Download on the canonical formulation of electrodynamics and wave mechanics

ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND
WAVE MECHANICS
By
DAVID JOHN MASIELLO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR 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 Öhrn 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 dynamics 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. Öhrn 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 Öhrn-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. Maurı́cio
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 Departments 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 scientific 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 quantum 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 Moléculaire 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 CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
CHAPTER
1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
1.4
1.5
1.6
2
Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Historical and Mathematical Background . . . . . . . . . . . . . . 2
1.2.1 Gauge Symmetry of Electrodynamics . . . . . . . . . . . . 3
1.2.2 Gauge Symmetry of Electrodynamics and Wave Mechanics
5
Approaches to the Solution of the Maxwell-Schrödinger Equations
7
Canonical Formulation of the Maxwell-Schrödinger Equations . . . 11
Format of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 14
Notation and Units . . . . . . . . . . . . . . . . . . . . . . . . . . 15
THE DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1
2.2
3
1
Lagrangian Formalism . . . . . . . . . . . . . . . .
2.1.1 Hamilton’s Principle . . . . . . . . . . . . . .
2.1.2 Example: The Harmonic Oscillator in (q k , q̇ k )
2.1.3 Geometry of TQ . . . . . . . . . . . . . . . .
Hamiltonian Formalism . . . . . . . . . . . . . . . .
2.2.1 Example: The Harmonic Oscillator in (q a , pa )
2.2.2 Symplectic Structure and Poisson Brackets .
2.2.3 Geometry of T∗ Q . . . . . . . . . . . . . . .
ELECTRODYNAMICS AND QUANTUM MECHANICS
3.1
3.2
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18
19
21
22
24
25
26
27
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Quantum Mechanics in the Presence of an Electromagnetic Field
3.1.1 Time-Dependent Perturbation Theory . . . . . . . . . . .
3.1.2 Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . .
3.1.3 Absorption of Electromagnetic Radiation by an Atom . .
3.1.4 Quantum Electrodynamics in Brief . . . . . . . . . . . . .
Classical Electrodynamics Specified by the Sources ρ and J . . .
3.2.1 Electromagnetic Radiation from an Oscillating Source . .
3.2.2 Electromagnetic Radiation from a Gaussian Wavepacket .
vi
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29
30
33
34
36
40
41
47
4
CANONICAL STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1
4.2
4.3
4.4
4.5
4.6
4.7
5
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56
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59
61
66
. 66
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69
70
71
78
79
81
NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . 84
5.1
5.2
5.3
5.4
5.5
6
Lagrangian Electrodynamics . . . . . . . . . . . . . . . . . . . .
4.1.1 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . . .
4.1.2 The Lorenz and Coulomb Gauges . . . . . . . . . . . . .
Hamiltonian Electrodynamics . . . . . . . . . . . . . . . . . . .
4.2.1 Hamiltonian Formulation of the Lorenz Gauge . . . . . .
4.2.2 Poisson Bracket for Electrodynamics . . . . . . . . . . . .
Hamiltonian Electrodynamics and Wave Mechanics in Complex
Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hamiltonian Electrodynamics and Wave Mechanics in Real Phase
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Coulomb Reference by Canonical Transformation . . . . . .
4.5.1 Symplectic Transformation to the Coulomb Reference . .
4.5.2 The Coulomb Reference by Change of Variable . . . . . .
Electron Spin in the Pauli Theory . . . . . . . . . . . . . . . . .
Proton Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
Maxwell-Schrödinger Theory in a Complex Basis . . . .
Maxwell-Schrödinger Theory in a Real Basis . . . . . .
5.2.1 Overview of Computer Program . . . . . . . . .
5.2.2 Stationary States: s- and p-Waves . . . . . . . .
5.2.3 Nonstationary State: Mixture of s- and p-Waves
5.2.4 Free Electrodynamics . . . . . . . . . . . . . . .
5.2.5 Analysis of Solutions in Numerical Basis . . . .
Symplectic Transformation to the Coulomb Reference .
5.3.1 Numerical Implementation . . . . . . . . . . . .
5.3.2 Stationary States: s- and p-Waves . . . . . . . .
5.3.3 Nonstationary State: Mixture of s- and p-Waves
5.3.4 Free Electrodynamics . . . . . . . . . . . . . . .
5.3.5 Analysis of Solutions in Coulomb Basis . . . . .
Asymptotic Radiation . . . . . . . . . . . . . . . . . .
Proton Dynamics in a Real Basis . . . . . . . . . . . .
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85
88
90
93
93
93
95
99
101
102
102
103
103
103
108
CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
APPENDIX
A
GAUGE TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . 113
A.1
A.2
Gauge Symmetry of Electrodynamics . . . . . . . . . . . . . . . . 113
Gauge Symmetry of Quantum Mechanics . . . . . . . . . . . . . . 115
vii
B
GREEN’S FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B.1
B.2
B.3
C
The Dirac δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . 117
The ∇2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
The ∂ 2 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
THE TRANSVERSE PROJECTION OF A(x, t) . . . . . . . . . . . . . 122
C.1
C.2
C.3
Tensor Calculus . .
T 0kk (x0 , t) Integrals
C.2.1 Inside Step .
C.2.2 Outside Step
Building AT (x, t) .
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125
127
129
131
133
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
viii
Figure
LIST OF FIGURES
page
2–1 The configuration manifold Q = S2 is depicted together with the
tangent plane Tqk Q at the point q k ∈ Q. . . . . . . . . . . . . . . . 22
3–1 The coefficient 1 of the unscattered plane wave exp(ik · x) is analogous to the 1 part of the S-matrix, while the scattering amplitude
fk (Ω) which modulates the scattered spherical wave exp(ikr)/r is
analogous to the iT part. . . . . . . . . . . . . . . . . . . . . . . . 38
3–2 In the radiation zone, the observation point x is located far from the
source J. In this case the distance |x − x0 | ≈ r − n̂ · x0 . . . . . . . . 44
3–3 The differential power dP/dΩ or radiation pattern corresponding to
an oscillating electric dipole verifies that no radiation is emitted in
the direction of the dipole moment. . . . . . . . . . . . . . . . . . . 46
3–4 The norms of J and A are plotted with different velocities along the
x-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 wavepacket
moves out on the smeared light cone with maximum at x = ct. . . . 50
3–7 The radiation pattern given by (3.63) shows the characteristic dipole
pattern at lowest order. . . . . . . . . . . . . . . . . . . . . . . . . 53
4–1 A limited but relevant portion of the gauge story in the Lagrangian
formalism is organized in this picture. . . . . . . . . . . . . . . . . 59
4–2 The Hamiltonian formulation of the gauge story is organized in this
picture with respect to the previous Lagrangian formulation. . . . . 65
4–3 Commutivity diagram representing the change of coordinates (q, p) to
(p̃, q̃) 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 and
its momentum Π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5–3 Phase space contour for the coefficients of the real-valued Schrödinger
field Q and its momentum P. . . . . . . . . . . . . . . . . . . . . . 94
ix
5–4 Phase space contour for the coefficients of the scalar potential Φ and
its momentum Θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5–5 Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 97
5–6 Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i,
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 Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 98
5–9 Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 104
5–10 Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i,
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 Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves. . . . . . . . . . . . . . . 105
5–13 Schematic picture of the local and asymptotic basis proposed for the
description of electromagnetic radiation and electron ionization. . . 107
B–1 The trajectory or world line r(t) of a massive particle moves from past
to future within the light cone. . . . . . . . . . . . . . . . . . . . . 120
C–1 Since à = h̃v, the transverse vector potential Ã⊥ = [v − k(k · v)/k 2 ]h̃
and the longitudinal vector potential Ãk = [k(k · v)/k 2 ]h̃, where h̃
is a scalar function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
x
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ON THE CANONICAL FORMULATION OF ELECTRODYNAMICS AND
WAVE MECHANICS
By
David John Masiello
May 2004
Chair: Nils Yngve Öhrn
Major 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 MaxwellSchrödinger 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 Schrödinger Lagrangians together with a Lorenz gauge fixing term, yields a
set of first order Hamilton equations which form a well-posed initial value problem. 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 Schrödinger wavefunction is chosen to be the
source for the electromagnetic field and the electromagnetic field reacts back upon
the wavefunction.
In practice, the Maxwell-Schrödinger 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-Schrödinger 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-Schrödinger system is investigated
for the spinless hydrogen atom interacting with the electromagnetic field.
xii
CHAPTER 1
INTRODUCTION
Chemistry encompasses a broad range of Nature that varies over orders of magnitude 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 Schrödinger equation [7, 8]
iΨ̇ = HΨ
(1.1)
and Maxwell’s equations [9]
∇ · E = 4πρ
∇×B =
4π
Ė
J+
c
c
∇·B = 0
∇×E+
Ḃ
= 0. (1.2)
c
As they stand these equations are uncoupled. The solutions of the Schrödinger
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-Schrödinger 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 MaxwellSchrödinger 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-Schrödinger equations dates back to the early twentieth century when the founding fathers of quantum mechanics worked out the theoretical 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 − Ȧ/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 mathematical 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
∇2 A −
h
Φ̇ i
4π
Ä
−
∇
∇
·
A
+
= − J
2
c
c
c
∇ · Ȧ
∇2 Φ +
= −4πρ.
c
(1.4a)
(1.4b)
The homogeneous Maxwell equations are identically satisfied. These potential equations enjoy a symmetry that is not present in the field equations (1.2). This symmetry is called the gauge symmetry and can be generated by the transformation
A → A0 = A + ∇F
