQ∇P − P∇Q− qQAQ/c− qPAP/c. (4.33)

4.5 The Coulomb Reference by Canonical Transformation

As was mentioned previously the numerical implementation of the theory can

be made to converge more quickly if the basis is chosen judiciously. Recall that

the electromagnetic field generated by any charge contains a Coulombic contribu-

tion. This monopole term accounts for a large portion of the local electromagnetic

field surrounding the charge. It would be advantageous to not describe this large

71

contribution in terms of the basis but rather to calculate it analytically. The re-

maining smaller portion of the radiative or dynamical electromagnetic field can then

be described in terms of the basis.

To this end, notice that the scalar potential Φ = ΦC+(Φ−ΦC) ≡ ΦC +ΦD may

be split into a Coulombic portion satisfying ∇2ΦC = −4πρ that can be calculated

analytically and a remainder ΦD regardless of the choice of gauge. The Coulombic

potential is not itself a dynamical variable but depends on the dynamical variables

Q and P. That is ΦC(x, t) =∫

Vd3x′|x− x′|−1q[Q(x′, t)Q(x′, t) + P (x′, t)P (x′, t)]/2.

The dynamical portion ΦD is a generalized coordinate and is represented in the

basis. Similarly, the momentum conjugate to A may be split into a Coulombic and

dynamical piece according to

Π ≡ ∂LMax

∂A= ΠC + ΠD =

∇ΦC

4πc+

1

4πc[A/c+ ∇ΦD]. (4.34)

Like ΦD, the dynamical portion ΠD is a generalized coordinate and is represented

in the basis.

4.5.1 Symplectic Transformation to the Coulomb Reference

The transformation to these new coordinates, i.e., ΦD and ΠD, is obtained by

the canonical or symplectic transformation

T :

A

Φ

Q

Π

Θ

P

→

A(A)

Φ(Φ, Q, P )

Q(Q)

Π(Π, Q, P )

Θ(Θ)

P (P )

=

A

Φ − ΦC(Q,P )

Q

Π − ΠC(Q,P )

Θ

P

, (4.35)

where Φ ≡ ΦD and Π ≡ ΠD. The variables Q, P, A, and Θ are unchanged by T.

Since both ΦC and ΠC are complicated functions of Q and P, the inversion of T may

72

be quite involved. However, it will be shown that the inverse of T does exist. In fact

both the T and T−1 are differentiable mappings on symplectic manifolds. Therefore

the canonical transformation is a symplectic diffeomorphism or symplectomorphism

[29].

The theory of restricted (i.e., explicitly time-independent) canonical transfor-

mations [27, 28] gives the general prescription for the transformation of the old

Hamilton equations (4.32) to the new Hamilton equations in terms of T (and TT )

only. In symbols, that is

η = ω−1∂H

∂η→ ˙η = ω−1∂H

∂η, (4.36)

where the new Hamiltonian H is equivalent to the old Hamiltonian H expressed in

terms of the new variables η. (For simplicity H will be written as H from this point

forward.) To this end, consider the time derivative of the new column matrix

˙ηi =∂ηi∂ηj

ηj ≡ Tij ηj or ˙η = Tη. (4.37)

Substituting η from (4.36) results in

˙ηi = Tijω−1jk

∂H

∂ηkor ˙η = Tω−1∂H

∂η. (4.38)

Lastly the column matrix ∂H/∂η can be written as

∂H

∂ηk=∂H

∂ηl

∂ηl∂ηk

≡ T Tkl∂H

∂ηlor

∂H

∂η= TT ∂H

∂η(4.39)

so that the new equations of motion (4.38) become

˙ηi = Tijω−1jk T

Tkl

∂H

∂ηlor ˙η = Tω−1TT ∂H

∂η≡ ω−1∂H

∂η. (4.40)

73

This canonical transformation on the equations of motion leaves only the com-

putation of ω−1 ≡ Tω−1TT since the Hamiltonian automatically becomes

H =[−4πcΠ + ΠC(Q, P )]2 + [∇× A]2 − [4πcΘ]2

8π+ q[Φ + ΦC(Q, P )]

Q2 + P 2

2

− c∇[Φ + ΦC(Q, P )] · [Π + ΠC(Q, P )] − cΘ∇ · A

+[∇Q+ qAP /c]2 + [−∇P + qAQ/c]2

4m+ V

Q2 + P 2

2

(4.41)

in terms of the new coordinates. However, the inversion of ω is not simple in practice.

It turns out that the equations of motion (4.36) are most practically written as

∂H

∂η= ω ˙η = (T−1)TωT−1 ˙η not ˙η = ω−1∂H

∂η= Tω−1TT ∂H

∂η, (4.42)

where the inverse transformation T−1 is the transformation of the inverse mapping,

i.e., T−1ij ≡ ∂ηi/∂ηj. It will be shown that detT 6= 0 so the mapping is well-defined.

These equations of motion are of the desired form because they involve ω and not

ω−1. That ω−1 is undesirable is seen by going to the basis. In the basis, the canonical

symplectic form becomes

(0 − 1

1 0

)

→

0 −∂2〈Q|P 〉∂qK∂pJ

∂2〈P |Q〉∂pK∂qJ

0

(4.43)

which is not easily inverted. As a result it is simpler to compute (T−1)TωT−1 than

Tω−1TT even though T−1 is needed in the former case. It will be shown that the

explicit evaluation of T−1 is not necessary.

To continue with the transformed equations of motion in (4.42), which only

require ω, the mapping (T−1)T : ∂/∂η → ∂/∂η must first be set up. The transposed

inverse transformation (T−1)T is defined on the vector fields themselves according

74

to

∂/∂A

∂/∂Φ

∂/∂Q

∂/∂Π

∂/∂Θ

∂/∂P

=

∂A/∂A 0 0 0 0 0

0 ∂Φ/∂Φ 0 0 0 0

0 ∂Φ/∂Q ∂Q/∂Q ∂Π/∂Q 0 0

0 0 0 ∂Π/∂Π 0 0

0 0 0 0 ∂Θ/∂Θ 0

0 ∂Φ/∂P 0 ∂Π/∂P 0 ∂P/∂P

∂/∂A

∂/∂Φ

∂/∂Q

∂/∂Π

∂/∂Θ

∂/∂P

.

(4.44)

Notice that

det(T−1)T = detT−1 = (detT)−1 ≡ ∂(A,Φ, Q,Π,Θ, P )

∂(A, Φ, Q, Π, Θ, P )

= (∂A/∂A)(∂Φ/∂Φ)(∂Q/∂Q)(∂Π/∂Π)(∂Θ/∂Θ)(∂P /∂P ) = 1

(4.45)

so that the transformation is canonical and symplectic or area preserving. In other

words, the new infinitesimal volume element dη is related to the old infinitesimal

volume element dη by

dη = detT dη = dη (4.46)

since the determinant of the Jacobian is unity. Thus, the volume element of phase

space is the same before and after the transformation. It is a canonical invariant.

75

With (T−1)T the similarity transformation of the canonical symplectic form in

(4.32) is

ω ≡ (T−1)TωT−1 =

(ΩM Ω>

Ω∨ ΩG

)

, (4.47)

where

Ω> =

(∂A/∂A)(−1)(∂Π/∂Π) 0 −(∂A/∂A)(∂Π/∂P )

0 (∂Φ/∂Φ)(−1)(∂Θ/∂Θ) 0

0 −(∂Φ/∂Q)(∂Θ/∂Θ) (∂Q/∂Q)(−1)(∂P/∂P )

(4.48)

Ω∨ =

(∂Π/∂Π)(1)(∂A/∂A) 0 0

0 (∂Θ/∂Θ)(1)(∂Φ/∂Φ) (∂Θ/∂Θ)(∂Φ/∂Q)

(∂Π/∂P )(∂A/∂A) 0 (∂P/∂P )(1)(∂Q/∂Q)

(4.49)

ΩM =

0 0 −(∂A/∂A)(∂Π/∂Q)

0 0 0

(∂Π/∂Q)(∂A/∂A) 0 0

(4.50)

ΩG =

0 0 0

0 0 (∂Θ/∂Θ)(∂Φ/∂P )

0 −(∂Φ/∂P )(∂Θ/∂Θ) 0

. (4.51)

The factors of 1 and −1 are explicitly written in Ω> and Ω∨ to bring out their

similarity to the canonical symplectic form in (4.32). After computing the derivates

76

in ω it can be shown that ω equals

0 0 −∂ΠC(Q, P )/∂Q −1 0 −∂ΠC(Q, P )/∂P

0 0 0 0 −1 0

∂ΠC(Q, P )/∂Q 0 0 0 −∂ΦC(Q, P )/∂Q −1

1 0 0 0 0 0

0 1 ∂ΦC(Q, P )/∂Q 0 0 ∂ΦC(Q, P )/∂P

∂ΠC(Q, P )/∂P 0 1 0 −∂ΦC(Q, P )/∂P 0

(4.52)

with

ΦC(x, t) =q

2

∫

V

Q(x′, t)2 + P (x′, t)2

|x − x′| d3x′ (4.53)

and

ΠC(x, t) =q

8πc∇

∫

V

Q(x′, t)2 + P (x′, t)2

|x − x′| d3x′. (4.54)

And so the new symplectic form contains extra elements that are not present in the

canonical ω. These extra elements add additional time-dependent couplings to the

theory. As before, the associated phase space is naturally endowed with the Poisson

bracket

F,G˜ = (∂F/∂η)T ω−1(∂G/∂η). (4.55)

77

The transformed equations of motion with symplectic form (4.52) may be writ-

ten in full as:

−∂ΠC

∂Q

˙Q− ˙Π− ∂ΠC

∂P

˙P =∂H

∂A=

∇× [∇× A]

4π+ c∇Θ

− q

2mc

Q∇P − P∇Q− q

cQAQ− q

cP AP

− ˙Θ =∂H

∂Φ= q

Q2 + P 2

2+ c∇ · [Π + ΠC ]

∂ΠC

∂Q· ˙A− ∂ΦC

∂Q

˙Θ − ˙P =∂H

∂Q=

−∇2Q− q∇ · (AP )/c− q∇P · A/c+ q2A2Q/c2

2m

+ V Q + q[Φ + ΦC ]Q

+ 4πc2[Π + ΠC ] − c∇[Φ + ΦC ] · ∂ΠC

∂Q

+

qQ2 + P 2

2+ c∇ · [Π + ΠC ]

∂ΦC

∂Q

˙A =

∂H

∂Π= 4πc2[Π + ΠC ] − c∇[Φ + ΦC ]

˙Φ +∂ΦC

∂Q

˙Q+∂ΦC

∂P

˙P =∂H

∂Θ= −4πc2Θ − c∇ · A

∂ΠC

∂P· ˙A + ˙Q− ∂ΦC

∂P

˙Θ =∂H

∂P=

−∇2P + q∇ · (AQ)/c+ qA · ∇Q/c+ q2A2P /c2

2m

+ V P + q[Φ + ΦC ]P

+ 4πc2[Π + ΠC ] − c∇[Φ + ΦC ] · ∂ΠC

∂P

+

qQ2 + P 2

2+ c∇ · [Π + ΠC ]

∂ΦC

∂P,

(4.56)

where ΠC ≡ ΠC(Q, P ) and ΦC ≡ ΦC(Q, P ). The forces appearing on the right

hand side of these equations have become more complicated, especially those in the

Schrodinger equations. There are new nonlinear terms. However, it is possible to

substitute these equations among themselves in order to simplify them. Notice that

parts of the ∂H/∂Φ and ∂H/∂Π equations appear in the forces of the Schrodinger

78

equations. Substitution of ∂H/∂Φ and ∂H/∂Π into the Schrodinger equations re-

sults in the following simplified equations:

−∂ΠC

∂Q

˙Q− ˙Π − ∂ΠC

∂P

˙P =∇× [∇× A]

4π+ c∇Θ

− q

2mc

Q∇P − P∇Q− q

cQAQ− q

cP AP

− ˙Θ = qQ2 + P 2

2+ c∇ · [Π + ΠC]

− ˙P = −∇2Q+ q∇ · (AP )/c+ q∇P · A/c− q2A2Q/c2

2m+ V Q

+ q[Φ + ΦC ]Q

˙A = 4πc2[Π + ΠC ] − c∇[Φ + ΦC ]

˙Φ +∂ΦC

∂Q

˙Q+∂ΦC

∂P

˙P = −4πc2Θ − c∇ · A

˙Q =−∇2P + q∇ · (AQ)/c+ qA · ∇Q/c+ q2A2P /c2

2m+ V P

+ q[Φ + ΦC ]P .