Φ → Φ0 = Φ − Ḟ /c,
(1.5)
where F is a well-behaved but otherwise arbitrary function called the gauge generator. 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 invariant 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)
∇2 A −
h
Ä
Φ̇ i
4π
−
∇
∇
·
A
+
=− J
2
c
c
c
∇ · Ȧ
= −4πρ
∇2 Φ +
c
(1.4)
form an ill-posed initial value problem. However, they can be converted to a welldefined initial value problem by adding an equation of constraint to them. For
example, adding the constraint Φ̇/c + ∇ · A = 0 leads to the well-defined Lorenz
gauge equations
∇2 A −
Ä
4π
=− J
2
c
c
∇2 Φ −
Φ̈
= −4πρ
c2
(1.6)
while adding ∇ · A = 0 leads to the well-defined Coulomb gauge equations
∇2 A −
Ä
4π
= − JT
2
c
c
∇2 Φ = −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 particular 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 Schrödinger 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
+ V + qΦ,
2m
(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 Schrödinger’s equation and Maxwell’s equations in potential form can be
generated by the transformation
A → A0 = A + ∇F
Φ → Φ0 = Φ − Ḟ /c
Ψ → Ψ0 = 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 wavefunction is indeterminate up to a global phase. Application of the gauge transformation (1.9) to the Schrödinger equation with Hamiltonian (1.8) and to Maxwell’s
equations in potential form leads to exactly the same equations after the transformation. Therefore, like the potential equations (1.4) by themselves, the system of
Maxwell-Schrödinger equations
iΨ̇ =
∇2 A −
[P − qA/c]2 Ψ
+ V Ψ + qΦΨ
2m
h
Φ̇ i
4π
Ä
−
∇
∇
·
A
+
= − J
2
c
c
c
∇ · Ȧ
= −4πρ
∇2 Φ +
c
(1.10)
(1.11a)
(1.11b)
is invariant under the gauge transformation (1.9). There are several other symmetries 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-Schrödinger equations enjoy charge, momentum, angular momentum, and energy conservation. That each continuous symmetry gives rise to
an associated conservation law was proven by Emmy Noether in 1918 (see Goldstein [27], José 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-Schrödinger equations are obtainable as the nonrelativistic limit of the Maxwell-Dirac equations
iΨ̇D = βmc2 ΨD + cα · [P − qA/c]ΨD + qΦΨD
h
4π
Ä
Φ̇ i
∇ A− 2 −∇ ∇·A+
= − J
c
c
c
∇
·
Ȧ
= −4πρ
∇2 Φ +
c
2
(1.12)
(1.13a)
(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 spin1/2. 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-Schrödinger equations and in addition is invariant under relativistic boosts.
1.3
Approaches to the Solution of the Maxwell-Schrödinger Equations
Solving the Maxwell-Schrödinger 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 commonly 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 the
electromagnetic field according to (1.2).
• The dynamics of the electromagnetic field is specified and modifies the dynamics
of the matter according to (1.1).
It is not surprising that either of these approaches is valid in many physical situations. 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-Schrödinger equations or its relativistic analog can be
closed and is coupled when the Schrödinger 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
Schrödinger equation according to
ρ = qΨ∗ Ψ
J = q Ψ∗ [−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 Schrödinger wavefunction as the source for the electromagnetic field was Schrödinger’s electromagnetic hypothesis, which dates back to
1926. The discovery of the quantum mechanical continuity equation and its similarity 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. Schrödinger’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 probabilistic nature of measurement of the wavefunction. In addition, certain properties of
electromagnetic radiation were found to be inconsistent with experiment.
9
Schrödinger’s electromagnetic hypothesis was extended by Fermi in 1927 and
later by Crisp and Jaynes in 1969 [36] to incorporate the unquantized electromagnetic self-fields into the theory. That is, the classical electromagnetic fields produced 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 theories 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 correct. 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 electrodynamics for Schrödinger, Pauli, Klein-Gordon, and Dirac matter theories. They
were able to eliminate all electromagnetic variables in favor of Green’s function integrals 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 Došlić and
10
Bosanac [48] argued that the instantaneous effects of the self interaction are unphysical. 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 interaction 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 electrons 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 semiclassical 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=
R
d3 xqΦ(x, t)Ψ∗ (x, t)Ψ(x, t) =
V
R
d3 x
V
R
d 3 x0
V
ρ(x, t)ρ(x0 , t)
.
|x − x0 |
(1.15)
As a result of the qΦ term, the Schrödinger equation (1.10) is nonlinear in Ψ. It
resembles the nonlinear Schrödinger equation [66]
iu̇ = −a(d2 u/dx2 ) + b|u|2 u
(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 equation.
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. Consequently, 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-Schrödinger 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 mechanics that has not yet been fully explored. To this end, the Hamiltonian or canonical formulation of the Maxwell-Schrödinger dynamics is constructed in this dissertation. (Canonical means according to the canons, i.e. standard or conventional.) The
associated work involves nonperturbative analytic and numerical methods for the solution of the coupled and closed nonlinear system of Maxwell-Schrödinger 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 Schrödinger Lagrangian
LSch = iΨ∗ Ψ̇ −
[i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ
− V Ψ∗ Ψ − qΦΨ∗ Ψ,
2m
(1.17)
and Maxwell Lagrangian together with a Lorenz gauge fixing term, i.e.,
[Φ̇/c + ∇ · A]2
8π
[−Ȧ/c − ∇Φ]2 − [∇ × A]2 [Φ̇/c + ∇ · A]2
=
−
.
8π
8π
LLMax = LMax −
(1.18)
This yields a set of coupled nonlinear first order differential equations of the form
ω η̇ = ∂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-Schrödinger 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
Schrödinger 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 ,
P
i.e., η(x, t) = K GK (x)ηK (t), where the time-dependent superposition coefficients
ηK (t) carry the dynamics, the time-dependent variational principle generates a hierarchy of approximations to the coupled Maxwell-Schrödinger equations. In the limit
of a complete basis these equations recover the exact Maxwell-Schrödinger 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 computer 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 = −∇2 Φ = 4πρ, once the initial conditions are provided) by applying a
certain canonical transformation to the dynamical equations. The canonical transformation 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 ω̃ η̃˙ = ∂ H̃/∂ η̃, have been added to the existing computer program
and the convergence of the solution of the Maxwell-Schrödinger 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 particular, 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 momentum 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 Schrödinger and Maxwell dynamics will be presented from the
point of view of perturbation theory. In the Schrödinger 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 transitions 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 Hamiltonian or canonical approach to the Maxwell-Schrödinger 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-Schrödinger 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-Schrödinger 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 addition, 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 = g αβ with elements g00 = g 00 = 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 = gµν Aν = (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., F̃ is the Fourier transform of F.
The representation independent Dirac notation |hi will be used in the discussion of
time-dependent perturbation theory, but for the most part functions h(x) = hx|hi
16
or h̃(k) = hk|hi 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 2
THE 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 equations, each having subtle differences [72]. However, all methods rely on the machinery inherent in variational calculus. Given a starting and ending point for the
dynamics, the calculus of variations determines the path connecting them. The dynamics 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 presented for discrete and continuous systems. The Lagrangian approach leads to second 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 equivalent in either case. However, the Hamiltonian approach enjoys a rich mathematical
foundation connecting differential geometry and dynamics [28, 29]. Much of the remainder of this dissertation will be devoted to the canonical formulation of Maxwell
and Schrödinger theories.
The time-dependent variational principle [73], which has its origin in nuclear
physics [74], is the variational approach to the determination of the Schrödinger
equation. The Hamiltonian dynamics associated with the Schrödinger 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 described 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 MaxwellSchrödinger 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 q k and q̇ k (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 symmetry 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 translation 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(q k , q̇ k , t) dependent upon the N position vectors q k , the
N velocity vectors q̇ k , and also the time t, the action I is defined by the path integral
I(q k , q̇ k , t) ≡
R t2
t1
L(q k , q̇ k , 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
R t2
t1
Ldt = 0,
(2.2)
where δI is the variation of the action I. Only those paths are varied for which
δq k (t1 ) = 0 = δq k (t2 ). A particular form of the variational path parametrized by the
infinitesimal parameter α is given by
q k (t, α) = q k (t, 0) + αη k (t),
(2.3)
where q k (t) = q k (t, 0) is the correct path of the motion and the vectors η k (t) are
well-behaved and vanish at the boundaries t1 and t2 . By continuously deforming
q k (t, α) until it is extremized, the correct path can be found.
This parametrization of the path in turn parametrizes the action itself. Equation (2.2) may now be rewritten more precisely as
δI(α) =
∂I(α) dα = 0
∂α α=0
(2.4)
20
which represents infinitesimal variations from the correct path. The calculus of
variations yields
∂L ∂ q̇ k o
∂I(α) R t2 n ∂L ∂q k
= t1 dt
+ k
∂α
∂q k ∂α
∂ q̇ ∂α
n ∂L
k t2
R
d ∂L o ∂q k
∂L ∂q t2
−
,
= k
+ t1 dt
∂ q̇ ∂α t1
∂q k dt ∂ q̇ k ∂α
(2.5)
where a partial integration was performed in the second line. Since δq k (t1 ) = 0 =
δq k (t2 ), the surface term vanishes. The stationary point of the variation is therefore
determined by
R t2 n ∂L
d ∂L o ∂q k = 0.
−
dt
t1
∂q k dt ∂ q̇ k ∂α α=0
(2.6)
But since the vectors ∂q k /∂α are arbitrary (choose in particular ∂q k /∂α > 0 and
continuous on [t1 , t2 ]), the integral is zero only when
∂L
d ∂L
−
=0
k
∂q
dt ∂ q̇ k
(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 (q k , q̇ k ). 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 EulerLagrange equations can be seen be expanding the total time derivative to give
∂L n ∂ 2 L l
∂2L l
∂2L o
−
q̈
+
q̇
+
= 0.
∂q k
∂ q̇ l ∂ q̇ k
∂q l ∂ q̇ k
∂t∂ q̇ k
(2.8)
It will always be assumed unless otherwise noted that the Hessian condition is satisfied. That is det{∂ 2 L/∂ q̇ l ∂ q̇ k } 6= 0.
Lastly, notice that the Lagrangian is arbitrary up to the addition of a total time
derivative. That is, if L → L0 = L + (d/dt)K for K a well-behaved function of the
21
dynamical variables, then the action
δI → δ
R t2
t1
{L + (d/dt)K}dt = δK(t2 ) − δK(t1 ) +
R t2
t1
δL dt =
R t2
t1
δL dt = δI
(2.9)
since δK(t2 ) = 0 = δK(t1 ). Thus the same Euler-Lagrange equations (2.7) are
generated for L0 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 (q k , q̇ k )
It is now useful to present a brief illustrative example. In two freedoms, the dynamics of a scalar mass subjected to the force of a harmonic potential with frequency
ωk is determined by the Lagrangian (no summation)
1
1
L(q k , q̇ k ) = mq̇ k q̇ k − mωk2 q k q k
2
2
k = 1, 2
(2.10)
which is a function of the real-valued vectors q k and q̇ k . Application of the calculus
of variations to the associated action functional leads to (2.7) with ∂L/∂ q̇ k = mq̇ k
and ∂L/∂q k = −mωk2 q k . The second order Euler-Lagrange equations of motion are
1 k 1
mq̈ + mωk2 q k = 0
2
2
k = 1, 2
(2.11)
with initial value solution q k (t) = q k (t0 ) cos(ωk t) + q̇ k (t0 ) sin(ωk t)/ωk . It is said that
q k is an integral curve of the dynamical equation (2.11). Once the initial values
q k (t0 ) and q̇ k (t0 ) are provided, the dynamics of the harmonic oscillator is known.
This dynamics occurs in a space whose coordinates are not just the q k , but both the
q k and q̇ k . Some geometric aspects of this space will now be presented.
g replacements
22
Tq k Q
qk
Q
Figure 2–1: The configuration manifold Q = S2 is depicted together with the tangent
plane Tqk Q at the point q k ∈ 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 (q k , q̇ k ). The position vectors q k 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 q k ∈ Q is called the tangent fiber above q k or
the tangent plane at q k and is denoted by Tqk Q. The union of the configuration
manifold Q and the collection of all fibers Tqk Q for each point q k ∈ Q (together
with local charts on Tqk Q) 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 q k reaches out of S2 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 ≡ q̇ k (∂/∂q k ) + q̈ k (∂/∂ q̇ k ), where q̇ k and q̈ k are
the components of ∆L and ∂/∂q k and ∂/∂ q̇ k form a local basis for ∆L . The time
dependence of a dynamical variable F (q k , q̇ k ), which is an implicitly time-dependent
function on TQ, is determined by its variation along the dynamics. That is
Ḟ (q k , q̇ k ) ≡ ∆L (F ) =
∂F k ∂F k
q̇ + k q̈ .
∂q k
∂ q̇
(2.12)
The accelerations q̈ k 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 geometrical 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 = A1a dq a + A2a dq̇ a is applied to the vector field X = X1b (∂/∂q b ) + X2b (∂/∂ q̇ b ), then
their inner product results in
hα|Xi = A1a X1b dq a (∂/∂q b ) + A1a X2b dq a (∂/∂ q̇ b ) + A2a X1b dq̇ a (∂/∂q b ) + A2a X2b dq̇ a (∂/∂ q̇ b )
= A1a X1b δ ab + A2a X2b δ ab
= A1a X1a + A2a X2a ,
(2.13)
where dq a (∂/∂q b ) = dq̇ a (∂/∂ q̇ b ) = δ ab and dq a (∂/∂ q̇ b ) = dq̇ a (∂/∂q b ) = 0, and where
Aa and X b are the local components, which are functions, of the one-form α and
vector field X. It is common to write hα|Xi ≡ α(X). It should be pointed out that
the differential of a function is a one-form. That is
dF =
∂F k ∂F k
dq + k dq̇
∂q k
∂ q̇
(2.14)
24
is a one-form and may be applied to the dynamical vector field ∆L to give
dF (∆L ) ≡ hdF |∆L i =
∂F k ∂F k
q̇ + k q̈ = Ḟ .
∂q k
∂ q̇
(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 q a and pa , which are governed
by the dynamical equations
q̇ a =
∂H
∂pa
and
− ṗa =
∂H
,
∂q a
(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 (q a , q̇ a ) to (q a , pa ) is accomplished by a Legendre transformation [27, 28]. The momentum conjugate to the vector q a is defined in terms of
the Lagrangian L by
pa ≡
∂L(q a , q̇ a )
.
∂ q̇ a
(2.17)
Notice that this conjugate momentum is not a vector as is the velocity q̇ a and does
not lie in the tangent manifold TQ. Rather the momentum pa is dual to the position
vector q a . 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(q a , pa ) = pa q̇ a (q a , pa ) − L(q a , pa ).
(2.18)
Here it is assumed that the relation (2.17) can be inverted to solve for the velocity
q̇ a . 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 = pa q̇ a − H from (2.18), i.e.,
I(q a , pa ) ≡
R t2
t1
[pa q̇ a (q a , pa ) − H(q a , pa )]dt,
(2.19)
then Hamilton’s equations follow in a straightforward manner.
2.2.1
Example: The Harmonic Oscillator in (q a , 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
1
1
L(q a , q̇ a ) = mq̇ a q̇ a − mωa2 q a q a
2
2
a = 1, 2.
(2.10)
The momentum conjugate to q a is pa ≡ ∂L/∂ q̇ a = mq̇ a and with it the Hamiltonian
becomes
H(q a , pa ) = pa q̇ a −
o p p
np p
1
mωa2 q a q a
a a
a a
− mωa2 q a q a =
+
.
2m
2
2m
2
(2.20)
With this Hamiltonian the equations of motion are:
∂H
= pa /m
∂pa
∂H
−ṗa = a = mωa2 q a
∂q
q̇ a =
(2.21)
and have the initial value solutions q a (t) = q a (t0 ) cos(ωa t) + q̇ a (t0 ) sin(ωa t)/ωa and
pa (t) = pa (t0 ) cos(ωa t) − mωa q a (t0 ) sin(ωa t). 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
ξ a = pa−N
(2.22)
a = N + 1, . . . , 2N.
Similarly, the forces become ∂H/∂pa ≡ ∂H/∂ξ a+N and ∂H/∂q a ≡ ∂H/∂ξ a so that
the equations of motion are:
∂H
ξ˙a = a+N
∂ξ
∂H
−ξ˙a = a−N
∂ξ
a = 1, . . . , N
(2.23)
a = N + 1, . . . , 2N.
These Hamilton equations may be written more compactly as
∂H
ωab ξ˙b = a ,
∂ξ
(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 − 1 N
,
1N
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:
ω 2 = −1
and
ω T = −ω,
(2.26)
or equivalently ω ab ωbc = δ ac and ωab = −ωba . The matrix element ω ab with both
indices up is the inverse of ωab .
27
In (2.12), the time derivative or variation of a (implicitly time-dependent) dynamical variable F on TQ was demonstrated. In a similar fashion, Ḟ can be viewed
in the momentum phase space T∗ Q, which will be discussed shortly. It is
Ḟ (q b , ṗb ) =
∂F ˙b ∂F ba ∂H
ξ = bω
,
∂ξ b
∂ξ
∂ξ a
(2.27)
where the equation of motion (2.24) was inverted and substituted for ξ˙b . 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 ∂G
∂F ∂G
∂F ba ∂G
ω
= a
−
.
b
a
∂ξ
∂ξ
∂q ∂pa ∂pa ∂q a
(2.28)
In particular, an alternative form of Hamilton’s equations is derived when the Poisson bracket is applied to the coordinate ξ. That is
ξ˙a = {ξ a , H}.
(2.29)
Since the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity
{f, gh} = g{f, h} + {f, g}h, 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 equations 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 different phase space whose components are the functions ω ba (∂H/∂ξ a ). The integral
curves of this vector field are the dynamics.
Recall that the points of TQ are made up of q k and q̇ k . The velocities q̇ k are the
local components of the vector field q̇ k (∂/∂q k ). However, the momenta are the local
components of the one-form pa dq a ≡ (∂L/∂ q̇ a )dq a , which are not the components
28
of a vector field. Since one-forms are dual to vector fields, pa dq a lies in the dual
space of Tqa Q. This space is the cotangent space at q a and is denoted by T∗qa Q. In
analogy with TQ, the cotangent bundle or cotangent manifold T∗ Q is made up of
Q together with its cotangent spaces T∗qa Q. 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 ≡ ξ˙b
∂
∂
∂H ∂
∂H ∂
a ∂
=
q̇
+
ṗ
=
−
,
a
∂ξ b
∂q a
∂pa
∂pa ∂q a ∂q a ∂pa
(2.30)
where Hamilton’s equations of motion (2.16) were substituted for the q̇ a and ṗa .
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 = X a (∂/∂ξ a ) and Y = Y b (∂/∂ξ b )
are vector fields on T∗ Q, then
ω(X, Y ) = X a Y b ω(∂/∂ξ a , ∂/∂ξ b ) = X a ωab Y b = X a Ya − Y a Xa .
(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
∂H
ωab ξ˙b = a .
∂ξ
More will be said on canonical transformations in Chapter 4.
(2.24)
CHAPTER 3
ELECTRODYNAMICS 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 the
electromagnetic 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 quantummatter/classical-field descriptions are available as well as fully quantum and relativistically 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 introduced 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 transitions between two stationary states |Ψk i and |Ψm i of an atomic system, and is
symbolized to lowest order by the matrix element Vkm = hΨk |V |Ψm i. Experimental observables such as the rate of transition or absorption cross section may be
calculated from Vkm . Additionally, time-dependent perturbation theory gives a prescription for calculating successively higher order corrections to Vkm , which may
in turn provide better and better agreement with experiment. This section concludes 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 Schrödinger equation (1.1) are those
which are eigenfunctions of the Hamiltonian operator H. These solutions |Ψm i satisfy
the time-independent Schrödinger equation
H|Ψm i = Em |Ψm i
(3.1)
and are called stationary states. A general solution |Ψ(t)i of the Schrödinger equation (1.1) may be constructed from these stationary states according to
|Ψ(t)i = e−iH(t−t0 ) |Ψ(t0 )i =
P
me
−iEm (t−t0 )
|Ψm ihΨm |Ψ(t0 )i,
(3.2)
31
where |Ψ(t0 )i is an initial state vector and where the sum over m may imply integration 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 i = Em |Ψm i,
(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(iH0 t) to |Ψ(t)i. That is
|ΨI (t)i = eiH0 t |Ψ(t)i
(3.5)
and the time-dependent Schrödinger equation (1.1) becomes
i(d/dt)|ΨI (t)i = VI (t)|ΨI (t)i,
(3.6)
32
where VI (t) = U0 V (t)U0† . 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)i = UI (t, t0 )|ΨI (t0 )i,
(3.7)
where UI is the time evolution operator which satisfies
UI (t, t0 ) = 1 − i
Rt
t0
dt0 VI (t0 )UI (t0 , t0 ).
(3.8)
The time evolution operator connects the (orthonormal) stationary states |Ψk i and
|Ψm i according to
where
P
Rt
dt0 hΨk |VI (t0 )UI (t0 , t0 )|Ψm i
P Rt
(3.9)
= δkm − i n t0 dt0 hΨk |VI (t0 )|Ψn ihΨn |UI (t0 , t0 )|Ψm i
P Rt
0
= δkm − i n t0 dt0 eiωkn t hΨk |V (t0 )|Ψn ihΨn |UI (t0 , t0 )|Ψm i,
hΨk |UI (t, t0 )|Ψm i = δkm − i
n
t0
|Ψn ihΨn | = 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
Rt
dt0 VI (t0 ) + (−i)2
t0
Rt
dt0 VI (t0 )
t0
R t0
t0
dt00 VI (t00 ) + · · · .
(3.10)