(4.57)

The generalized forces appearing on the right hand side are now very similar to the

forces in (4.31). In fact, the equations of motion (4.57) can be further simplified as:

−[ Π + ΠC(Q,P )] = ∂H/∂A A = ∂H/∂Π

− Θ = ∂H/∂Φ Φ + ΦC(Q,P ) = ∂H/∂Θ

−P = ∂H/∂Q Q = ∂H/∂P ,

(4.58)

where the tildes were omitted to show the resemblance between (4.58) and (4.31).

4.5.2 The Coulomb Reference by Change of Variable

It can be shown that the new equations of motion ω ˙η = ∂H/∂η, which were

obtained by a symplectic transformation in phase space, may also be obtained by a

change of variable in the Lagrangians (4.28)-(4.30). The new Lagrangian density is:

79P

Sfrag

replacem

ents

L(p, q) L(p, q)

ωη = ∂H/∂η ω ˙η = ∂H/∂η

Canonical Transformation

δ∫

[pq−H

(p,q

)]dt=

0

δ∫

[p˙q−

H(p,q)]d

t=

0

Change of Variables

Figure 4–3: Commutivity diagram representing the change of coordinates (q, p) to(p, q) at both the Lagrangian and equation of motion levels.

LMax =1

2[(Π+ΠC) · ˙

A−( ˙Π+ ˙

ΠC) ·A]−

[−4πc(Π + ΠC)]2 + [∇× A]2/8π

−c∇[Φ + ΦC ] · [Π + ΠC]

(4.59)

LSch =1

2[P ˙Q− ˙PQ] −

[∇Q+ qAP /c]2 + [−∇P + qAQ/c]2/4m

+[V + q(Φ + ΦC)]Q2 + P 2/2

(4.60)

Lgauge =1

2[Θ( ˙Φ + ΦC) − ˙Θ(Φ + ΦC)] − −2πc2Θ2 − cΘ∇ · A. (4.61)

That the transformation to the Coulomb reference holds at both Lagrangian and

equation of motion level demonstrates the commutivity of the diagram in Figure

4–3.

4.6 Electron Spin in the Pauli Theory

The electron field used so far in the nonrelativistic Schrodinger theory is a field

of spin zero, i.e., a scalar field. It is a simple generalization of the theory to add in

the electron’s spin. The electron field would then be a two component spinor field,

80

i.e., a spin-1/2 field, and would be of the form

ΨP (x, t) =

(Ψ↑(x, t)

Ψ↓(x, t)

)

. (4.62)

The first component Ψ↑ is spin up and the second component Ψ↓ is spin down. The

dynamics of ΨP is governed by the Pauli equation [30]

iΨP =[−i∇− qA/c]2ΨP

2m+ VΨP + qΦΨP − q

2mcσ · [∇× A]ΨP (4.63)

which is the nonrelativistic limit of the Dirac equation

iΨD = βmc2ΨD + cα · [−i∇− qA/c]ΨD + qΦΨD (4.64)

in terms of the four component spinor ΨD, where the β and α matrices are

β =

(I 0

0 − I

)

α =

(0 σ

σ 0

)

(4.65)

and

σx =

(0 1

1 0

)

σy =

(0 − i

i 0

)

σz =

(1 0

0 − 1

)

. (4.66)

Notice that taking the nonrelativistic limit of the Dirac equation involves the elim-

ination of the two component positron field from ΨD. Also note that the current

density associated with the Pauli theory [90] is different from that in the Schrodinger

theory (see (1.14)). It is

JP = qΨ†P [−i∇− qA/c]ΨP + ΨP [i∇− qA/c]Ψ†P + ∇× [Ψ†PσΨP ]

/2m, (4.67)

where Ψ†P = (Ψ∗↑ Ψ∗↓) is the adjoint of ΨP . This can be derived by taking the

nonrelativistic limit of the Dirac current density. The last term in (4.67) is only

present in the Pauli current. This term does not effect the continuity equation

ρ = −∇ · J since ∇ · ∇ × [Ψ†PσΨP ] = 0.

81

4.7 Proton Dynamics

In the theory set up so far, the matter dynamics was entirely described by

the electronic wavefunction Ψ. The proton had no dynamics whatsoever. Only

the electrostatic scalar potential Φq = q/|x| of the structureless proton of charge

q entered so as to bind the electron in the hydrogen atom. A first step in the

direction of atomic and molecular collisions requires the dynamics of the proton as

well (and eventually a few other particles). Suppose the proton is described by its

own wavefunction Ω and Lagrangian density

LqSch = iΩ∗Ω − [i∇− qA/c]Ω∗ · [−i∇− qA/c]Ω

2mq

− qΦΩ∗Ω, (4.68)

where (Φ,A) are the scalar and vector potentials arising from the charge and current

densities

ρ = qΨ∗Ψ + qΩ∗Ω (4.69)

J = qΨ∗[−i∇− qA/c]Ψ + Ψ[i∇− qA/c]Ψ∗

/2mq

+ qΩ∗[−i∇− qA/c]Ω + Ω[i∇− qA/c]Ω∗

/2mq.

(4.70)

These densities are just the sum of the individual electronic and proton densities.

The proton density is not a delta function. Thus, the proton wavefunction is not a

delta function either. Rather it is described by a wavepacket and has some structure.

With (4.68) the total Lagrangian is

LMax =1

2[Π · A− Π · A] −

[−4πcΠ]2 + [∇× A]2

8π− c∇Φ ·Π

(4.71)

LqSch =i

2[Ψ∗Ψ − Ψ∗Ψ] −

[i∇− qA/c]Ψ∗ · [−i∇− qA/c]Ψ

2mq

+ qΦΨ∗Ψ

(4.72)

82

LqSch =i

2[Ω∗Ω − Ω∗Ω] −

[i∇− qA/c]Ω∗ · [−i∇− qA/c]Ω

2mq

+ qΦΩ∗Ω

(4.73)

Lgauge =1

2[ΘΦ − ΘΦ] − −2πc2Θ2 − cΘ∇ · A. (4.74)

Notice that the electron Lagrangian (4.72) does not explicitly contain the static pro-

ton potential energy V = qq/|x| as did the previous Schrodinger Lagrangian (1.17).

The two matter fields are coupled entirely through electrodynamics. That is, the

electron-proton interaction is mediated by the electrodynamics. The Coulombic

potential is included implicitly in qΦΨ∗Ψ and qΦΩ∗Ω in the above matter Hamil-

tonians. In other words, the scalar potential Φ contains (in any gauge) a Coulomb

piece of the form

ΦC(x, t) = Φ + (ΦC − Φ) =∫

V

ρ(x′, t)

|x − x′|d3x′

=∫

V

qΨ∗(x′, t)Ψ(x′, t)

|x − x′| d3x′ +∫

V

qΩ∗(x′, t)Ω(x′, t)

|x − x′| d3x′.

(4.75)

With this potential, the qΦΨ∗Ψ term in the electron Hamiltonian contains the

electron-proton attraction as well as electron-electron self interaction. Similarly

the qΦΩ∗Ω in the proton Hamiltonian contains the electron-proton attraction and

proton-proton self interaction.

The self-energies that are computed from the aforementioned self interactions

are finite because Ψ and Ω are square integrable functions. That is

Eint =∫

Vρ(x, t)ΦC(x, t)d3x =

∫

Vd3x

∫

Vd3x′

ρ(x, t)ρ(x′, t)

|x − x′| <∞ (4.76)

for both the cross terms (electron-proton attraction) and the direct terms (electron-

electron and proton-proton repulsion). Note that in the relativistic quantum theory

the direct terms are infinite and there are infinitely many Coulomb states of the

83

bare problem to sum over [40]. These infinities do not arise in the semiclassical

theory presented in this dissertation. While the self interactions do appear in the

above matter Hamiltonians, the resulting self-energies are finite and moreover do

not even contribute to the electron or proton portions of the energy. This is due to

−c∇Φ ·Π in the above Maxwell Hamiltonian. After a partial integration this term

becomes cΦ∇ · Π. Substitution of −Θ = ρ + c∇ · Π = 0 from (4.31) turns cΦ∇ ·Π

into −ρΦ, which cancels +ρΦ in the electron and proton energies. However, the self

interactions do remain in the Coulomb energy E2/8π of the electromagnetic field.

Note that the self interactions do appear in the Hamiltonians and therefore do make

a contribution to the overall dynamics.

It should be mentioned that this theory of electron-proton dynamics can be

applied to electron-positron dynamics as well. While there is a 2000-fold difference

in mass between the proton and the positron, the two theories are otherwise identical.

In either case, the theory may be rich enough to capture bound states of hydrogen

or positronium.

CHAPTER 5NUMERICAL IMPLEMENTATION

The formal theory of Maxwell-Schrodinger dynamics was constructed in the

previous chapter. In particular, the coupled and nonlinear Maxwell-Schrodinger

equations

iΨ =[P− qA/c]2Ψ

2m+ VΨ + qΦΨ (5.1)

∇2A − A

c2−∇

[

∇ · A +Φ

c

]

= −4π

cJ (5.2a)

∇2Φ +∇ · Ac

= −4πρ (5.2b)

were recognized to be ill-posed unless an extra equation of constraint is added to

them. Using the Hamiltonain approach to dynamics, this extra equation was auto-

matically generated by adding a Lorenz gauge fixing term at the Lagrangian level.

It was emphasized in Chapter 4 that the resulting Hamiltonian system of differential

equations, which are of first order in time, form a well-defined initial value problem.

That is, the Maxwell-Schrodinger dynamics are known in principle once the initial

values are specified for each of the dynamical variables.

The details of converting the formal mathematics of Chapter 4 to a form that

can be practically implemented in a computer are presented in this chapter. The

Hamiltonian system of partial differential equations will be reduced to a Hamilto-

nian system of ordinary differential equations in time by introducing a spatial basis

for each of the dynamical variables. The resulting basis equations are coded in a

Fortran 90 computer program. With this program, various pictures are made

to depict the dynamics of the hydrogen atom interacting with the electromagnetic

field.

84

85

5.1 Maxwell-Schrodinger Theory in a Complex Basis

Each of the Maxwell-Schrodinger dynamical variables, which are themselves

fields, may be expanded into a complete basis of functions GK according to

Ψ(x, t) =∑

KGK(x)ψK(t) Ψ∗(x, t) =∑

KGK(x)ψ∗K(t)

Ak(x, t) =∑

KGK(x)akK(t) Πk(x, t) =∑

KGK(x)πkK(t)

Φ(x, t) =∑

KGK(x)φK(t) Θ(x, t) =∑

KGK(x)θK(t),

(5.3)

where the index K runs over the basis and the index k runs over 1, 2, 3 or x, y, z.

Any complete set of functions such as the oscillator eigenstates will suffice. In the

following work the set of gaussian functions of the form

GK(x) = G∗K(x) = NK exp(−`K[x − rK]2) (5.4)

are used. These functions are centered on rK, normalized to unity by NK, and are

real-valued. Additionally, they span L2 so that any square integrable function may

be represented in this basis. In principle the sums in (5.3) are to infinity. However,

a complete basis cannot be realized in practice. But for all practical purposes the

numerical results can be shown to converge to within an arbitrary accuracy in a finite

basis. In fact with a smart choice of basis, the numerical results may converge with

just a few terms. Here the basis coefficients, which are complex- and real-valued as

well as time-dependent, carry the dynamics.

The basis representation of the previous Lagrangians is

LMax =∑

KM

1

2[(∂/∂amM)amM − (∂/∂πmM)πmM]SMax −HMax (5.5)

LSch =∑

K

i

2[(∂/∂ψK)ψK − (∂/∂ψ∗K)ψ∗K]SSch −HSch (5.6)

Lgauge =∑

KM

1

2[(∂/∂φK)φK − (∂/∂θK)θK]Sgauge −Hgauge (5.7)

86

with integrals

SMax =∫

VΠ · Ad3x SSch =

∫

VΨ∗Ψd3x Sgauge =

∫

VΘΦd3x. (5.8)

The calculus of variations leads to the following dynamical equations:

∂2SMax

∂πmM∂anNanN =

∂H

∂πmMor MmM,nN anN = ∇πmM

H (5.9)

∂2Sgauge

∂θI∂φLφL =

∂H

∂θIor NILφL = ∇θIH (5.10)

∂2iSSch

∂ψ∗I∂ψLψL =

∂H

∂ψ∗Ior iCILψL = ∇ψ∗

IH (5.11)

− ∂2SMax

∂anN∂πmMπmM =

∂H

∂anNor −MT

nN ,mMπmM = ∇anNH (5.12)

−∂2Sgauge

∂φJ ∂θKθK =

∂H

∂φJor −NT

JKθK = ∇φJH (5.13)

− ∂2iSSch

∂ψJ ∂ψ∗Kψ∗K =

∂H

∂ψJor − iC∗JKψ

∗K = ∇ψJ

H (5.14)

which are of the Hamiltonian form ωη = ∂H/∂η. The summation convention is used

throughout. These equations may be written more compactly as

Ma = ∇πH −MT π = ∇aH

Nφ = ∇θH −NT θ = ∇φH

iCψ = ∇ψ∗H − iC∗ψ∗ = ∇ψH

(5.15)

87

and can be cast into matrix form as

iC 0 0 0 0 0

0 −iC∗ 0 0 0 0

0 0 0 0 −MT 0

0 0 0 0 0 −NT

0 0 M 0 0 0

0 0 0 N 0 0

ψ

ψ∗

a

φ

π

θ

=

∂H/∂ψ∗

∂H/∂ψ

∂H/∂a

∂H/∂φ

∂H/∂π

∂H/∂θ

, (5.16)

where the matrices M, N, and C and defined in (5.9)-(5.14). This symplectic form

almost has the canonical structure of (4.25). In a basis of rank N, the contractions

involving a and π run to 3N while the contractions involving the remaining dy-

namical variables run to N. This is because a and π are spatial vectors that have

(x, y, z)-components whereas the remaining dynamical variables are scalars.

With the choice of representation in (5.3) and the choice of basis in (5.4) all

approximations are specified. The equations of motion in (5.16) are the basis rep-

resentation of the coupled Maxwell-Scrodinger equations. They are automatically

obtained by applying the time-dependent variational principle to the Lagrangians

(5.5)-(5.7). In the limit of a complete basis these equations are exact.

The complex phase space that carries the associated dynamics is endowed with

the Poisson bracket

F,G =

∂F/∂ψ∗

∂F/∂ψ

∂F/∂a

∂F/∂φ

∂F/∂π

∂F/∂θ

T

iC 0 0 0 0 0

0 −iC∗ 0 0 0 0

0 0 0 0 −MT 0

0 0 0 0 0 −NT

0 0 M 0 0 0

0 0 0 N 0 0

−1

∂G/∂ψ∗

∂G/∂ψ

∂G/∂a

∂G/∂φ

∂G/∂π

∂G/∂θ

.

(5.17)

88

Even though the symplectic form is not canonical, its inversion is simple. The matrix

elements in ω involve gaussian overlap integrals like 〈GI|GK〉 =∫

V〈GI|x〉〈x|GK〉d3x.

5.2 Maxwell-Schrodinger Theory in a Real Basis

As was done previously, each dynamical variable may be expanded into a com-

plete basis of functions GK as

Q(x, t) =∑

KGK(x)qK(t) P (x, t) =∑

KGK(x)pK(t)

Ak(x, t) =∑

KGK(x)akK(t) Πk(x, t) =∑

KGK(x)πkK(t)

Φ(x, t) =∑

KGK(x)φK(t) Θ(x, t) =∑

KGK(x)θK(t),

(5.18)

where the index K runs over the basis and the index k runs over 1, 2, 3 or x, y, z.

Unlike in (5.3), the coefficients in (5.18) that carry the dynamics are all real-valued.

In this basis, the real Lagrangian densities become

LMax =∑

KM

1

2[(∂/∂amM)amM − (∂/∂πmM)πmM]SMax −HMax (5.19)

LSch =∑

K

1

2[(∂/∂qK)qK − (∂/∂pK)pK]SSch −HSch (5.20)

Lgauge =∑

KM

1

2[(∂/∂φK)φK − (∂/∂θK)θK]Sgauge −Hgauge (5.21)

with integrals

SMax =∫

VΠ ·Ad3x SSch =

∫

VPQd3x Sgauge =

∫

VΘΦd3x. (5.22)

Applying the calculus of variations to the above Lagrangians leads to the equa-

tions of motion:

∂2SMax

∂πmM∂anNanN =

∂H

∂πmMor MmM,nN anN = ∇πmM

H (5.23)

∂2Sgauge

∂θI∂φLφL =

∂H

∂θIor NILφL = ∇θIH (5.24)

89

∂2SSch

∂pI∂qLqL =

∂H

∂pIor CILqL = ∇pIH (5.25)

− ∂2SMax

∂anN∂πmMπmM =

∂H

∂anNor −MT

nN ,mMπmM = ∇anNH (5.26)

−∂2Sgauge

∂φJ ∂θKθK =

∂H

∂φJor −NT

JKθK = ∇φJH (5.27)

− ∂2SSch

∂qJ ∂pKpK =

∂H

∂qJor − CT

JKpK = ∇qJH (5.28)

which are of the Hamiltonian form ωη = ∂H/∂η. These equations may be written

more compactly as

Ma = ∇πH −MT π = ∇aH

Nφ = ∇θH −NT θ = ∇φH

Cq = ∇pH − CT p = ∇qH

(5.29)

and can be cast into matrix form as

0 0 0 −MT 0 0

0 0 0 0 −NT 0

0 0 0 0 0 −CT

M 0 0 0 0 0

0 N 0 0 0 0

0 0 C 0 0 0

a

φ

q

π

θ

p

=

∂H/∂a

∂H/∂φ

∂H/∂q

∂H/∂π

∂H/∂θ

∂H/∂p

, (5.30)

where the matrices M, N, and C and defined in (5.23)-(5.28). Again they are the

basis representation of the coupled Maxwell-Schrodinger equations of motion.

90

The real phase space that carries the associated dynamics is endowed with the

Poisson bracket

F,G =

∂F/∂a

∂F/∂φ

∂F/∂q

∂F/∂π

∂F/∂θ

∂F/∂p

T

0 0 0 −MT 0 0

0 0 0 0 −NT 0

0 0 0 0 0 −CT

M 0 0 0 0 0

0 N 0 0 0 0

0 0 C 0 0 0

−1

∂G/∂a

∂G/∂φ

∂G/∂q

∂G/∂π

∂G/∂θ

∂G/∂p

.

(5.31)

Even though the symplectic form is not canonical, its inversion is simple once

again. The matrix elements in ω involve gaussian overlap integrals like 〈GI |GK〉 =∫

V〈GI|x〉〈x|GK〉d3x.

5.2.1 Overview of Computer Program

The equations of motion (5.30) are coded in Fortran 90. The computer pro-

gram is called Electron Nuclear Radiation Dynamics or ENRD. Each matrix element

in the symplectic form and in the forces is performed analytically. The program is

flexible enough to handle a rank N basis of s-gaussians, each with an adjustable

width and an arbitrary center. A numerical solution to (5.30) is determined once

the initial value data is specified for ηb. The forces ∂H/∂ηa are constructed from

this data. The symplectic form ωab is then inverted with the LAPACK [91] sub-

routine DGESVX, which is the expert driver for the AX = B solver DGESV. This

establishes a first order system of differential equations of the form ηb = ω−1ab ∂H/∂η

a

which may be solved, for example, with an Euler stepping method. That is

ηb(t + ∆t) = ηb(t) + (∆t)ηb(t) = ηb(t) + (∆t)[ω−1ab ∂H/∂η

a](t). (5.32)

In practice, the Euler method is not accurate enough so the more sophisticated RK4

method [92] is implemented in the code. The equations of motion are advanced

91

at a fixed stepsize of 10−3 au. For typical basis function widths and centers, the

estimated condition number reported by DGESVX is about 30.

Lastly, it should be pointed out that the equations of motion (5.30) are nu-

merically implemented in terms of the electric field E ≡ −4πcΠ rather than the

momentum Π. It was found that working in terms of this new (scaled) coordinate

provides a more balanced set of dynamical equations. Nevertheless, the electromag-

netic radiation is still quite small compared to the dynamics of the matter. An

overview of the ENRD program is presented in Figure 5–1.

92

PSfrag replacements

enrd.f90

RHS EM.f90

RHS SH.f90

• Create library for four index

• Create library for vector

• Input deck.

• RHS

• RHS Maxwell.

• RHS Gauge.

matrix elements.

DERIVS.f90

MAXENERGY.f90

GAENERGY.f90

SCHENERGY.f90

• Maxwell energy.

• Gauge energy.

∂H∂Π, ∂H

∂A

∂H∂Θ, ∂H

∂Φ

∂H∂P, ∂H

∂Q

• Schrodinger energy.Schrodinger.

• Build forces ∂H∂η

on RHS.

RHS GA.f90

RIRJ.f90

RK4.f90

• Runge-Kutta 4 ODE solver.

η(t+ ∆t) = η(t) + (∆t)η(t)

• Invert ωη = ∂H/∂η to

get η = ω−1∂H/∂η.

METRIC.f90

• Build symplectic form ω.

• Call LAPACK routineDGESVX.

arguments.

• Symplecticform ω.

• Build vector arguments.

• Call integrator.

• Write η(t+ ∆t) to PS.dat.

• Compute 〈ηM|η(t)〉.

• Compute 〈ηM|η(t)〉.

• Compute 〈ηM|η(t)〉.

Figure 5–1: Schematic overview of ENRD computer program.

93

5.2.2 Stationary States: s- and p-Waves

The ENRD program was first tested with a stationary state of the hydrogen

atom. In a basis of six gaussians an s-wave was constructed as well as the corre-

sponding basis representation of the Coulombic scalar potential and the Coulombic

electric field. In fact, any spherically symmetric distribution of charge along with

the corresponding Coulomb fields would suffice. This delicate balance of charges

and fields proved to be a stationary state of the combined system. No electromag-

netic radiation was produced. The total charge q =∫

Vρ(x)d3x remained constant.

A px-wave and its associated Coulombic fields were also created in the same basis.

This again is a stationary state.