And so at first order the transition amplitude between two distinct states of energy
Ek and Em (with k 6= m) is
hΨk |UI (t, t0 )|Ψm i = −i
Rt
t0
0
dt0 eiωkm t hΨk |V (t0 )|Ψm i.
(3.11)
33
Assuming that the perturbation is sufficiently small, the probability of finding the
system in the state |Ψk i is given by
2
2 Rt
0
Pk←m (t) = hΨk |UI (t, t0 )|Ψm i = − i t0 dt0 eiωkm t hΨk |V (t0 )|Ψm i .
(3.12)
If the perturbation is localized in time then t0 and t may be naively extended to
infinity to yield the transition probability
2
2 R
∞
Pk←m (+∞) = hΨk |UI (+∞, −∞)|Ψm i = − i −∞ dt eiωkm t hΨk |V (t)|Ψm i (3.13)
which involves a Fourier integral of the matrix element Vkm = hΨk |V (t)|Ψm i.
3.1.2
Fermi Golden Rule
The formalism set up thus far is also applicable for time-independent perturbations V 6= V (t). In this case the transition probability can be obtained from (3.12)
as
Pk←m (t) = 2|hΨk |V |Ψm i|2
1 − cos(ωkm t)
(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
P
k∈F Pk←m (t) =
R Em +∆E/2
Em −∆E/2
2|hΨk |V |Ψm i|2
1 − cos(ωkm t)
ρF (Ek )dEk ,
(Ek − Em )2
(3.15)
where the sum runs over all states |Ψk i belonging to the near continuum of final
states. The quotient [1−cos(ωkm t)]/(Ek −Em )2 is sharply peaked at Ek = Em which
confirms that the dominant transitions are those that conserve the unperturbed
energy. Since both |hΨk |V |Ψm i|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
P
k∈F Pk←m (t)
≈ 2|hΨk |V |Ψm i|2 ρF (Ek )
2
R ∞ 1 − cos(ωkm t)
dωkm
2
−∞
ωkm
(3.16)
= 2πt|hΨk |V |Ψm i| ρF (Ek )
which increases linearly with time.
The total transition probability per unit time or transition rate Γ is given by
Γ = (d/dt)
P
k∈F Pk←m (t)
= 2π|hΨk |V |Ψm i|2 ρF (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 transitions 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 harmonically in frequency ω. An electromagnetic disturbance of a charge would be an
example. In this case the golden rule generalizes to
Γ = 2π|hΨk |V |Ψm i|2 ρF (Em + ω).
3.1.3
(3.18)
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
(3.19)
−i∇ → −i∇ − qA/c,
where Aα = (Φ, A) are the dynamical variables of the electromagnetic field. Applying this transformation to the Schrödinger equation iΨ̇ = P2 Ψ/2m + V0 Ψ results in
the Schrödinger equation coupled to the electromagnetic field
iΨ̇ =
[P − qA/c]2
Ψ + V0 Ψ + qΦΨ
2m
(3.20)
35
with Hamiltonian
H=
[P − qA/c]2
+ V0 + qΦ.
2m
(3.21)
In the Coulomb gauge (see Appendix A for details) this Hamiltonian becomes
H=
q
q2
P2
+ V0 −
A·P+
A2 .
2m
mc
2mc2
(3.22)
The external free electromagnetic field evolves according to ∇2 A − Ä/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
+ V0
2m
(3.23)
V =−
q
A · P.
mc
(3.24)
plus the perturbation
It should be pointed out here that substantial confusion has existed in the literature 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 harmonically in frequency as a plane wave, then the perturbation becomes
V (x, t) = −
q A0 ei(k·x−ωt) + A∗0 e−i(k·x−ωt) ˆ · P
mc
(3.25)
where ˆ is the field’s polarization. The rate of energy absorption by the atom is
2
q2ω
2
ik·x
Γω = 2 2 |A0 | hΨk |e ˆ · P|Ψm i ρF (Em + ω).
mc
(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) =
2
q 2 |A0 |2 hΨk |eik·x ˆ · P|Ψm i δ(ωkm + ω)/m2 c2
ω|A0 |2 /2πc
2
4π 2 q 2
2
ik·x
|A0 | hΨk |e ˆ · P|Ψm i δ(ωkm + ω),
=
2
ωm c
(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) =
2
4π 2 q 2
2
−ik·x
ˆ
|A
|
hΨ
|e
·
P|Ψ
i
0
k
m δ(ωkm − ω).
ωm2 c
(3.29)
Notice that the time-dependent perturbation theory gives properties of the solution 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 = −Jµ Aµ /c operates. The free QED Lagrangian density
Lfree QED = ψ̄[iγ µ ∂µ − mc]ψ −
1
Fµν F µν
16π
(3.30)
gives the equations of motion for the free electron
[iγ µ ∂µ − mc]ψ = 0
(3.31)
and the equations of motion for the free electromagnetic field
∂µ F µν = ∂ 2 Aν − ∂ ν (∂ · A) = 0ν ,
(3.32)
where the Dirac γ-matrices are related to the Pauli spin matrices (4.66), ψ̄ = ψ † γ 0
is the Dirac adjoint of the four component spinor ψ, and Fµν = ∂µ Aν − ∂ν Aµ is
the electromagnetic field tensor. This noninteracting theory sets up the free unperturbed in-state |p1 · · · pn iin and out-state |k1 · · · km iout , which will be connected
R
by UI (+∞, −∞) = T{exp[i d4 xLint ]}, 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 perturbation theory is used to construct the scattering matrix or S-matrix
out
hk1 · · · km |p1 · · · pn iin = in hΩ|a(k1 ) · · · a(km )UI (+∞, −∞)a† (p1 ) · · · a† (pn )|Ωiin
(3.33)
where S = 1 + iT and |Ωiin 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)|Ωi = |ki and a(k)|Ωi = 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
hk1 · · · km |UI (+∞, −∞)|p1 · · · pn iin
= (2π)4 δ(p1 + · · · + pn − k1 − · · · − km ) · iM(p1 , · · · , pn → k1 , · · · , km ).
(3.34)
This invariant amplitude is analogous to the scattered wavefunction of quantum
PSfrag replacements
mechanics, i.e.,
h
eikr i
Ψk (x) ∼ N eik·x + fk (Ω)
,
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
Scattered field
Incident field
Figure 3–1: The coefficient 1 of the unscattered plane wave exp(ik · x) is analogous
to the 1 part of the S-matrix, while the scattering amplitude fk (Ω) which modulates
the 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 iout in |p1 · · · pn iin is given by
2 3
d3 km
out
in d k1
·
·
·
P (+∞) ∼ hk1 · · · km |p1 · · · pn i (2π)3
(2π)3
2 d3 k
d3 km
1
·
·
·
= in hΩ|a(k1 ) · · · a(km )UI (+∞, −∞)a† (p1 ) · · · a† (pn )|Ωiin (2π)3
(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
2 d 3 k
d3 km
1
·
·
·
(2π)4 δ(p1 + p2 − k1 − · · · − km )
dσ ∼ iM(p1 , p2 → k1 , · · · , km )
(2π)3
(2π)3
(3.37)
which is analogous to (3.28). As in quantum mechanics, time-dependent perturbation 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
d3 k P n
e−ik·x
eik·x o
∗
†
Aµ (x, t) = V
µ (k, λ)c(k, λ) √
+ µ (k, λ)c (k, λ) √
(2π)3 λ
2k
2k
−ik·x
ik·x o
R d3 k P n
e
e
ψi (x, t) = V
+ vi (k, s)b† (k, s) √
s ui (k, s)a(k, s) √
3
(2π)
2ω
2ω
R
(3.38)
act. In (3.38), µ and ∗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 timedependent perturbation theory was introduced by Feynman [55]. These so called
Feynman diagrams provide a pictorial version of the invariant amplitude
iM = C · out hk1 · · · km |UI (+∞, −∞)|p1 · · · pn iin
n R 4
o
out
= C · hk1 · · · km |T ei d xLint |p1 · · · pn iin
h
i
R 4 out
in
=
hk1 · · · km | 1 + i d xT Lint + · · · |p1 · · · pn i
(3.39)
connected
.
40
For example, the invariant amplitude for Bhabha scattering, i.e., e+ e− → e+ e− , is
iMBhabha =
h
out
p2
i
R
hk1 k2 |p1 p2 iin + out hk1 k2 |i d4 xT Lint |p1 p2 iin + O(L2int )
p2
k2
k2
+ O(L4int ),
+
=
p1
k1
connected
p1
k1
(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 diagrammatically 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 resulting 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 produced. 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 corresponding 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 investigate 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 timedependence. The general solutions to the wave equations of (A.5) are given by
|x − x0 | i
ρ(x0 , t0 ) 0 h
δ t − t+
Φ(x, t) =
|x − x0 |
c
1 R 3 0R ∞ 0 J(x0 , t0 ) 0 h
|x − x0 | i
A(x, t) =
d
x
dt
δ
t
−
t
+
,
−∞
c V
|x − x0 |
c
R
R
3 0 ∞
d
x
dt0
V
−∞
(3.42)
where G(+) (x, t; x0 , t0 ) = δ(t0 − [t + |x − x0 |/c])/4π|x − x0 | 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
0
A(x) =
1 R 3 0 eik|x−x |
d x
J(x0 ),
c V
|x − x0 |
(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 Ampére’s
law. That is
B = ∇×A
(3.45a)
E =
(3.45b)
i
∇×B
k
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:
drλ
The intermediate zone:
d r∼λ
The far or radiation zone:
dλr
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 − x0 |) ∼ 1 and the vector potential becomes
1 R 3 0 J(x0 )
dx
c V
|x − x0 |
o
nP
1R 3 0
4π r 0l
∗
0
0
Y
(Ω)Y
(Ω
)
=
d
x
J(x
)
lm
lm
lm
c V
2l + 1 r l+1
ikp
+··· ,
=−
r
Anear (x) =
(3.46)
where |x − x0 |−1 has been expanded into the spherical harmonics Ylm and an integration 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
R
of the dipole moment p = V d3 xxρ(x). From (3.45), the resulting magnetic and
electric fields are:
ik
n̂ × p + · · ·
r2
3n̂(n̂ · p) − p
=
+··· ,
r3
Bnear =
(3.47a)
Enear
(3.47b)
where n̂ 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
p
(x − x0 ) · (x − x0 )
s
2x · x0 |x0 |2
+
= |x| 1 −
|x|2
|x|2
x · x0 ≈ |x| 1 −
|x|2
|x − x0 | =
(3.48)
PSfrag replacements
= r − n̂ · x0
since |x0 | |x| = r|n̂| = r. A picture of the corresponding situation is shown in
Figure 3–2, where the x0 -integration runs over the domain of the source J. With the
x
r = |x|
n̂
d
O
Figure 3–2: In the radiation zone, the observation point x is located far from the
source J. In this case the distance |x − x0 | ≈ r − n̂ · x0 .
approximation (3.48), the far zone vector potential becomes
eikr 1 R 3 0
0
d x J(x0 )e−ikn̂·x
V
r c
∞
ikr X
e
(−ik)m 1 R 3 0
=
d x J(x0 )(n̂ · x0 )m
r m=0 m! c V
Afar (x) =
= −ikp
(3.49)
eikr
+··· ,
r
where |x − x0 |−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
R 3 0
d x J(x0 )(−ik n̂ · x0 )m /cm!. From (3.45), the corresponding fields are:
V
Bfar
Efar
ikr
h
1 i
(n̂ × p) 1 −
= k
+···
r
ikr
h1
eikr
ik i
(n̂ × p) × n̂ + 3 − 2 eikr [3n̂(n̂ · p) − p] + · · · .
= k2
r
r
r
2e
(3.50a)
(3.50b)
45
The magnetic field is transverse to the radius vector x = r n̂ 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 timeaveraged differential power radiated per unit solid angle
i
dP
1 h
c
= Re r 2 n̂ · E × B∗
dΩ
2
4π
4
ck
=
|n̂ × p|2
8π
ck 4 2 2
=
|p| sin θ
8π
(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.,
ck 4 2
dP
=
|p| .
P = dΩ
dΩ
3
R
(3.52)
The corresponding radiation pattern is shown in Figure 3–3. In general, the power
radiated by an l-pole goes like k 2(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 r 2 in the measure factor d3 x = r 2 drdΩ. In two dimensions, it is the
r −1/2 -fields whose power makes it to infinity since d2 x = 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 understanding 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
x
p
θ
Figure 3–3: The differential power dP/dΩ or radiation pattern corresponding to
an oscillating electric dipole verifies that no radiation is emitted in the direction
of the dipole moment. Rather the dipole radiation is a maximum in the direction
transverse 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) =
h 2`2 i3/4
π
e−`
2 [x−(r+vt)]2
eimv·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
[Ψ∗ (−i∇Ψ) + Ψ(i∇Ψ∗ )].
2m
(3.54)
In Fourier space this current becomes J̃(k, t) = qv exp(−ik · [r + vt] − k 2 /8`2 ) and
(+)
the vector potential is obtained by integrating against the Green’s function Dk
for
the wave operator (see Appendix B). The vector potential becomes
d3 k ik·x R ∞ 0 (+)
e
dt Dk (t, t0 )4πcJ̃(k, t0 )
V (2π)3
−∞
R d3 k ik·(x−r)−k2 /8`2 R ∞ 0 Θ(t − t0 ) sin ck(t − t0 ) −ik·vt0
e
dt
e
= 4πcqv V
(3.55)
−∞
(2π)3
ck
R d3 k eik·[x−(r+vt)]−k2 /8`2
= 4πcqv V
(2π)3 c2 k 2 − (k · v)2
A(x, t) =
R
which is difficult to perform analytically due to the complicated angular dependence
of the integrand. For nonrelativistic velocities, A can be approximated by
2
2
d3 k eik·[x−(r+vt)]−k /8`
V
(2π)3 c2 k 2 [1 − (v/c)2 cos2 θ]
qv R d3 k
2
2
≈ 2 V 2 eik·[x−(r+vt)]−k /8`
2π c √k
qv erf 2`|x − (r + vt)|
=
,
c
|x − (r + vt)|
A(x, t) = 4πcqv
R
(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
1
0.5
0
-5
0
5
10
15
20
x
Figure 3–4: The norms of J and A are plotted with different velocities along the
x-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
J̃(k, t) = qv(t)e−ik·(r+vt)−k
= qve
2 /8`2
−ik·(r+vt)−k 2 /8`2
(3.57)
Θ(t − t0 )Θ(t1 − t),
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
ct
r(t)
future
PSfrag replacements
elsewhere
t2
t1
x
past
Figure 3–5: The trajectory or world line r(t) of the charge is plotted. Electromagnetic 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 ,
√
qv n erf 2`|x − (r + vt)|
A(x, t) ≈
c
|x − (r + vt)|
√
1 erf 2`[c(t − t0 ) − |x − (r + vt0 )|]
+
2
|x − (r + vt0 )|
√
1 erf 2`[c(t − t0 ) + |x − (r + vt0 )|] o
−
2
|x − (r + vt0 )|
and for t ≥ t1 ,
√
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 )|] o
−
.
|x − (r + vt1 )|
qv n erf
A(x, t) ≈
2c
(3.58)
(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
acements
50
2
0
-2
-10
-5
x
0
5
10
0
5
10
t
Figure 3–6: The bremsstrahlung radiation from a charged gaussian wavepacket
moves 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 instantaneously 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 components. 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
Ȧ
c r
2
2
2
2
qv` 2 h e−2` [c(t−t0 )−R(t0 )] − e−2` [c(t−t0 )+R(t0 )]
=−
c
π
R(t0 )
E≈−
+
qv`
=−
c
r
e−2`
2 [c(t−t
2
− e−2`
R(t1 )
1 )+R(t1 )]
2 [c(t−t
1 )−R(t1 )]
2
2 −
g0 (t) + g1− (t)
π|
{z
}
i
(3.60)
h− (t)
neglecting the purely longitudinal −∇Φ, and
B=∇×A
q`
= −v ×
c
r h
2
2
2
2
e−2` [c(t−t1 )+R(t1 )] + e−2` [c(t−t1 )−R(t1 )]
2
û(t1 )
π
R(t1 )
e−2`
2 [c(t−t
2
+ e−2`
− û(t0 )
R(t0 )
r
i
h
q` 2
=−
v × û(t1 )g1+ (t) − û(t0 )g0+ (t) ,
c π
0 )−R(t0 )]
2 [c(t−t
0 )+R(t0 )]
2
i
(3.61)
where R(t) ≡ |R(t)| = |x − (r + vt)| and where the unit vectors û(t0 ) = R(t0 )/R(t0 )
and û(t1 ) = R(t1 )/R(t1 ). With (3.60) and (3.61), the Poynting vector is
c
E×B
4π
io
n
h
q 2 `2
= 2 h− (t) v × v × û(t1 )g1+ (t) − û(t0 )g0+ (t) .
2π c
S=
(3.62)
The differential power radiated into the solid angle dΩ at time t becomes
dP (x, t)
= R(t)2 n̂ · S(t)
dΩ
nh
i
q 2 `2
v · û(t1 ) g1+ (t) − v · û(t0 ) g0+ (t) n̂ · v
= 2 h− (t)R(t)2
2π c
h
io
− v 2 n̂ · û(t1 ) g1+ (t) − n̂ · û(t0 ) g0+ (t)
n
h
i
q 2 `2
= 2 h− (t)R(t)2 g0+ (t) v × n̂ · v × û(t0 )
2π c
h
io
+
− g1 (t) v × n̂ · v × û(t1 )
(3.63)
52
where the unit vector n̂ = R(t)/R(t) is normal to the surface of the ball that
emanates from the radiation source. The vectors v, n̂, û(t0 ), and û(t1 ) are all
constant in time. By choosing the z-axis along the velocity v, the angles between
v and the unit vectors û(t0 ) and û(t1 ) are δ0 and δ1 respectively. With a little
geometry, it can be verified that
sin θ
sin θ
sin δ0 ≈ p
and sin δ1 ≈ p
(3.64)
1 + (v/ct)(t1 − t0 ) cos θ
1 − (v/ct)(t1 − t0 ) cos θ
by suppressing terms of quadratic order and higher in c−1 , where θ is the angle
between v and n̂. In terms of the angles δ0 , δ1 , and θ, the differential power becomes
h
i
q 2 `2 v 2 −
dP
2
+
+
=
h
(t)R(t)
g
(t)
sin
δ
sin
θ
−
g
(t)
sin
δ
sin
θ
0
1
0
1
dΩ
2π 2 c
(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
h
i
sin2 θ
v
2
2
2
p
= sin θ 1 ∓
(t1 − t0 ) cos θ + O (v/c) cos θ . (3.66)
2ct
1 ± (v/ct)(t1 − t0 ) cos θ
These contributions are more significant at higher velocities. The quadrupole pattern in Figure 3–7 is obviously overemphasized. By integration over the unit sphere,
the total power is found to be
n
o
R
q 2 `2 v 2 −
2 π
+
2
+
2
P (x, t) =
h (t)R(t) 0 dθ g0 (t) sin δ0 sin θ − g1 (t) sin δ1 sin θ
πc
(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
PSfrag replacements
v
x
Figure 3–7: The radiation pattern given by (3.63) shows the characteristic dipole
pattern at lowest order. Keeping O(v/c) terms reveals the quadrupole pattern.
Higher order multipole patterns are generated by O(v 2 /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) =
2q 2
v̇(t)2
3c3
(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
q2
dP (t)
=
v̇(t)2 sin2 θ
dΩ
4πc3
(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
`
2 2
δ(x) = lim √ e−` x ,
`→+∞
π
(3.70)
√
where ` = 1/ 2b, the differential power in (3.65) becomes
h
i2
q2
dP
2
=
sin
θ
v{δ(t
−
t
)
−
δ(t
−
t
)}
.
0
1
dΩ
4πc3
|
{z
}
(3.71)
a2
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 4
CANONICAL STRUCTURE
The governing equation of quantum mechanics is the Schrödinger equation [7, 8].
In the minimal coupling prescription it is
iΨ̇ =
[−i∇ − qA/c]2 Ψ
+ V Ψ + qΦΨ.
2m
(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 Schrödinger 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 =
Ė
4π
J+
c
c
∇·B = 0
∇×E+
Ḃ
= 0. (1.2)
c
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 Schrödinger 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 Schrödinger equation does not say anything about the electrodynamics.
55
56
It is possible to couple the linear Maxwell and Schrödinger equations. The
resulting nonlinear Maxwell-Schrödinger theory accounts for the dynamics of the
charges and the electromagnetic field as well as their mutual interaction. For example, 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 generated. It will be demonstrated that these coupled nonlinear Maxwell-Schrödinger
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 =
[−Ȧ/c − ∇Φ]2 − [∇ × A]2
J·A
− ρΦ +
8π
c
(4.1)
with external sources ρ and J. Variation of this Lagrangian leads to the governing
equations of electrodynamics, i.e.,
∇2 A −
h
Φ̇ i
4π
Ä
−
∇
∇
·
A
+
=− J
2
c
c
c
∇ · Ȧ
= −4πρ
∇2 Φ +
c
(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 transformation of the solutions, i.e., Φ → Φ0 = Φ − Ḟ /c and A → A0 = A + ∇F, maps
them to new solutions that satisfy the gauge condition.
57
2. At the equation level, the set consisting of (1.4) together with a gauge constraint
has only solutions that satisfy the gauge condition.
3. At the Lagrangian level, a gauge fixing term can be added to (4.1) so that the
resulting 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 ∇2 F − F̈ /c2 = −[Φ̇/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 − Ḟ /c
A → ALorenz = A + ∇F.
(4.2)
Alternatively, adding the gauge constraint Φ̇/c + ∇ · A = 0 to (1.4) leads to the
Lorenz gauge equations of motion:
∇2 A −
4π
Ä
=− J
2
c
c
∇2 Φ −
Φ̈
= −4πρ.
c2
(4.3)
With ρ and J specified throughout space-time, the Lorenz gauge equations of motion are well-defined once the initial values for A, Ȧ, Φ, and Φ̇ are known. There
is some symmetry left in the solutions to these equations. Namely, the residual
gauge freedom left in the homogeneous equation ∇2 F − F̈ /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 ∇2 G = −∇·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 − Ġ/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:
∇2 A −
4π
∇Φ̇
Ä
=
−
J
+
c2
c
c
∇2 Φ = −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, Ȧ, Φ, and Φ̇ are known. As
before, there remains a symmetry or residual gauge freedom from the homogeneous
equation ∇2 G = 0. Note that in the Coulomb gauge Gauss’s law reduces to ∇2 Φ =
−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 sgaussians, 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 PTab = δ ab −
∂ 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.,
[Φ̇/c + ∇ · A]2
8π
[−Ȧ/c − ∇Φ]2 − [∇ × A]2
J · A [Φ̇/c + ∇ · A]2
=
− ρΦ +
−
.
8π
c
8π
LLMax = LMax −
(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 Φ̇/c + ∇ · A = 0.
LMax
g replacements
LC
Max
59
LLMax
d ∂LMax
dt ∂ ξ˙
d ∂LMax
d ∂LMax
Max
Max
Max
=0
=0
=0
− ∂L∂ξ
− ∂L∂ξ
− ∂L∂ξ
dt ∂ ξ˙
dt ∂ ξ˙
gauge invariant
∇ · A + Φ̇/c = 0
∇·A=0
add
add
well-posed IVP constraint
ill-posed IVP constraint
well-posed IVP
unique
solution
many
solutions
gauge
gauge
transformation
transformation
unique
solution
Maxwell’s equations
well-posed IVP
Figure 4–1: A limited but relevant portion of the gauge story in the Lagrangian
formalism is organized in this picture. The middle column (i.e., the column below
LMax ) enjoys full gauge freedom. The far left (Coulomb gauge) and far right (Lorenz
gauge) columns have limited gauge freedom. That is, there are a limited class of
gauge transformations that can be made on the solutions such that they remain in
the same gauge. This symmetry is due to the residual gauge freedom. Note that
these solutions correspond to different initial conditions within the gauge. Also note
that the Euler-Lagrange equations together with a particular gauge constraint are
equivalent to the Euler-Lagrange equations derived from that particular gauge fixed
Lagrangian.
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 = −Ȧ/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
1
[Ȧ/c + ∇Φ].
=
4πc
∂ Ȧ
(4.6)
60
The momentum conjugate to Φ is identically zero, i.e.,
Θ≡
∂LMax
= 0.
∂ Φ̇
(4.7)
A Hamiltonian density can still be defined as the time-time component of the
αβ
Maxwell stress-energy tensor TMax
= {∂LMax /∂(∂α ξ)}∂ β ξ − g αβ LMax . It is
00
HMax ≡ TMax
= Π · Ȧ + ΘΦ̇ − LMax =
[−4πcΠ]2 + [∇ × A]2
J·A
− c∇Φ · Π + ρΦ −
8π
c
(4.8)
and the resulting equations of motion are:
Ȧ ≡
∂HMax
= 4πc2 Π − c∇Φ
∂Π
−Π̇ ≡
∂HMax
∇[∇ · A] − ∇2 A J
=
− + c∇Θ
∂A
4π
c
(4.9)
Φ̇ ≡
∂HMax
=0
∂Θ
−Θ̇ ≡
∂HMax
= ρ + c∇ · Π.
∂Φ
Since the momentum Θ defined in (4.7) is identically zero, so is its time derivative
Θ̇ 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
Φ̇ = 0. It turns out that this extra equation fixes a particular gauge where Φ̇ = 0.
This gauge can always be fixed by a gauge transformation whose generator satisfies
F̈ /c = Φ̇. The residual gauge freedom left in the homogeneous equation F̈ = 0
does allow for a gauge transformation on the solutions to (4.9). These new gauge
transformed solutions do not leave the Φ̇ = 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 Φ̇ = 0.
61
4.2.1
Hamiltonian Formulation of the Lorenz Gauge
Rather than fixing the Coulomb gauge at the equation level it may be beneficial 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 =
[−Ȧ/c − ∇Φ]2 − [∇ × A]2
J · A [Φ̇/c + ∇ · A]2
− ρΦ +
−
.
8π
c
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
1
=
[Ȧ/c + ∇Φ]
4πc
∂ Ȧ
(4.10)
and the momentum conjugate to Φ is
Θ≡
1
∂LLMax
[Φ̇/c + ∇ · A].
=−
4πc
∂ Φ̇
(4.11)
With these momenta and coordinates, electrodynamics is given a symplectic structure. The Hamiltonian density is
L
HMax
=
[−4πcΠ]2 + [∇ × A]2 − [4πcΘ]2
J·A
− c∇Φ · Π − cΘ∇ · A + ρΦ −
(4.12)
8π
c
and the resulting equations of motion are:
Ȧ = 4πc2 Π − c∇Φ
−Π̇ =
∇[∇ · A] − ∇2 A J
− + c∇Θ
4π
c
(4.13)
Φ̇ = −4πc2 Θ − c∇ · A
−Θ̇ = 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 function F leading to the Lorenz gauge. That is, from Θ = −[Φ̇/c + ∇ · A]/4πc and
F̈ /c2 − ∇2 F = Φ̇/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
Ȧ