5.2.3 Nonstationary State: Mixture of s- and p-Waves

After identifying some stationary states, a nonstationary state that is a mixture

of s- and px-waves was constructed in the same rank six basis. Both the Coulom-

bic scalar potential and the Coulombic electric field that are associated with this

charge distribution were created as well. Electromagnetic radiation was produced as

the electron oscillated between stationary states. Energy, momentum, and angular

momentum were exchanged between the electron and the electromagnetic field. It

was shown that the total energy and total Hamiltonian are conserved to four deci-

mal places. The total charge q =∫

Vρ(x)d3x remained constant. The phase space

contours for the electromagnetic field, matter field, and gauge field are presented in

Figures 5–2, 5–3, and 5–4 respectively.

5.2.4 Free Electrodynamics

Lastly a free electromagnetic field was constructed. In this case no charge was

created. Energy, momentum, and angular momentum were exchanged only between

the electromagnetic and gauge degrees of freedom. The total energy remained con-

stant.

94

PSfrag

replacem

ents

akK(t)

πkK(t)

0.5-0.5 0

0

10

-10

20

-20

Figure 5–2: Phase space contour for the coefficients of the vector potential A andits momentum Π.

PSfrag

replacem

ents

qK(t)

pK(t)

0

0

5

5

-5

-5

Figure 5–3: Phase space contour for the coefficients of the real-valued Schrodingerfield Q and its momentum P.

95

PSfrag

replacem

ents

φK(t)

θK(t)

0.0005

-0.0005

0

0

10-10

20

-20-30

Figure 5–4: Phase space contour for the coefficients of the scalar potential Φ and itsmomentum Θ.

5.2.5 Analysis of Solutions in Numerical Basis

The solutions η(t) of the equations of motion (5.30) are further analyzed by

expansion into the basis eigenstates ηM. The Schrodinger eigenstates are found by

diagonalizing the time-independent Schrodinger equation

HC = εSC, (5.33)

where H is the basis representation of the Hamiltonian H = −∇2/2m + V, C is

the matrix of basis expansion coefficients, S is the basis overlap matrix, and ε is

the matrix of energy eigenvalues. Similarly, the Maxwell eigenstates are found by

diagonalizing the free wave equation ∇2φ − φ/c2 = 0, where φ can be the scalar

potential Φ or any component of the vector potential A. Fourier inversion of the

free wave equation results in −c2∇2φ = ω2φ, where ω is the frequency. In a basis

this equation turns into the matrix equation

HC = ω2SC, (5.34)

96

where H is the basis representation of the Hamiltonian-like quantity H = −c2∇2,

C is the matrix of basis expansion coefficients, S is the basis overlap matrix, and

ω2 is the matrix of frequencies squared. Recall that energy is related to frequency

by E = ~ω, so that in atomic units energy is equivalent to frequency.

Both of these basis equations (5.33) and (5.34) are recognized as belonging to

the generalized eigenvalue problem Aη = λBη, which can be inverted with the

LAPACK routine DSYGV. The ENRD program employs DSYGV to solve both

(5.33) and (5.34) for their corresponding eigenvalues λM and eigenvectors ηM.

With the eigenvectors ηM, the evolving state vector η(t) can be expanded ac-

cording to

|η(t)〉 =∑

M|ηM〉〈ηM|η(t)〉, (5.35)

where CM(t) ≡ 〈ηM|η(t)〉 are the basis expansion coefficients. The real and imag-

inary parts of the Schrodinger coefficients for a superposition of s- and px-waves

are plotted versus time in Figures 5–5 and 5–6 respectively. The squares of these

coefficients are plotted versus time in Figure 5–7. Notice in Figures 5–5 and 5–6

that there are three frequencies involved in the dynamics, which correspond to exci-

tations of the s-, px-, and dy2−z2-waves. Figure 5–7 suggests that the electron decays

from px to s in under 10 au of time. However, due to the finite size of the basis,

the electron is excited back to the px-state as the electromagnetic fields reflect off of

the artificial basis boundaries. Lastly, the phase space contour of the Schrodinger

coefficients are presented in Figure 5–8.

97

PSfrag

replacem

ents

t0

0

10 20 30

Re〈ΨM|Ψ(t)〉

spxpypzdx2

dy2−z2

1

−1

Figure 5–5: Real part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, where η(t)is a superposition of s- and px-waves.

PSfrag

replacem

ents

t0

0

10 20 30

Im〈ΨM|Ψ(t)〉

spxpypzdx2

dy2−z2

1

−1

Figure 5–6: Imaginary part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves.

98

PSfrag

replacem

ents

t

∑

M|〈ΨM|Ψ(t)〉|2spx

|〈ΨM|Ψ(t)〉|2

spx

py

pz

dx2

dy2−z2

0

0

10 20 30

−0.5

0.2

0.4

0.6

0.8

−10

−20

−30

0.001−

0.001

1

−1

Re〈Ψ

N |Ψ(t)〉Im〈ΨM|Ψ(t)〉

qK(t)p

K(t)

Figure 5–7: Probability for the electron to be in a particular basis eigenstate.

PSfrag

replacem

entst

spxpypzdx2

dy2−z2

0

0

10202530

−0.5

−0.5

0.5

0.5

−10

−20

−30

0.001−

0.001

1

1

−1

−1

Re〈ΨM|Ψ(t)〉

Im〈ΨM|Ψ(t)〉

qK(t)p

K(t)

Figure 5–8: Phase space of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves.

99

5.3 Symplectic Transformation to the Coulomb Reference

Recall the basis representation of the Maxwell-Schrodinger equations of motion

in (5.30). They are

0 0 0 −MT 0 0

0 0 0 0 −NT 0

0 0 0 0 0 −CT

M 0 0 0 0 0

0 N 0 0 0 0

0 0 C 0 0 0

a

φ

q

π

θ

p

=

∂H/∂a

∂H/∂φ

∂H/∂q

∂H/∂π

∂H/∂θ

∂H/∂p

, (5.30)

where

MKM ≡ ∂2〈Π|A〉∂πK∂aM

NKM ≡ ∂2〈Θ|Φ〉∂θK∂φM

CKM ≡ ∂2〈P |Q〉∂pK∂qM

(5.36)

and where the integrals 〈·|·〉 involve only gaussian functions.

In analogy to the transformation (T−1)T in (4.44) that was defined on the vector

fields ∂/∂η in function space, a basis representation of (T−1)T can be made. This

basis representation is defined in terms of the coefficients according to

∂/∂a

∂/∂φ

∂/∂q

∂/∂π

∂/∂θ

∂/∂p

=

∂a/∂a 0 0 0 0 0

0 ∂φ/∂φ 0 0 0 0

0 ∂φ/∂q ∂q/∂q ∂π/∂q 0 0

0 0 0 ∂π/∂π 0 0

0 0 0 0 ∂θ/∂θ 0

0 ∂φ/∂p 0 ∂π/∂p 0 ∂p/∂p

∂/∂a

∂/∂φ

∂/∂q

∂/∂π

∂/∂θ

∂/∂p

(5.37)

100

so that the symplectic form in (5.30) transforms as

0 0 0 −MT 0 0

0 0 0 0 −NT 0

0 0 0 0 0 −CT

M 0 0 0 0 0

0 N 0 0 0 0

0 0 C 0 0 0

→

0 0 −MAQ −MAΠ 0 −MAP

0 0 0 0 −NΦΘ 0

MQA 0 0 0 −NQΘ −CQPMΠA 0 0 0 0 0

0 NΘΦ NΘQ 0 0 NΘP

MP A 0 CP Q 0 −NP Θ 0

,

(5.38)

where the new matrix elements are:

MXY ≡ ∂π

∂XM

∂a

∂Y=

∂π

∂X

∂2〈Π + ΠC|A〉∂π∂a

∂a

∂Y=∂2〈Π + ΠC|A〉

∂X∂Y

NXY ≡ ∂θ

∂XN∂φ

∂Y=

∂θ

∂X

∂2〈Θ|Φ + ΦC〉∂θ∂φ

∂φ

∂Y=∂2〈Θ|Φ + ΦC〉

∂X∂Y

CXY ≡ ∂p

∂XC∂q

∂Y=

∂p

∂X

∂2〈P |Q〉∂p∂q

∂q

∂Y=∂2〈P |Q〉∂X∂Y

(5.39)

for X and Y an arbitrary dynamical variable. The remaining elements are de-

termined by transposition. Again the extra terms in ω add new time-dependent

couplings to the theory. These new terms can all be performed analytically. The

101

resulting equations of motion are

0 0 −MAQ −MAΠ 0 −MAP

0 0 0 0 −NΦΘ 0

MQA 0 0 0 −NQΘ −CQPMΠA 0 0 0 0 0

0 NΘΦ NΘQ 0 0 NΘP

MP A 0 CP Q 0 −NP Θ 0

˙a

˙φ

˙q

˙π

˙θ

˙p

=

∂H/∂a

∂H/∂φ

∂H/∂q

∂H/∂π

∂H/∂θ

∂H/∂p

.

(5.40)

5.3.1 Numerical Implementation

Recall that the forces appearing in the canonical transformed equations (4.57)

could be simplified by substituting these equations among themselves. As a result

ω →

0 0 −MAQ −MAΠ 0 −MAP

0 0 0 0 −NΦΘ 0

0 0 0 0 0 −CQPMΠA 0 0 0 0 0

0 NΘΦ NΘQ 0 0 NΘP

0 0 CP Q 0 0 0

. (5.41)

Note that (5.41) is not a symplectic form. After making this substitution, the

Hamiltonian structure is lost. However, the numerical implementation is greatly

facilitated with the simplified equations (4.57) instead of those in (4.56). Since the

ENRD program does not rely on a symplectic integrator scheme to advance the

dynamics, the symplectic structure is not numerically important anyway.

The equations of motion (4.57) has been added to the ENRD code. The

Coulomb reference can be conveniently turned on or off (resulting in (4.32)) with an

optional flag. As before, the program is flexible enough to handle a rank N basis

of s-gaussians, each with an adjustable width and an arbitrary center. A solution

102

to (4.57) may be obtained once the initial value data is specified for ηb. The forces

∂H/∂ηa are constructed from this data. The new terms appearing on left hand side

of (4.57) are coded analytically. Notice that these terms make up the elements of a

matrix that is not a symplectic form. Nevertheless, the resulting matrix equations

are integrated with the same RK4 stepping method and the DGESVX subroutine

of LAPACK. For typical basis function widths and centers, the condition number

reported by DGESVX is on the order of one thousand.

5.3.2 Stationary States: s- and p-Waves

The ENRD program with the Coulomb reference was first tested with a sta-

tionary state of the hydrogen atom. In a basis of six gaussians an s-wave was

constructed. The corresponding basis representation of the Coulombic scalar poten-

tial and the Coulombic electric field were not needed. All Coulombic properties are

treated analytically once the Coulomb reference is chosen. Again, it was found that

any spherically symmetric distribution of charge will suffice to produce an s-wave

that is a stationary state of the combined system. No electromagnetic radiation was

produced. The total charge q =∫

Vρ(x)d3x remained constant. A px-wave was also

created in the same basis. This again was a stationary state of the combined system.

5.3.3 Nonstationary State: Mixture of s- and p-Waves

After identifying some stationary states, a nonstationary state that is a mixture

of s- and px-waves was constructed in the same rank six basis. Both the Coulombic

scalar potential and the Coulombic electric field that are associated with this charge

distribution were done analytically by the canonical transformation to the Coulomb

reference. Electromagnetic radiation was produced as the electron oscillated between

stationary states. Energy, momentum, and angular momentum were exchanged

between the electron and the electromagnetic field. It was shown that the total

energy and total Hamiltonian are conserved to two decimal places. The total charge

q =∫

Vρ(x)d3x remained constant.

103

5.3.4 Free Electrodynamics

Lastly a free electromagnetic field was constructed. In this case no charge was

created. Energy, momentum, and angular momentum were exchanged only between

the electromagnetic and gauge degrees of freedom. The total energy remained con-

stant.

5.3.5 Analysis of Solutions in Coulomb Basis

As done previously, the evolving state vector η(t) in the Coulomb basis is ex-

panded in terms of the stationary eigenbasis ηM ≡ ηM according to

|η(t)〉 =∑

M|ηM〉〈ηM|η(t)〉. (5.35)

As before, the real and imaginary parts of the Schrodinger coefficients for a superpo-

sition of s- and px-waves are plotted versus time in Figures 5–9 and 5–10 respectively.

The squares of these coefficients are plotted versus time in Figure 5–11. Notice in

Figures 5–9 and 5–10 that there are again three frequencies involved in the dynam-

ics, which correspond to excitations of the s-, px-, and dy2−z2-waves. Figure 5–11

suggests that the electron decays from px to s in under 15 au of time. However,

due to the same aforementioned basis effects, the electron oscillates between the s-

and px-states. Lastly, the phase space contour of the Schrodinger coefficients are

presented in Figure 5–12.

5.4 Asymptotic Radiation

It has been demonstrated that the dynamics of the hydrogen atom’s electron

in the presence of the electromagnetic field was quasiperiodic. This unphysical be-

havior is due to the fact the electromagnetic radiation produced by the electron

cannot escape to infinity and carry away energy, momentum, and angular momen-

tum. Rather, the radiation reflects off of the artificial boundaries of the finite spatial

basis and reexcites the electron.

104

PSfrag

replacem

ents

t0

0

10 20 30

Re〈ΨM|Ψ(t)〉

spxpypzdx2

dy2−z2

1

−1

Figure 5–9: Real part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, where η(t)is a superposition of s- and px-waves.

PSfrag

replacem

ents

t0

0

10 20 30

Im〈ΨM|Ψ(t)〉

spxpypzdx2

dy2−z2

1

−1

Figure 5–10: Imaginary part of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉,where η(t) is a superposition of s- and px-waves.

105

PSfrag

replacem

ents

t

∑

M|〈ΨM|Ψ(t)〉|2spx

|〈ΨM|Ψ(t)〉|2

spx

py

pz

dx2

dy2−z2

0

0

10 20 30

−0.5

0.2

0.4

0.6

0.8

−10

−20

−30

0.001−

0.001

1

−1

Re〈Ψ

N |Ψ(t)〉Im〈ΨM|Ψ(t)〉

qK(t)p

K(t)

Figure 5–11: Probability for the electron to be in a particular basis eigenstate.

PSfrag

replacem

entst

spxpypzdx2

dy2−z2

0

0

10202530

−0.5

−0.5

0.5

0.5

−10

−20

−30

0.001−

0.001

1

1

−1

−1

Re〈ΨM|Ψ(t)〉

Im〈ΨM|Ψ(t)〉

qK(t)p

K(t)

Figure 5–12: Phase space of the Schrodinger coefficients CM(t) ≡ 〈ηM|η(t)〉, whereη(t) is a superposition of s- and px-waves.

106

The asymptotic problem, be it electromagnetic radiation or free (ionized) elec-

trons, has posed a difficult numerical challenge. Free electromagnetic radiation in

vacuum does not spread in time, since there is no dispersion, but does travel at the

speed of light c ≈ 137 au. However, the velocity of the sources of charge and current,

e.g., the electron in the hydrogen atom, is on the order of 1 au. This drastically

different velocity scale makes a numerical description of the time-dependent theory

in direct space quite demanding.

On the other hand, the description of the free nonrelativistic electron is made

difficult by a combination of its large velocity v (v < c), the spreading of its

wavepacket, and the rapid oscillation of its phase. Even in vacuum, the Schrodinger

equation is dispersive so that the electronic wavepacket width grows proportionally

with time and its phase grows quadratically with the distance from the center of the

wavepacket. Several techniques have been developed to partially treat these prob-

lems. In 1947, Wigner and Eisenbud [93] developed the R-matrix method, which

provides a technique for matching the solutions on some surface separating the inner

bound state region and outer scattering state region. More recently, masking func-

tions, repetitive projection and complex rotation methods, and Siegert pseudostates

are common theoretical tools. These techniques are discussed by Yoshida, Watan-

abe, Reinhold, and Burgdorfer in [94] and by Tolstikhin, Ostrovsky, and Nakamura

in [95]. A scaling transformation method that eliminates the rapid phase variation

and wavepacket expansion and requires no matching at infinity has been presented

by Sidky and Esry in [96]. Lastly, McCurdy and collaborators [97–99] have effectively

implemented an exterior complex scaling method [100] in the time-independent for-

mulation of scattering theory. The exterior complex scaling method maps all coor-

dinates beyond a certain radius to a contour that is rotated by some fixed angle into

the complex plane. This technique damps all purely outgoing scattered waves to

zero exponentially which permits a numerical treatment on a finite domain or grid.

107

PSfrag

replacem

ents

Local basis

Asymptotic basis

t

φ(t)

Figure 5–13: Schematic picture of the local and asymptotic basis proposed for thedescription of electromagnetic radiation and electron ionization. The amplitudefrom the asymptotic basis is dumped into the free field φ, which acts as a storagetank for energy and probability.

A formulation of the asymptotic numerical problem that falls more in line with

the canonical treatment presented in this dissertation would begin at the Lagrangian

level with a Lagrangian of the form

L = LENRD + Lcoupling + Lfree. (5.42)

The ENRD Lagrangian LENRD would be the Maxwell-Schrodinger Lagrangian from

(1.17) and (1.18). The dynamics of this system would be described by two different

types of basis functions. As pictured in Figure 5–13, the atomic or molecular system

would have a local basis representation in terms of real gaussian basis functions of

the form

GK(x) = G∗K(x) = NK exp(−`K[x − rK]2). (5.4)

Further away, a set of complex basis functions of the form

108

Gk(x) =∑

lmClmYlm(x)eikr

rexp(−ar2) (5.43)

would be used, where the wavevector magnitude k = ω/c could be chosen to lie in

some range kmin ≤ k ≤ kmax and most likely only a few l would be necessary. These

complex basis functions will require the calculation of new matrix elements. The free

Lagrangian Lfree would be the free particle Lagrangian iψ∗ψ − i∇ψ∗ · −i∇ψ/2m

or the free field Lagrangian ∂αφ∗∂αφ. The solutions of the free equations of motion

derived from these free Lagrangians are known analytically and are of the form

exp(i[k · x− ωt]). The coupling Lagrangian should be a Lorentz scalar that is made

up of a certain combination or product of dynamical variables of LENRD and of

Lfree. If amplitude is put into the coefficients of the asymptotic basis functions Gk,

then the amplitude will transfer to the free solutions ψ or φ. This amplitude will

provide an initial condition for the free fields, thereby defining ψ or φ throughout

spacetime. The free fields will store the energy and probability (and momentum and

angular momentum) radiated at infinity, which is needed to maintain the various

conservation laws.

5.5 Proton Dynamics in a Real Basis

The previous complex Schrodinger Lagrangians may be written in real form by

taking the electronic wavefunction Ψ = [Q+ iP ]/√

2 and the protonic wavefunction

Ω = [U + iW ]/√

2. In terms of these real dynamical variables the Hamiltonian

density becomes

H =[−4πcΠ]2 + [∇× A]2 − [4πcΘ]2

8π− c∇Φ · Π− cΘ∇ · A

+[∇Q + qAP/c]2 + [−∇P + qAQ/c]2

4mq+ qΦ

Q2 + P 2

2

+[∇U + qAW/c]2 + [−∇W + qAU/c]2

4mq

+ qΦU2 +W 2

2

(5.44)

109

As in (5.3), each of the dynamical variables may be expanded into a basis. In

particular, the real and imaginary components of Ψ and Ω are expanded as

Q(x, t) =∑

KGK(x)qK(t) U(x, t) =∑

KGK(x)uK(t)

P (x, t) =∑

KGK(x)pK(t) W (x, t) =∑

KGK(x)wK(t).(5.45)

The basis functions GK are chosen to be simple s-gaussians. The Hamilton equations

of motion associated with these real dynamical variables are

0 0 0 −MT 0 0 0

0 0 0 0 −NT 0 0

0 0 0 0 0 −CT 0

0 0 0 0 0 0 −KT

M 0 0 0 0 0 0

0 N 0 0 0 0 0

0 0 C 0 0 0 0

0 0 0 K 0 0 0

a

φ

q

u

π

θ

p

w

=

∂H/∂a

∂H/∂φ

∂H/∂q

∂H/∂u

∂H/∂π

∂H/∂θ

∂H/∂p

∂H/∂w

, (5.30)

where

MKM ≡ ∂2〈Π|A〉∂πK∂aM

NKM ≡ ∂2〈Θ|Φ〉∂θK∂φM

CKM ≡ ∂2〈P |Q〉∂pK∂qM

KKM ≡ ∂2〈W |U〉∂wK∂uM

(5.46)

and where the integrals 〈·|·〉 involve only gaussian functions.

CHAPTER 6CONCLUSION

Nonperturbative analytical and numerical methods for the solution of the non-

linear Maxwell-Schrodinger equations have been presented including the complete

coupling of both systems. The theory begins by applying the calculus of variations

to the Maxwell and Schrodinger Lagrangians together with a gauge fixing term for

the Lorenz gauge. Within the Hamiltonian or canonical prescription, this yields a

set of first order differential equations in time of the form

ωabηb = ∂H/∂ηa. (6.1)

These Maxwell-Schrodinger equations are closed when the Schrodinger wavefunction

is chosen as a source for the electromagnetic field and the electromagnetic field acts

back upon the wavefunction. Moreover, this system of equations forms a well-

defined initial value problem. That is, the entire dynamics is known in principle

once the initial values for each of the dynamical variables η are specified. The

resulting dynamics enjoys conservation of energy, momentum, angular momentum,

and charge between the matter and the electromagnetic field.

In practice, the Maxwell-Schrodinger equations are represented in a finite basis

of gaussian functionsGK(x) and solved numerically. That is, each dynamical variable

is expanded in this basis according to

η(x, t) =∑

KGK(x)ηK(t), (6.2)

where the time-dependent coefficients ηK(t) carry the dynamics. As a result, a hier-

archy of approximate equations of motion are generated that basis-represent the ex-

act Maxwell-Schrodinger equations and can be made systematically more and more

110

111

accurate by enriching the basis. In the limit of a complete basis, these approximate

equations would be exact since the gaussian functions span L2.

The basis representation of the Maxwell-Schodinger equations of motion has

been numerically implemented in a Fortran 90 computer program. This program

allows for an arbitrary rank basis of s-gaussians of varying widths and centers.

It has been used to explore the dynamics of the hydrogen atom interacting with

the electromagnetic field. In particular, stationary states of the combined atom-

field system have been constructed as well as nonstationary states that radiate.

This radiation carries away energy, momentum, and angular momentum from the

hydrogen atom. A series of plots are presented to document the radiative decay of

hydrogen’s electron from a superposition of s and px states to the s ground state.

In order to improve numerical convergence, a canonical transformation was

performed on the Maxwell-Schrodinger equations to isolate the Coulombic or elec-

trostatic contribution to the scalar potential ΦC and electric field EC. This portion

of the fields can be performed analytically once the source ρ is specified by solving

the Poisson equation ∇2ΦC = −4πρ and then calculating EC = −∇ΦC . By remov-

ing the burden of describing both the Coulombic and radiative contributions to the

electrodynamics, the efforts of the basis are focused entirely on the description of the

radiation. The canonical transformed equations of motion have been represented in

a gaussian basis as done previously and have been added to the existing Fortran

90 computer program. With an optional flag the Coulomb reference can be used.