  ∇ × [∇ × A]/4π − J/c + c∇Θ


 
 0 0 0 −1   Φ̇  
ρ + c∇ · Π


 


=


 
0   Π̇  
4πc2 Π − c∇Φ
 1 0 0


 
0 1 0
0
Θ̇
−4πc2 Θ − c∇ · A





,



(4.14)
where 1 is the 3 × 3 identity matrix. Notice that (4.14) is of the Hamiltonian form
ω η̇ = ∂H/∂η.
(4.15)
More specifically ωab η̇ b = ∂H/∂η a , where η b is a column matrix of the generalized
positions and momenta, i.e.,

k

 A 


 Φ 


η b (x, t) = 
,
 k 
Π




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 substitution, these first order Hamiltonian equations of motion can be shown to be equivalent 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 + 4π Θ̇
∇×B=
4π
Ė
J + − 4πc∇Θ
c
c
(4.17)
∇·B =0
∇×E+
Ḃ
= 0.
c
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 ≡ ∇2 Θ −
Θ̈
1
= 2 [ρ̇ + ∇ · J].
2
c
c
(4.18)
If the matter theory is such that the equation of continuity ρ̇ = −∇ · J is satisfied,
then Θ = 0. So if Θ(t = 0) = Θ̇(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 Φ̇ = 0.
PSfrag replacements
LC
Max
LLMax
LMax
d ∂LMax
d ∂LMax
Max
Max
Max
− ∂L∂ξ
=0
− ∂L∂ξ
=0
− ∂L∂ξ
=0
dt ∂ ξ˙
dt ∂ ξ˙
gauge invariant
∇ · A + Φ̇/c = 0
∇·A=0
add
add
ill-posed IVP constraint
well-posed IVP
well-posed IVP constraint
d ∂LMax
dt ∂ ξ˙
unique
solution
gauge
transformation
many
solutions
gauge
transformation
unique
solution
unique
solution
gauge
transformation
Maxwell’s equations
well-posed IVP
∂HC
ω η̇ = ∂ηMax
well-posed IVP
C
HMax
=
∂LC
Max ˙
ξ
∂ ξ˙
− LC
Max
∂HL
ω η̇ = ∂ηMax
well-posed IVP
L
HMax
=
∂LL
Max ˙
ξ
∂ ξ˙
− LLMax
ω η̇ = ∂H∂ηMax
well-posed IVP
HMax =
∂LMax ˙
ξ
∂ ξ˙
− LMax
Figure 4–2: The Hamiltonian formulation of the gauge story is organized in this picture with respect to the previous Lagrangian
formulation. Figure 4–1 is depicted in the box with dotted borders. It can now be seen how the Coulomb and Lorenz gauges
connect in both formalisms.
65
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)η̇. That is
−1
∆H (ξ) ≡ (d/dt)ξ = (∂ξ/∂η b )η̇ b = (∂ξ/∂η b )ωab
(∂H/∂η a ) ≡ {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

T 
0
∂F/∂A
 

 

 ∂F/∂Φ   0
 

{F, G} = 
 
 

 ∂F/∂Π   1
 

0
∂F/∂Θ
0 −1
0
0
0
0
1
0
0
−1 


−1 



0 

0

∂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 = ω T = −ω.
Also notice that ω 2 = −1, ω T ω = 1, and det ω = 1.
4.3
Hamiltonian Electrodynamics and Wave Mechanics in Complex
Phase Space
Consider the matter theory associated with the Schrödinger Lagrangian (~ = 1)
LSch = iΨ∗ Ψ̇ −
[i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ
− V Ψ∗ Ψ − qΦΨ∗ Ψ
2m
(1.17)
where Ψ is the wavefunction for a single electron, V = q q̄/|x| is the static Coulomb
potential energy of a proton, and (Φ, A) are the electron’s scalar and vector potentials. Notice that this Lagrangian is already written in phase space. The momentum
conjugate to the wavefunction Ψ is iΨ∗ . Together with the previous Maxwell Lagrangian, the coupled nonlinear dynmical theory arising from the Lagrangians
LMax
n [−4πcΠ]2 + [∇ × A]2
o
1
= [Π · Ȧ − Π̇ · A] −
− c∇Φ · Π
2
8π
(4.21)
67
n [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ
o
i
+ V Ψ∗ Ψ + qΦΨ∗ Ψ
LSch = [Ψ∗ Ψ̇ − Ψ̇∗ Ψ] −
2
2m
(4.22)
1
Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A}
2
(4.23)
yields the following equations of motion:
Ȧ = 4πc2 Π − c∇Φ
Φ̇ = −4πc2 Θ − c∇ · A
2
iΨ̇ =
[−i∇ − qA/c] Ψ
+ V Ψ + qΦΨ
2m
−Π̇ =
∇[∇ · A] − ∇2 A J
− + c∇Θ
4π
c
−Θ̇ = ρ + c∇ · Π
−iΨ̇∗ =
(4.24)
2
∗
[i∇ − qA/c] Ψ
+ V Ψ∗ + qΦΨ∗ .
2m
Surface terms of the form (d/dt){pq/2} have been added in the above Lagrangians
in order to symmetrize them, i.e., L = pq̇ − H − (d/dt){pq/2} becomes L = [pq̇ −
R
R
ṗq]/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 Schrödinger 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


 

2
0   Ψ̇   [−i∇ − qA/c] Ψ/2m + V Ψ + qΦΨ 
 i 0 0 0 0


 

 0 −i 0 0 0
  Ψ̇∗   [i∇ − qA/c]2 Ψ∗ /2m + V Ψ∗ + qΦΨ∗ 
0


 



 



 

 0 0 0 0 −1 0   Ȧ  
∇ × [∇ × A]/4π − J/c + c∇Θ



=
,


 

 0 0 0 0 0 −1   Φ̇  

ρ + c∇ · Π


 



 

 0 0 1 0 0
  Π̇  

2
0
4πc
Π
−
c∇Φ


 



 

0 0 0 1 0
0
Θ̇
−4πc2 Θ − c∇ · A
(4.25)
where the symplectic form is canonical. The electromagnetic sector of it is identical to (4.14). These dynamical equations define the coupled Maxwell-Schrödinger
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 Schrödinger equation in (4.24) implies the continuity equation
(d/dt)qΨ∗ Ψ = −∇ · q Ψ∗ [−i∇ − qA/c]Ψ + Ψ[i∇ − qA/c]Ψ∗ /2m
(4.26)
which may be written more compactly as ρ̇ = −∇ · 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
1
−1
[(d/dt)Φ/c + ∇ · A] =
[(d/dt)4πρ/c + ∇ · 4πJ/c] = 2 [ρ̇ + ∇ · J] = 0
4πc
4πc
c
(4.27)
by appealing to (4.26). So if Θ(t = 0) = Θ̇(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) = Θ̇(t = 0) = 0 is Θ(t) = 0.
It is worth mentioning that if Θ̇(t = 0) = 0 for all time, then the electronelectron self interaction makes no contribution to the Schrödinger energy. This
is true since the self interaction term qΦΨ∗ Ψ in the above Schrödinger 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 Θ̇(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:
n [−4πcΠ]2 + [∇ × A]2
o
1
− c∇Φ · Π
LMax = [Π · Ȧ − Π̇ · A] −
2
8π
LSch


 {[∇Q + qAP/c]2 + [−∇P + qAQ/c]2 }/4m
1
= [P Q̇ − Ṗ Q] −

2

+V [Q2 + P 2 ]/2 + qΦ[Q2 + P 2 ]/2
1
Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A}
2
(4.28)



(4.29)


(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:
Ȧ = 4πc2 Π − c∇Φ
−Π̇ =
∇[∇ · A] − ∇2 A J
− + c∇Θ
4π
c
(4.31a)
Φ̇ = −4πc2 Θ − c∇ · A
Q̇ =
−Θ̇ = ρ + c∇ · Π
−∇2 P + q∇ · (AQ)/c + qA · ∇Q/c + q 2 A2 P/c2
+ V P + qΦP
2m
(4.31b)
−Ṗ = −
∇2 Q + q∇ · (AP )/c + q∇P · A/c − q 2 A2 Q/c2
+ V Q + qΦQ.
2m
70
These