Otherwise the raw numerical basis is used by default. As before, a series of plots

are presented to document the dynamics of the hydrogen atom interacting with the

electromagnetic field. The results in both cases are analyzed.

The work presented in this dissertation is particularly applicable to physical sit-

uations where the dynamics of the sources of charge and current occurs on the same

timescale as the dynamics of the electromagnetic field. In these situations, adiabatic

112

and perturbative approaches may be insufficient to describe the strongly coupled

matter-field dynamics. Possible applications of the Maxwell-Schrodinger theory lie

in photon-electron-phonon dynamics in semiconductor quantum wells [10], sponta-

neous emission in cold atom collisions [11, 12], atom-photon interaction in single

atom laser cavities [14, 15], and photon-exciton dynamics in fluorescent polymers

[16].

APPENDIX AGAUGE TRANSFORMATIONS

A.1 Gauge Symmetry of Electrodynamics

The basic equations of electrodynamics [9] are:

∇ · E = 4πρ ∇× B =4π

cJ +

E

c

∇ · B = 0 ∇× E +B

c= 0,

(A.1)

where ρ and J are the charge and current densities. These Maxwell equations may

be rewritten in terms of the scalar and vector potentials Φ and A as

∇2Φ +∇ · Ac

= −4πρ (A.2a)

∇2A− A

c2−∇

[

∇ ·A +Φ

c

]

= −4π

cJ (A.2b)

by defining the electric field E = −A/c − ∇Φ and the magnetic field B = ∇ × A.

Maxwell’s equations written in either field or potential forms are invariant under the

gauge transformation

Φ → Φ′ = Φ − G

c

A → A′ = A + ∇G,(A.3)

where G is an arbitrary and well-behaved function called the gauge function or

gauge generator. It is said that Maxwell’s equations enjoy the full gauge freedom.

However, a particular gauge may be chosen with an appropriate choice of the gauge

function. For example, a gauge transformation can always be made on the potentials

so that the Lorenz gauge condition is satisfied. That is

0 = ∇ · A′ + Φ′

c= ∇ · A + ∇2G +

Φ

c− G

c(A.4)

113

114

implies that the Lorenz gauge function satisfies the equation G = −[∇ ·A + Φ/c].

This choice of gauge function leads to the manifestly Lorentz invariant equations of

motion:

∇2Φ − Φ

c2= −4πρ (A.5a)

∇2A − A

c2= −4π

cJ. (A.5b)

Another choice is the Coulomb (radiation) gauge, which leads to the equations of

motion:

∇2Φ = −4πρ (A.6a)

∇2AT − AT

c2= −4π

cJT (A.6b)

with the gauge function satisfying ∇2G = −∇ · A. The Coulomb gauge vector

potential AT is the transverse projection of the Lorenz gauge vector potential A, as

is the current JT . That is

AaT (x, t) =[

δab − ∂a∂b

∇2

]

Ab(x, t) =∫

V

d3k

(2π)3eik·x

[

δab − kakb

k2

]

Ab(k, t), (A.7)

where P abT = δab − ∂a∂b/∇2 is the transverse projection operator (see Appendix

C) and 1/∇2 is shorthand for the Green’s function∫

Vd3x′[−4π|x − x′|]−1 of the

Laplacian operator (see Appendix B), where (∇2/∇2)g = g for g a well-behaved

function. More precisely

1

∇2∇2g(x) =

∫

Vd3x′g(x′)∇′2 −1

4π|x − x′| = g(x). (A.8)

The dynamical radiation fields associated with AT are almost separated from the in-

stantaneous or static fields associated with Φ, as seen in (A.6). A closer examination

of the transverse projection operator P abT will show that even AT contains instan-

taneous components. That these instantaneous effects exactly cancel between AT

115

and Φ to produce the causal E and B is one of the often misunderstood properties

of the Coulomb gauge [101].

Notice that the Lorenz gauge equations of motion (A.5) enjoy a limited gauge

freedom known as the residual gauge symmetry. In other words, within the Lorenz

gauge there is still a limited family of gauge transformations that can be made on

the potentials that leave them in the Lorenz gauge. Such gauge generators satisfy

the homogeneous equation G = 0. Similarly, there is residual gauge freedom left in

the Coulomb gauge. That is, there is a limited class of gauge transformations that

can be made on (A.6) that leave them unchanged as well. Such gauge generators

satisfy the homogeneous equation ∇2G = 0.

Many other choices of gauge function are possible, each leading to a different

gauge. A particular gauge is often chosen in accordance with a given physical situa-

tion so as to simplify the associated mathematics. See Cohen-Tannoudji et. al. [30]

for a discussion of other gauges pertinent in the context of atomic and molecular

physics.

A.2 Gauge Symmetry of Quantum Mechanics

In addition to the electrodynamic gauge symmetries of (A.3), quantum mechan-

ics exhibits the additional symmetry

Ψ → Ψ′ = ΨeiqG/c. (A.9)

The coupled system (3.20) and (A.2) are invariant under gauge transformations

(A.3) and (A.9) with the same gauge function G. While there are other gauge in-

variant coupling schemes, the minimal coupling prescription embodied in (3.20) is

the simplest.

APPENDIX BGREEN’S FUNCTIONS

The Green’s function or fundamental solution G associated with the inhomoge-

neous (partial) differential equation Lu(k) = h(k) satisfies the equation

LG(k,k′) = δ(n)(k − k′) (B.1)

where L is a differential operator, u and h are C∞ and L2 ∩ L1 functions, and δ

is the Dirac δ-function defined below. The inhomogeneous term h is often called a

source for u. In this language, the Green’s function G(k,k′) = G(k′,k) is a solution

of the differential equation corresponding to a point-like source, i.e., the δ-function.

Loosely speaking the Green’s function is the inverse of the operator L. The particular

solution u of the differential equation Lu = h may be obtained in principle by

integration against G over all of space. That is

u(k) =∫d(n)k′ h(k′)G(k′,k) (B.2)

which along with the solution of Lu = 0 constitutes the full solution. In other words,

the differential equation Lu = h has been transformed into an integral equation in

which the Green’s function is the kernel. Substitution of this solution into the

differential equation leads to

h(k) = L∫d(n)k′ h(k′)G(k′,k)

=∫d(n)k′ h(k′)δ(n)(k′ − k).

(B.3)

A brief discussion of the δ-function is presented in the next section. With this

knowledge, the Green’s functions for the Laplacian ∇2 and the wave operator ∂2 =

−2 = ∂2/∂(ct)2 −∇2 are derived.

116

117

B.1 The Dirac δ-Function

The Dirac δ-function may be defined in n-dimensions by the volume integral

∫ d(n)k

(2π)n1eik·x = δ(n)(x), (B.4)

where k·x is the Euclidean scalar product. This integral is just the Fourier inverse of

the unit 1. Notice that the δ-function is not a function but rather it is a distribution

which is zero everywhere except at x = 0. It may be used as an integration kernel

to “pluck out” the value of a function at a particular point. For example

g(x) =∫d(n)y g(y)δ(n)(y − x), (B.5)

where δ(n)(y − x) = δ(n)(x − y). A particular choice for the function g such as

g(y) = 1 shows that

∫

Ωd(n)y δ(n)(y − x) =

0, if x lies outside Ω

1, if x lies inside Ω, (B.6)

where Ω is a closed region of integration. Furthermore that if g is well-behaved in

Ω, then∫

Ωd(n)y g(y)

∂

∂yδ(n)(y − x) = −∂g(y)

∂y

∣∣∣y=x

(B.7)

after an integration by parts. If Ω is all of space, then the integral in (B.6) is always

1 and the function g in (B.7) need only vanish at infinity.

Notice that the dimensions of the δ-function must cancel the dimensions of

the differential d(n)y to give a dimensionless result. Hence δ(n) has dimensions of

(length)−n. There are a number of additional properties of the δ-function which will

not be elaborated on here. The interested reader is referred to [7] for a detailed

discussion.

118

B.2 The ∇2 Operator

Consider the Poisson equation from electrodynamics

∇2Φ = −4πρ, (B.8)

where Φ is the scalar potential and ρ is the charge density. The corresponding

Green’s function satisfies the equation

∇2G(x,x′) = δ(x − x′), (B.9)

where the dimensionality of the δ-function has been omitted. By going to the Fourier

space (B.9) is diagonalized and becomes (ik)2G = exp(−ik ·x′). By another Fourier

inversion

G(x,x′) = −∫

V

d3k

(2π)3

eik·(x−x′)

k2=

−1

4π|x− x′| . (B.10)

With (B.10) the Poisson equation (B.8) is recast as the integral equation

Φ(x, t) =∫

Vd3x′[−4πρ(x′, t)]G(x′,x)

=∫

Vd3x′

ρ(x′, t)

|x − x′| .(B.11)

Together with the homogeneous solutions of the Laplace equation ∇2Φ = 0, the

total solution is obtained.

B.3 The ∂2 Operator

Consider the wave equation from electrodynamics

−∂2Φ = Φ = ∇2Φ − Φ

c2= −4πρ, (B.12)

where Φ is the scalar potential and ρ is the charge density as before. The corre-

sponding Green’s function satisfies

[

∇2 − 1

c2∂2

∂t2

]

G(x, t;x′, t′) = δ(x − x′)δ(t− t′). (B.13)

119

The solution of (B.13) will require the Fourier inversion in both x and t to obtain

[k2 −ω2/c2] ˜G = − exp(−i[k ·x′−ωt′]). Division by [k2 −ω2/c2] followed by another

Fourier inversion in both k and ω will give the desired result.

Rather than performing this task in one step, it is beneficial to split the effort

in half. First consider the Fourier inverse of (B.12) in x. That is

¨Φ + (ck)2Φ = 4πc2ρ, (B.14)

with Green’s function satisfying

[

(d2/dt2) + (ck)2]

Dk(t, t′) = δ(t− t′). (B.15)

Fourier inversion in t results in [ω2− c2k2]Dk = − exp(iωt′) from which the retarded

Green’s function becomes

D(+)k (t, t′) = −

∫∞

−∞

dω

2π

e−iω(t−t′)

ω2 − c2k2= Θ(t− t′)

sin ck(t− t′)

ck, (B.16)

where the integration was performed in lower half complex plane. Similarly the

advanced solution is D(−)k (t, t′) = Θ(t′ − t) sin ck(t − t′)/ck by integrating in the

upper half plane.

The solution to (B.13) is more clear, now that this first step has been accom-

plished. It has already been shown that [k2−ω2/c2] ˜G = − exp(−i[k ·x′−ωt′]) must

be Fourier inverted twice in order to obtain G. The resulting Green’s function is

G(x, t;x′, t′) = c2∫

V

d3k

(2π)3eik·(x−x′)

∫∞

−∞

dω

2π

e−iω(t−t′)

ω2 − c2k2

=cΘ(t− t′)

2π2|x − x′|∫∞

0dk sin ck(t− t′) sin(k|x − x′|)

=cΘ(t− t′)

4π|x − x′|[

δ(|x − x′| − c(t− t′)

)− δ

(|x − x′| + c(t− t′)

)]

,

(B.17)

where the retarded Green’s function D(+) in (B.16) was used in the first equality

and a change of variables was used in the last equality. The first term in (B.17) is

120

referred to as the retarded solution

G(+)(x, t;x′, t′) =cΘ(t− t′)

4π|x − x′|δ(|x − x′| − c(t− t′)

)(B.18)

while the second term is just zero since both |x − x′| and c(t − t′) are positive.

An analogous computation with the advanced Green’s function D(−) leads to the

advanced solution

G(−)(x, t;x′, t′) =cΘ(t′ − t)

4π|x− x′|δ(|x − x′| + c(t− t′)

). (B.19)

The retarded solution exhibits the causal properties of field propagation (see Figure

B–1). That is, a disturbance that is observed at the point (x, t) in space-timeP

Sfrag

replacem

ents

r(t)

ct

x

past

elsewhere

future

Figure B–1: The trajectory or world line r(t) of a massive particle moves from pastto future within the light cone. A massless particle such as a photon propagates onthe light cone.

originated from a point that is a distance |x − x′| = c(t − t′) away and at a time

t′ = t − |x − x′|/c earlier. The opposite is true for the advanced solution. With

(B.18) or (B.19) the wave equation (B.12) is recast as the integral equation

Φ(x, t) =∫

Vd3x′

∫∞

−∞dt′[−4πρ(x′, t)]G(±)(x′, t′;x, t)

=∫

Vd3x′

∫∞

−∞dt′

ρ(x′, t′)

|x − x′|δ(

t′ −[

t∓ |x − x′|c

])

.(B.20)

121

Together with the homogeneous solutions of the wave equation Φ = 0, the total

solution is obtained.

APPENDIX CTHE TRANSVERSE PROJECTION OF A(x, t)

It can be seen in both (3.56) and (3.59) that the full vector potential generated

by a charge q moving with velocity v also points in the direction of v. When working

in the Coulomb gauge it is not the full vector potential that is needed but the

transverse projection thereof. The transverse vector potential AT does not flow

in the direction perpendicular to v, but rather the direction perpendicular to the

wavevector k as seen in Figure C–1. The true meaning of transverse and longitudinal

PSfrag

replacem

ents

r(t)

v

k

v⊥

−k(k · v)/k2

v‖ ∼ k(k · v)/k2

Figure C–1: Since A = hv, the transverse vector potential A⊥ = [v− k(k · v)/k2]hand the longitudinal vector potential A‖ = [k(k·v)/k2]h, where h is a scalar function.

is easily visualized by going to the Fourier space. There ik · A⊥ = 0 and ik× A‖ =

0, where A⊥ is the Fourier inverse of AT and A‖ is the Fourier inverse of the

longitudinal AL. The transverse and longitudinal projections satisfy AT +AL = A.

Each component AiT of the AT can be obtained by contraction of the transverse

122

123

projection tensor T ij with the velocity vj. That is

AiT (x, t) = T ij(x, t)vj (C.1)

where T ij(x, t) is related to the transverse projection operator P ijT = δij − ∂i∂j/∇2

according to

P ijT h(x, t)vj

︸ ︷︷ ︸

Aj(x,t)

=[

δij − ∂i∂j

∇2

]

h(x, t)vj = [∂i∂j −∇2δij]−h(x, t)

∇2

︸ ︷︷ ︸

g(x,t)

vj = T ij(x, t)vj.

(C.2)

Note that P ikT = P ij

T PjkT and P† = P. In Fourier space T ij is of the form

T ij(k, t) = [kikj − k2δij]g(k, t)

= [3kikj − k2δij]g(k, t)︸ ︷︷ ︸

Qij(k,t)

+ [−2kikj]g(k, t)︸ ︷︷ ︸

M ij(k,t)

(C.3)

where g distinguishes between the Fourier inverses of A(0), A(1), or A(2). These

vector potentials are the pieces which make up the full potential

A = Θ(t− t1)Θ(t2 − t)[A(0) + A(1)] + Θ(t− t2)[A(1) − A(2)] (C.4)

generated by the current J = qvΘ(t− t1)Θ(t2 − t) exp(−ik · [r + vt] − k2/8`2). The

first instantaneous or Coulomb-like piece is

A(0)(x, t) =v

cΦ(0)(x, t) =

qv

c

erf(√

2`|x − (r + vt)|)

|x − (r + vt)| (C.5)

while the remaining three pairs of radiating terms are of the form

A(k)(x, t) =v

cΦ(k)(x, t) =

qv

2c

erf(√

2`[c(t− tk) − |x − (r + vtk)|])

|x − (r + vtk)|

− erf(√

2`[c(t− tk) + |x − (r + vtk)|])

|x − (r + vtk)|

(C.6)

124

for k = 1, 2. The first term in (C.3) resembles the traceless quadrupole moment

tensor and is labeled Qij, while the second term M ij is the remainder. It will be

seen that the M ij tensor can be completely determined from Qij.

Solving for each component of T ij first and then contracting with the velocity

eliminates the Fourier inversion of

A⊥(k, t) = [k(k · v) − k2v]g(k, t) (C.7)

in favor of the angularly inferior

T ij(x, t) =∫

V

d3k

(2π)3[kikj − k2δij]g(k, t)eik·x, (C.8)

followed by a simple multiplication of vj. In matrix notation the transverse vector

potential is just

AT = Tv = [Q + M]v. (C.9)

Looking only at Qij (although the same is true for both M ij and T ij), it can

be shown that while Qij 6= 0 for all i and j, its Fourier inverse Qij is diagonal in a

certain frame. For example (for A(0))

Q(0)12 (k, t) = 3kxkyg

(0)C (k, t) = 3kxky

−4πq

ck4e−ik·(r+vt)−k2/8`2 6= 0, (C.10)

while

Q(0)12 (x, t) =

∫

V

d3k

(2π)33kxky

−4πq

ck4eik·[x−(r+vt)]−k2/8`2 = 0 (C.11)

in the boosted frame of origin r + vt where x− (r + vt) is rotated about the origin

to lie along the ez-axis. As expected, if x − (r + vt) is placed in general along the

constant vector n0 = ex sin θ0 cosφ0 + ey sin θ0 sinφ0 + ez cos θ0, then the angular

part of Q(0)12 becomes

∫dΩ sin θ cos φ sin θ sinφ eik|x−(r+vt)|[cos θ cos θ0+sin θ sin θ0 cos(φ−φ0)] 6= 0. (C.12)

125

C.1 Tensor Calculus

It will be seen that the transverse projection tensor T ij in (C.8) is diagonal in

the boosted frame of origin r + vt where the general vector x − (r + vt) → x is

rotated about the origin to the new vector x′ = r(0, 0, 1) which lies along the ez-axis.

This spatial rotation is performed via the rotation matrix

Λkl(Ω) =

cos θ cosφ cos θ sinφ − sin θ

− sinφ cosφ 0

sin θ cosφ sin θ sinφ cos θ

, (C.13)

where x′k = Λklxl or in matrix notation x′ = Λx. With Λk

l, the transverse projection

tensor in (C.8) may be rotated to the diagonal frame by

T kl(x) → T ′kl(x′) =∂x′k

∂xi∂x′l

∂xjT ij(Λ−1x′)

= ΛkiΛ

ljT

ij(x)

(C.14)

where ΛkiΛ

ljT

ij = ΛkiT

ij[Λ−1] lj or T′ = ΛTΛ−1. The explicit time-dependence

which is unaffected by the rotation has been dropped for brevity. Applying this

similarity transformation to T ij results in

T ′kl(x′) = ΛkiT

ij(x)[Λ−1] lj =∫

V

d3k

(2π)3eik·xΛk

iTij(k)[Λ−1] lj =

∫

V

d3k′

(2π)3eik

′·x′

T ′kl(k′),

(C.15)

where d3k′ = detΛ d3k = d3k and k′ · x′ = k · x. In this frame the three diagonal

elements of T ′kl are known. They are

T ′11(x′) 0 0

0 T ′22(x′) 0

0 0 T ′33(x′)

=∫

V

d3k′

(2π)3eik

′·x′

T ′11(k′) T ′12(k′) T ′13(k′)

T ′21(k′) T ′22(k′) T ′23(k′)

T ′31(k′) T ′32(k′) T ′33(k′)

.

(C.16)

Each of the elements T ′ii(x′) will be computed in the diagonal frame. How-

ever the physics is not correct until x′ is rotated back to the general position

126

x = r(sin θ cos φ, sin θ sin φ, cos θ). The reverse rotation is obtained by inversion of

the transformation (C.14). The resulting transverse projection tensor becomes

T kl(x) =

cos2 θ cos2 φ cos2 θ sin φ cosφ − sin θ cos θ cos φ

cos2 θ sin φ cosφ cos2 θ sin2 φ − sin θ cos θ sinφ

− sin θ cos θ cosφ − sin θ cos θ sinφ sin2 θ

T ′11(x′)

+

sin2 φ − sin φ cosφ 0

− sinφ cosφ cos2 φ 0

0 0 0

T ′22(x′)

+

sin2 θ cos2 φ sin2 θ sin φ cosφ sin θ cos θ cosφ

sin2 θ sinφ cosφ sin2 θ sin2 φ sin θ cos θ sinφ

sin θ cos θ cosφ sin θ cos θ sin φ cos2 θ

T ′33(x′)

(C.17)

which may be written in matrix form as T = Λ−1T′Λ. The transverse vector poten-

tial AkT (x) is now obtained by the simple contraction T kl(x)vl. Note that the T ′kk(x′)

appearing in (C.17) are scalar functions that only depend upon the norm of x′.

A proper treatment of the tensor calculus reveals that ∇′ · A′T (x′, t) = ∇ ·

AT (x, t). Componentwise that is

∂′kT′kl(x′, t)v′l = ∂iT

ij(x, t)vj. (C.18)

It must still be verified that AT is divergenceless in either frame. It is sufficient to

show that ∂iTijvj = 0. This result is most easily shown by working in the (r, θ, φ)

basis with unit vectors

ex = er sin θ cosφ+ eθ cos θ cosφ− eφ sinφ

ey = er sin θ sinφ+ eθ cos θ sinφ+ eφ cosφ

ez = er cos θ − eθ sin θ.

(C.19)

127

In these spherical coordinates the the velocity becomes

v =

vx

vy

vz

= v1

sin θ cosφ

cos θ cosφ

− sinφ

+ v2

sin θ sin φ

cos θ sinφ

cos φ

+ v3

cos θ

− sin θ

0

(C.20)

and the transverse vector potential is just

AT = v1

T 33 sin θ cos φ

T 11 cos θ cosφ

−T 22 sinφ

+ v2

T 33 sin θ sinφ

T 11 cos θ sinφ

T 22 cosφ

+ v3

T 33 cos θ

−T 11 sin θ

0

. (C.21)

It is not difficult to verify that ∇ · AT = 0 using the spherical divergence.

To summarize the work so far, it was stated that a frame exists where the

transverse projection tensor T ij is diagonal. A spatial rotation was performed to go

to that diagonal frame. The tensor was then rotated back to the arbitrary frame.

There the matrix elements of the general T ij involve the diagonal elements T ′ii as

seen in (C.21). In the following section the three T ′ii terms corresponding to the three

terms in the vector potential A = Θ(t−t1)Θ(t2−t)[A(0)+A(1)]+Θ(t−t2)[A(1)−A(2)]

will be computed.

C.2 T ′kk(x′, t) Integrals

In (C.3) and (C.9) it was shown that the transverse vector potential may be

obtained by contraction of the transverse projection tensor T = Q + M with the

velocity. That is

AiT (x, t) =∫

V

d3k

(2π)3[Qij(k, t) + M ij(k, t)]eik·xvj

=∫

V

d3k

(2π)3[3kikj − k2δij]g(k, t)eik·xvj +

∫

V

d3k

(2π)3[−2kikj]g(k, t)eik·xvj,

(C.22)

128

where it was noticed that Qij is analogous to the traceless quadrupole moment tensor

from electrodynamics [9]. In the diagonal frame Q′ij looks like

Q′ij(x′, t) =

Q′11(x′, t) 0 0

0 Q′22(x′, t) 0

0 0 Q′33(x′, t)

(C.23)

with Q′11 + Q′22 + Q′33 = 0. The elements of the M tensor can all be found from Q.