 0

 0



 0


 1


 0


0
dynamical equations may be put into matrix form as


0 0 −1 0
0   Ȧ 


 Φ̇ 
0 0 0 −1 0 






0 0 0
0 −1   Q̇ 




 Π̇ 
0 0 0
0
0 






1 0 0
0
0   Θ̇ 



Ṗ
0 1 0
0
0

∇ × [∇ × A]/4π − J/c + c∇Θ



ρ + c∇ · Π



 −[∇2 Q + q∇ · (AP )/c + q∇P · A/c − q 2 A2 Q/c2 ]/2m + V Q + qΦQ
=


4πc2 Π − c∇Φ



−4πc2 Θ − c∇ · A


[−∇2 P + q∇ · (AQ)/c + qA · ∇Q/c + q 2 A2 P/c2 ]/2m + V P + qΦP








,







(4.32)
where the symplectic form is again canonical. Note that the equation of continuity
ρ̇ = −∇ · J still holds with the real charge and current densities
ρ = q[Q2 + P 2 ]/2
4.5
J=
q
{Q∇P − P ∇Q − qQAQ/c − qP AP/c}. (4.33)
2m
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 contribution. 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 remaining 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 ∇2 ΦC = −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
R
Q and P. That is ΦC (x, t) = V d3 x0 |x − x0 |−1 q[Q(x0 , t)Q(x0 , t) + P (x0 , t)P (x0 , 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≡
1
∇ΦC
∂LMax
+
[Ȧ/c + ∇ΦD ].
= ΠC + ΠD =
4πc
4πc
∂ Ȧ
(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



Ã(A)

 A 



 Φ̃(Φ, Q, P )
 Φ 









Q̃(Q)

 Q 
 → 
T:



 Π̃(Π, Q, P )
 Π 







 Θ 
Θ̃(Θ)






P̃ (P )
P


A
 
 
  Φ − Φ (Q, P )
C
 
 
 
Q
 
=
 
  Π − ΠC (Q, P )
 
 
 
Θ
 
 
P








,







(4.35)
where Φ̃ ≡ ΦD and Π̃ ≡ 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 transformations [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
η̇ = ω −1
∂H
∂η
∂ H̃
→ η̃˙ = ω̃ −1
,
∂ η̃
(4.36)
where the new Hamiltonian H̃ is equivalent to the old Hamiltonian H expressed in
terms of the new variables η̃. (For simplicity H̃ will be written as H from this point
forward.) To this end, consider the time derivative of the new column matrix
∂ η̃i
η̃˙ i =
η̇j ≡ Tij η̇j
∂ηj
or η̃˙ = Tη̇.
(4.37)
Substituting η̇ from (4.36) results in
−1 ∂H
η̃˙ i = Tij ωjk
∂ηk
∂H
or η̃˙ = Tω −1
.
∂η
(4.38)
Lastly the column matrix ∂H/∂η can be written as
∂H
∂H ∂ η̃l
∂H
=
≡ TklT
∂ηk
∂ η̃l ∂ηk
∂ η̃l
or
∂H
∂H
= TT
∂η
∂ η̃
(4.39)
so that the new equations of motion (4.38) become
−1 T ∂H
Tkl
η̃˙ i = Tij ωjk
∂ η̃l
∂H
∂H
or η̃˙ = Tω −1 TT
≡ ω̃ −1
.
∂ η̃
∂ η̃
(4.40)
73
This canonical transformation on the equations of motion leaves only the computation of ω̃ −1 ≡ Tω −1 TT since the Hamiltonian automatically becomes
H=
[−4πc{Π̃ + ΠC (Q̃, P̃ )}]2 + [∇ × Ã]2 − [4πcΘ̃]2
Q̃2 + P̃ 2
+ q[Φ̃ + ΦC (Q̃, P̃ )]
8π
2
− c∇[Φ̃ + ΦC (Q̃, P̃ )] · [Π̃ + ΠC (Q̃, P̃ )] − cΘ̃∇ · Ã
+
Q̃2 + P̃ 2
[∇Q̃ + q ÃP̃ /c]2 + [−∇P̃ + q ÃQ̃/c]2
+V
4m
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
= ω̃ η̃˙ = (T−1 )T ωT−1 η̃˙
∂ η̃
∂H
∂H
not η̃˙ = ω̃ −1
= Tω −1 TT
,
∂ η̃
∂ η̃
(4.42)
where the inverse transformation T−1 is the transformation of the inverse mapping,
i.e., Tij−1 ≡ ∂ηi /∂ η̃j . It will be shown that det T 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 hP |Qi
∂pK ∂qJ
2
i
− ∂∂qKhQ|P
∂pJ
0



(4.43)
which is not easily inverted. As a result it is simpler to compute (T−1 )T ωT−1 than
Tω −1 TT 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 → ∂/∂ η̃ must first be set up. The transposed
inverse transformation (T−1 )T is defined on the vector fields themselves according
74
to


∂/∂ Ã 

∂/∂ Φ̃ 



∂/∂ Q̃ 


∂/∂ Π̃ 


∂/∂ Θ̃ 


∂/∂ P̃

0
0
0
0
0
 ∂A/∂ Ã


0
∂Φ/∂ Φ̃
0
0
0
0



0
∂Φ/∂ Q̃ ∂Q/∂ Q̃ ∂Π/∂ Q̃
0
0

=


0
0
0
∂Π/∂ Π̃
0
0



0
0
0
0
∂Θ/∂ Θ̃
0


0
∂Φ/∂ P̃
0
∂Π/∂ P̃
0
∂P/∂ P̃
































∂/∂A 

∂/∂Φ 



∂/∂Q 
.

∂/∂Π 


∂/∂Θ 


∂/∂P
(4.44)
Notice that
det(T−1 )T = det T−1 = (det T)−1 ≡
∂(A, Φ, Q, Π, Θ, P )
∂(Ã, Φ̃, Q̃, Π̃, Θ̃, P̃ )
(4.45)
= (∂ Ã/∂A)(∂ Φ̃/∂Φ)(∂ Q̃/∂Q)(∂ Π̃/∂Π)(∂ Θ̃/∂Θ)(∂ P̃ /∂P ) = 1
so that the transformation is canonical and symplectic or area preserving. In other
words, the new infinitesimal volume element dη̃ is related to the old infinitesimal
volume element dη by
dη̃ = det T 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
−1 T
ω̃ ≡ (T ) ωT
−1
=
ΩM Ω>
,
Ω∨ ΩG
(4.47)
where


0
−(∂A/∂ Ã)(∂Π/∂ P̃ ) 
 (∂A/∂ Ã)(−1)(∂Π/∂ Π̃)



Ω> = 
0
(∂Φ/∂
Φ̃)(−1)(∂Θ/∂
Θ̃)
0




0
−(∂Φ/∂ Q̃)(∂Θ/∂ Θ̃) (∂Q/∂ Q̃)(−1)(∂P /∂ P̃ )
(4.48)


0
0

 (∂Π/∂ Π̃)(1)(∂A/∂ Ã)


∨

Ω =
0
(∂Θ/∂
Θ̃)(1)(∂Φ/∂
Φ̃)
(∂Θ/∂
Θ̃)(∂Φ/∂
Q̃)




(∂Π/∂ P̃ )(∂A/∂ Ã)
0
(∂P /∂ P̃ )(1)(∂Q/∂ Q̃)
(4.49)
ΩM



=





ΩG = 


0

0 −(∂A/∂ Ã)(∂Π/∂ Q̃) 


0
0
0


(∂Π/∂ Q̃)(∂A/∂ Ã) 0
0

0
0
0


0
0
(∂Θ/∂ Θ̃)(∂Φ/∂ P̃ ) 
.

0 −(∂Φ/∂ P̃ )(∂Θ/∂ Θ̃)
0
(4.50)
(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 ω̃ it can be shown that ω̃ equals


0
0 −∂ΠC (Q̃, P̃ )/∂ Q̃ −1
0
−∂ΠC (Q̃, P̃ )/∂ P̃ 





0
0
0
0
−1
0







0
0 −∂ΦC (Q̃, P̃ )/∂ Q̃
−1
 ∂ΠC (Q̃, P̃ )/∂ Q̃ 0






1
0
0
0
0
0






0
1
∂Φ
(
Q̃,
P̃
)/∂
Q̃
0
0
∂Φ
(
Q̃,
P̃
)/∂
P̃
C
C




∂ΠC (Q̃, P̃ )/∂ P̃ 0
1
0 −∂ΦC (Q̃, P̃ )/∂ P̃
0
(4.52)
with
ΦC (x, t) =
and
ΠC (x, t) =
q R Q(x0 , t)2 + P (x0 , t)2 3 0
dx
2 V
|x − x0 |
q R Q(x0 , t)2 + P (x0 , t)2 3 0
∇ V
d x.
8πc
|x − x0 |
(4.53)
(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/∂ η̃)T ω̃ −1 (∂G/∂ η̃).
(4.55)
77
The transformed equations of motion with symplectic form (4.52) may be written in full as:
∂ΠC ˙
˙ − ∂ΠC P̃˙ = ∂H = ∇ × [∇ × Ã] + c∇Θ̃
Q̃ − Π̃
4π
∂ Q̃
∂ P̃
∂ Ã
o
q n
q
q
−
Q̃∇P̃ − P̃ ∇Q̃ − Q̃ÃQ̃ − P̃ ÃP̃
2mc
c
c
2
2
Q̃
+
P̃
∂H
˙ =
+ c∇ · [Π̃ + ΠC ]
=q
− Θ̃
2
∂ Φ̃
∂ΠC ˙
∂ΦC ˙
−∇2 Q̃ − q∇ · (ÃP̃ )/c − q∇P̃ · Ã/c + q 2 Ã2 Q̃/c2
∂H
· Ã −
=
Θ̃ − P̃˙ =
2m
∂ Q̃
∂ Q̃
∂ Q̃
−
+ V Q̃ + q[Φ̃ + ΦC ]Q̃
∂ΠC
∂ Q̃
n Q̃2 + P̃ 2
o ∂Φ
C
+ q
+ c∇ · [Π̃ + ΠC ]
2
∂ Q̃
+ {4πc2 [Π̃ + ΠC ] − c∇[Φ̃ + ΦC ]} ·
˙ = ∂H = 4πc2 [Π̃ + Π ] − c∇[Φ̃ + Φ ]
Ã
C
C
∂ Π̃
˙ + ∂ΦC Q̃˙ + ∂ΦC P̃˙ = ∂H = −4πc2 Θ̃ − c∇ · Ã
Φ̃
∂ Q̃
∂ P̃
∂ Θ̃
−∇2 P̃ + q∇ · (ÃQ̃)/c + q à · ∇Q̃/c + q 2 Ã2 P̃ /c2
∂H
∂ΠC ˙
∂ΦC ˙
· Ã + Q̃˙ −
=
Θ̃ =
2m
∂ P̃
∂ P̃
∂ P̃
+ V P̃ + q[Φ̃ + ΦC ]P̃
∂ΠC
∂ P̃
n Q̃2 + P̃ 2
o ∂Φ
C
,
+ q
+ c∇ · [Π̃ + ΠC ]
2
∂ P̃
+ {4πc2 [Π̃ + ΠC ] − c∇[Φ̃ + ΦC ]} ·
(4.56)
where ΠC ≡ ΠC (Q̃, P̃ ) and ΦC ≡ ΦC (Q̃, P̃ ). The forces appearing on the right
hand side of these equations have become more complicated, especially those in the
Schrödinger 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/∂ Φ̃ and ∂H/∂ Π̃ equations appear in the forces of the Schrödinger
78
equations. Substitution of ∂H/∂ Φ̃ and ∂H/∂ Π̃ into the Schrödinger equations results in the following simplified equations:
−
∂ΠC ˙
˙ − ∂ΠC P̃˙ = ∇ × [∇ × Ã] + c∇Θ̃
Q̃ − Π̃
4π
∂ Q̃
∂ P̃
o
q
q
q n
Q̃∇P̃ − P̃ ∇Q̃ − Q̃ÃQ̃ − P̃ ÃP̃
−
2mc
c
c
2
2
˙ = q Q̃ + P̃ + c∇ · [Π̃ + Π ]
− Θ̃
C
2
∇2 Q̃ + q∇ · (ÃP̃ )/c + q∇P̃ · Ã/c − q 2 Ã2 Q̃/c2
−P̃˙ = −
+ V Q̃
2m
+ q[Φ̃ + ΦC ]Q̃
˙ = 4πc2 [Π̃ + Π ] − c∇[Φ̃ + Φ ]
Ã
C
C
˙ + ∂ΦC Q̃˙ + ∂ΦC P̃˙ = −4πc2 Θ̃ − c∇ · Ã
Φ̃
∂ Q̃
∂ P̃
−∇2 P̃ + q∇ · (ÃQ̃)/c + q à · ∇Q̃/c + q 2 Ã2 P̃ /c2
Q̃˙ =
+ V P̃
2m
+ q[Φ̃ + ΦC ]P̃ .
(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:
−[ Π̇ + Π̇C (Q, P )] = ∂H/∂A
Ȧ = ∂H/∂Π
− Θ̇ = ∂H/∂Φ
Φ̇ + Φ̇C (Q, P ) = ∂H/∂Θ
−Ṗ = ∂H/∂Q
Q̇ = ∂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 ω̃ η̃˙ = ∂H/∂ η̃, 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:
R
δ [pq̇ − H(p, q)]dt = 0
ω η̇ = ∂H/∂η
Change of Variables
L̃(p̃, q̃)
R
δ [p̃q̃˙ − H(p̃, q̃)]dt = 0
L(p, q)
ω̃ η̃˙ = ∂H/∂ η̃
Canonical Transformation
Figure 4–3: Commutivity diagram representing the change of coordinates (q, p) to
(p̃, q̃) at both the Lagrangian and equation of motion levels.
L̃Max