Recall from (3.58)-(3.59) or (C.4)-(C.6) that the piecewise vector potential

A = Θ(t− t1)Θ(t2 − t)[A(0) + A(1)] + Θ(t− t2)[A(1) − A(2)] (C.24)

is made up of a Coulomb-like piece

A(0)(x, t) =v

cΦ(0)(x, t) =

qv

c

erf(√

2`|x − (r + vt)|)

|x − (r + vt)| (C.25)

and three pairs of radiating terms

A(k)(x, t) =v

cΦ(k)(x, t) =

qv

2c

erf(√

2`[c(t− tk) − |x − (r + vtk)|])

|x − (r + vtk)|

− erf(√

2`[c(t− tk) + |x − (r + vtk)|])

|x − (r + vtk)|

(C.26)

for k = 1, 2. For each of these pieces there is a corresponding transverse projection

tensor. For example, when t1 ≤ t ≤ t2 there is a T(0) = Q(0) + M(0) associated with

A(0) and a T(1) = Q(1) + M(1) associated with A(1). Each of these tensors involve a

Fourier inversion. The resulting integrals are computed below.

129

C.2.1 Inside Step

When t1 ≤ t ≤ t2, the vector potential Ain = A(0) + A(1). The traceless part of

the transverse projection tensor is

Qij(x, t) =∫

V

d3k

(2π)3[3kikj − k2δij]g(0)

C (k, t) + g(1)R (k, t)eik·x

=∫

V

d3k

(2π)3[3kikj − k2δij]

−4πq

ck4e−ik·(r+vt)−k2/8`2

+4πq

ck4e−ik·(r+vt1)−k2/8`2 cos ck(t− t1)

eik·x

(C.27)

and

M ij(x, t) =∫

V

d3k

(2π)3[−2kikj]g(0)

C (k, t) + g(1)R (k, t)eik·x (C.28)

where −k2g(0)C v is the Fourier inverse of A(0) and −k2g

(k)R v is the Fourier inverse of

A(k). Higher order terms in v/c are omitted from (C.27).

Taking g in (C.22) as g(0)C , the Q

(0)ij tensor becomes

Q(0)ij =

∫

V

d3k

(2π)3[3kikj − k2δij]

−4πq

ck4eik·[x−(r+vt)]−k2/8`2 . (C.29)

Using cylindrical symmetry it is found that all off-diagonal elements of Q(0)ij are zero

and Q(0)11 = Q

(0)22 when the vector x − (r + vt) is rotated to the frame where it lies

along the z-axis. But since Q(0)ij is traceless, Q

(0)33 = −2Q

(0)11 . Thus, solving for Q

(0)33

determines Q(0) entirely. That is

Q(0)33 =

∫

V

d3k

(2π)3[3k2

z − k2]−4πq

ck4eik·[x−(r+vt)]−k2/8`2

= −3q

πc

∫∞

0dz cos(|x − (r + vt)|z)e−z2/16`2W−1,−1/2

( z2

8`2

)

+Φ(0)

c

= − 2q`√2πc

2F2

(1

2,3

2;1

2,5

2;−2`2|x − (r + vt)|2

)

+Φ(0)

c

= − 3q

8`2c

erf(√

2`|x − (r + vt)|)

|x − (r + vt)|3 +3q√8π`c

e−2`2[x−(r+vt)]2

|x − (r + vt)|2 +Φ(0)

c,

(C.30)

130

where the primes which denote this diagonal frame have been temporarily omitted

and where Gradshteyn and Ryzhik [87] was used. Notice that Q(0)33 not only de-

termines Q(0)11 and Q

(0)22 , it also determines M

(0)11 , M

(0)22 , and M

(0)33 . Taking M

(0)ij as

M(0)ij = [−2kikj]

−4πq

ck4e−ik·(r+vt)−k2/8`2 (C.31)

it is found that M(0)11 = M

(0)22 = [cQ

(0)33 + 2Φ(0)]/3c and M

(0)33 = −2[cQ

(0)33 − Φ(0)]/3c

with M(0)ij = 0 for i 6= j. With this, the vector potential transverse to A(0) becomes

A(0)T = [Q(0) + M(0)]v. (C.32)

Taking g in (C.22) as g(1)R , the Q

(1)ij tensor becomes

Q(1)ij =

∫

V

d3k

(2π)3[3kikj − k2δij]

4πq

ck4eik·[x−(r+vt1)]−k2/8`2 cos ck(t− t1). (C.33)

Using spherical symmetry it is again found that all off-diagonal elements of Q(1)ij are

zero and Q(1)11 = Q

(1)22 when the vector x− (r + vt1) is chosen to lie along the z-axis.

Again since Q(1)ij is traceless, Q

(1)33 = −2Q

(1)11 . Thus, solving for Q

(1)33 determines Q(1)

entirely. That is

Q(1)33 =

∫

V

d3k

(2π)3[3k2

z − k2]4πq

ck4eik·[x−(r+vt1)]−k2/8`2 cos ck(t− t1)

=12q`√2πc

e−2`2c2(t−t1)2∫ 1

0du u2e−2`2[x−(r+vt1)]2u2

· cosh(4`2c(t− t1)|x − (r + vt1)|u

)+

Φ(1)

c

=3q√8π`c

[c(t− t1) − |x − (r + vt1)|]e−2`2[c(t−t1)+|x−(r+vt1)|]2

|x − (r + vt1)|3

− [c(t− t1) + |x − (r + vt1)|]e−2`2[c(t−t1)−|x−(r+vt1)|]2

|x − (r + vt1)|3

+3q

8`2c

1 + 4`2c2(t− t1)2

|x − (r + vt1)|3

erf(√

2`[c(t− t1) + |x − (r + vt1)|])

− erf(√

2`[c(t− t1) − |x − (r + vt1)|])

+Φ(1)

c,

(C.34)

131

where Gradshteyn and Ryzhik [87] was used. Similarly Q(1)33 determines M

(1)11 , M

(1)22 ,

and M(1)33 . Taking M

(1)ij as

M(1)ij = [−2kikj]

4πq

ck4e−ik·(r+vt1)−k2/8`2 cos ck(t− t1) (C.35)

it is found that M(1)11 = M

(1)22 = [cQ

(1)33 + 2Φ(1)]/3c and M

(1)33 = −2[cQ

(1)33 − Φ(1)]/3c

with M(1)ij = 0 for i 6= j. With this, the vector potential transverse to A(1) becomes

A(1)T = [Q(1) + M(1)]v (C.36)

so that within the step ATin = A

(0)T + A

(1)T .

C.2.2 Outside Step

When t > t2, the vector potential Aout = A(1) −A(2). The traceless part of the

transverse projection tensor is

Qij(x, t) =∫

V

d3k

(2π)3[3kikj − k2δij]g(1)

R (k, t) − g(2)R (k, t)eik·x

=∫

V

d3k

(2π)3[3kikj − k2δij]

4πq

ck4e−ik·(r+vt1)−k2/8`2 cos ck(t− t1)

−4πq

ck4e−ik·(r+vt2)−k2/8`2 cos ck(t− t2)

eik·x

(C.37)

and

M ij(x, t) =∫

V

d3k

(2π)3[−2kikj]g(1)

R (k, t) − g(2)R (k, t)eik·x, (C.38)

where −k2g(k)R v is the Fourier inverse of A(k).

132

Taking g in (C.22) as g(1)R , the zz-component of the tensor Q

(1)ij is

Q(1)33 =

3q√8π`c

[c(t− t1) − |x − (r + vt1)|]e−2`2[c(t−t1)+|x−(r+vt1)|]2

|x − (r + vt1)|3

− [c(t− t1) + |x − (r + vt1)|]e−2`2[c(t−t1)−|x−(r+vt1)|]2

|x − (r + vt1)|3

+3q

8`2c

1 + 4`2c2(t− t1)2

|x − (r + vt1)|3

erf(√

2`[c(t− t1) + |x − (r + vt1)|])

− erf(√

2`[c(t− t1) − |x − (r + vt1)|])

+Φ(1)

c

(C.39)

and as before M(1)11 = M

(1)22 = [cQ

(1)33 +2Φ(1)]/3c and M

(1)33 = −2[cQ

(1)33 −Φ(1)]/3c with

M(1)ij = 0 for i 6= j.

Lastly, taking g in (C.22) as g(2)R , the zz-component of Q

(2)ij is found to be

Q(2)33 =

3q√8π`c

[c(t− t2) − |x − (r + vt2)|]e−2`2[c(t−t2)+|x−(r+vt2)|]2

|x − (r + vt2)|3

− [c(t− t2) + |x − (r + vt2)|]e−2`2[c(t−t2)−|x−(r+vt2)|]2

|x − (r + vt2)|3

+3q

8`2c

1 + 4`2c2(t− t2)2

|x − (r + vt2)|3

erf(√

2`[c(t− t2) + |x − (r + vt2)|])

− erf(√

2`[c(t− t2) − |x − (r + vt2)|])

+Φ(2)

c.

(C.40)

Taking M(2)ij as

M(2)ij = [−2kikj]

4πq

ck4e−ik·(r+vt1)−k2/8`2 cos ck(t− t1) (C.41)

it is found that M(2)11 = M

(2)22 = [cQ

(2)33 + 2Φ(2)]/3c and M

(2)33 = −2[cQ

(2)33 − Φ(2)]/3c

with M(2)ij = 0 for i 6= j. With this, the vector potential transverse to A(2) becomes

A(2)T = [Q(2) + M(2)]v (C.42)

so that outside the step ATout = A

(1)T − A

(2)T .

133

C.3 Building AT (x, t)

With the result of equations (C.30), (C.34), (C.39), and (C.40), the transverse

vector potential can be determined by contraction of Qij +Mij with the velocity vj.

As a result AT becomes

AT = [Q + M]v

= Θ(t− t1)Θ(t2 − t)[A(0)T + A

(1)T ] + Θ(t− t2)[A

(1)T − A

(2)T ].

(C.43)

These results are presented in [88].

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BIOGRAPHICAL SKETCH

David John Masiello was born on October 8th, 1977, in Providence, Rhode

Island and was the only child of John Alfred Masiello and Norma Jean Masiello.

Although his immediate family was small, David was part of a large Italian-American

family that gathered religiously every Sunday for dinner. David’s many cousins

were more like brothers and sisters. Together they spent endless days catching

butterflies, building tree forts, and practicing their artwork under the supervision of

their grandfather.

In search of warm sunshine and blue skies, David’s parents decided to leave

New England. They moved to sunny Florida just in time for David to begin high

school. While in high school David became interested in chemistry and biology, and

entered the University of Florida in 1995 with the intentions of pursuing a career in

medicine. These intentions quickly changed as David found that his deeper questions

could not be answered by these disciplines.

In May of 1999, David received a B.S. degree in mathematics from the Univer-

sity of Florida. During his undergraduate career, David became interested in the

applications of mathematics in the physical sciences. This interest led him to carry

out research in optical physics over three university campuses worldwide.

Always striving for a deeper more fundamental understanding of Nature, David

decided to stay at the University of Florida to earn a Ph.D. under the advisement of

Prof. Yngve Ohrn and Dr. Erik Deumens at the Quantum Theory Project. During

his third year of graduate school David married his college sweetheart, Kathryn

Allida Masiello.

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