 {[−4πc(Π̃ + ΠC )]2 + [∇ × Ã]2 }/8π
1
˙
˙
˙
= [(Π̃ + ΠC ) · Ã − ( Π̃ + Π̃C ) · Ã] −

2

−c∇[Φ̃ + ΦC ] · [Π̃ + ΠC ]





(4.59)
L̃Sch


 {[∇Q̃ + q ÃP̃ /c]2 + [−∇P̃ + q ÃQ̃/c]2 }/4m
1 ˙
˙
= [P̃ Q̃ − P̃ Q̃] −

2

+[V + q(Φ̃ + ΦC )]{Q̃2 + P̃ 2 }/2
1
˙ + Φ̇ ) − Θ̃(
˙ Φ̃ + Φ )] − {−2πc2 Θ̃2 − cΘ̃∇ · Ã}.
L̃gauge = [Θ̃( Φ̃
C
C
2



(4.60)


(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 Schrödinger 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,
g replacements
79
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]
iΨ̇P =
[−i∇ − qA/c]2 ΨP
q
+ V ΨP + qΦΨP −
σ · [∇ × A]ΨP
2m
2mc
(4.63)
which is the nonrelativistic limit of the Dirac equation
iΨ̇D = βmc2 ΨD + cα · [−i∇ − qA/c]ΨD + qΦΨD
(4.64)
in terms of the four component spinor ΨD , where the β and α matrices are
β=
and
σx =
0 1
1 0
I 0
0 −I
σy =
α=
0 −i
i 0
0 σ
σ 0
σz =
(4.65)
1 0
.
0 −1
(4.66)
Notice that taking the nonrelativistic limit of the Dirac equation involves the elimination 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 Schrödinger
theory (see (1.14)). It is
JP = q Ψ†P [−i∇ − qA/c]ΨP + ΨP [i∇ − qA/c]Ψ†P + ∇ × [Ψ†P σΨP ] /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
ρ̇ = −∇ · J since ∇ · ∇ × [Ψ†P σΨP ] = 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 Φq̄ = q̄/|x| of the structureless proton of charge
q̄ 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
Lq̄Sch = iΩ∗ Ω̇ −
[i∇ − q̄A/c]Ω∗ · [−i∇ − q̄A/c]Ω
− q̄ΦΩ∗ Ω,
2mq̄
(4.68)
where (Φ, A) are the scalar and vector potentials arising from the charge and current
densities
ρ = qΨ∗ Ψ + q̄Ω∗ Ω
J = q Ψ∗ [−i∇ − qA/c]Ψ + Ψ[i∇ − qA/c]Ψ∗ /2mq
+ q̄ Ω∗ [−i∇ − q̄A/c]Ω + Ω[i∇ − q̄A/c]Ω∗ /2mq̄ .
(4.69)
(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
n [−4πcΠ]2 + [∇ × A]2
o
1
− c∇Φ · Π
= [Π · Ȧ − Π̇ · A] −
2
8π
(4.71)
o
n [i∇ − qA/c]Ψ∗ · [−i∇ − qA/c]Ψ
i ∗
∗
∗
= [Ψ Ψ̇ − Ψ̇ Ψ] −
+ qΦΨ Ψ
2
2mq
(4.72)
LMax
LqSch
82
n [i∇ − q̄A/c]Ω∗ · [−i∇ − q̄A/c]Ω
o
i
Lq̄Sch = [Ω∗ Ω̇ − Ω̇∗ Ω] −
+ q̄ΦΩ∗ Ω
2
2mq̄
1
Lgauge = [ΘΦ̇ − Θ̇Φ] − {−2πc2 Θ2 − cΘ∇ · A}.
2
(4.73)
(4.74)
Notice that the electron Lagrangian (4.72) does not explicitly contain the static proton potential energy V = q q̄/|x| as did the previous Schrödinger 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 q̄ΦΩ∗ Ω in the above matter Hamiltonians. In other words, the scalar potential Φ contains (in any gauge) a Coulomb
piece of the form
ρ(x0 , t) 3 0
dx
V
|x − x0 |
R qΨ∗ (x0 , t)Ψ(x0 , t) 3 0 R q̄Ω∗ (x0 , t)Ω(x0 , t) 3 0
d x + V
d x.
= V
|x − x0 |
|x − x0 |
ΦC (x, t) = Φ + (ΦC − Φ) =
R
(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 q̄ΦΩ∗ 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 =
R
ρ(x, t)ΦC (x, t)d3 x =
V
R
d3 x
V
R
d 3 x0
V
ρ(x, t)ρ(x0 , t)
<∞
|x − x0 |
(4.76)
for both the cross terms (electron-proton attraction) and the direct terms (electronelectron 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 −Θ̇ = 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 5
NUMERICAL IMPLEMENTATION
The formal theory of Maxwell-Schrödinger dynamics was constructed in the
previous chapter. In particular, the coupled and nonlinear Maxwell-Schrödinger
equations
[P − qA/c]2 Ψ
+ V Ψ + qΦΨ
2m
h
Ä
Φ̇ i
4π
2
∇ A− 2 −∇ ∇·A+
= − J
c
c
c
∇ · Ȧ
∇2 Φ +
= −4πρ
c
iΨ̇ =
(5.1)
(5.2a)
(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 automatically 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-Schrödinger 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 Hamiltonian 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-Schrödinger Theory in a Complex Basis
Each of the Maxwell-Schrödinger dynamical variables, which are themselves
fields, may be expanded into a complete basis of functions GK according to
Ψ(x, t) =
Ak (x, t) =
Φ(x, t) =
P
P
P
K GK (x)ψK (t)
Ψ∗ (x, t) =
K GK (x)akK (t)
Πk (x, t) =
K GK (x)φK (t)
Θ(x, t) =
P
P
P
∗
K GK (x)ψK (t)
K GK (x)πkK (t)
(5.3)
K GK (x)θK (t),
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 =
P
KM
LSch =
Lgauge =
1
[(∂/∂amM )ȧmM − (∂/∂πmM )π̇mM ]SMax − HMax
2
P i
∗
∗
K [(∂/∂ψK )ψ̇K − (∂/∂ψK )ψ̇K ]SSch − HSch
2
P
KM
1
[(∂/∂φK )φ̇K − (∂/∂θK )θ̇K ]Sgauge − Hgauge
2
(5.5)
(5.6)
(5.7)
86
with integrals
SMax =
R
V
Π · Ad3 x
SSch =
R
V
Ψ∗ Ψd3 x
Sgauge =
R
V
ΘΦd3 x.
(5.8)
The calculus of variations leads to the following dynamical equations:
∂ 2 SMax
∂H
ȧnN =
∂πmM ∂anN
∂πmM
−
(5.9)
∂H
∂ 2 Sgauge
φ̇L =
∂θI ∂φL
∂θI
or NIL φ̇L = ∇θI H
(5.10)
∂H
∂ 2 iSSch
ψ̇L =
∗
∂ψI ∂ψL
∂ψI∗
or iCIL ψ̇L = ∇ψI∗ H
(5.11)
∂ 2 SMax
∂H
π̇mM =
∂anN ∂πmM
∂anN
or
T
− MnN
,mM π̇mM = ∇anN H
(5.12)
∂H
∂ 2 Sgauge
θ̇K =
∂φJ ∂θK
∂φJ
or
− NJT K θ̇K = ∇φJ H
(5.13)
∂ 2 iSSch ∗
∂H
ψ̇ =
∗ K
∂ψJ ∂ψK
∂ψJ
or
∗
− iCJ∗ K ψ̇K
= ∇ ψJ H
(5.14)
−
−
or MmM,nN ȧnN = ∇πmM H
which are of the Hamiltonian form ω η̇ = ∂H/∂η. The summation convention is used
throughout. These equations may be written more compactly as
M ȧ = ∇π H
− M T π̇ = ∇a H
N φ̇ = ∇θ H
− N T θ̇ = ∇φ H
iC ψ̇ = ∇ψ∗ H
− iC ∗ ψ̇ ∗ = ∇ψ H
(5.15)
87
and can be cast into matrix form as

0
0 0
0
0
 iC

 0 −iC ∗ 0 0
0
0



0
0 0 −M T
0
 0


 0
0
0 0
0
−N T


 0
0
M 0
0
0


0
0
0 N
0
0


















ψ̇  
 

ψ̇ ∗ 
 
 
 
ȧ  
=
 

φ̇ 
 
 

π̇ 
 
 
θ̇
∗

∂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 dynamical 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 representation of the coupled Maxwell-Scrödinger 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/∂ψ

 ∂F/∂ψ



 ∂F/∂a
{F, G} = 

 ∂F/∂φ


 ∂F/∂π


∂F/∂θ
T 






























iC
0
0
0
0
0
0
−iC ∗
0
0
0
0
0
0
0
0
−M T
0
0
0
0
0
0
−N T
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
R
elements in ω involve gaussian overlap integrals like hGI |GK i = V hGI |xihx|GK id3 x.
5.2
Maxwell-Schrödinger Theory in a Real Basis
As was done previously, each dynamical variable may be expanded into a complete basis of functions GK as
P
Q(x, t) =
P
Ak (x, t) =
P
Φ(x, t) =
K GK (x)qK (t)
P (x, t) =
K GK (x)akK (t)
Πk (x, t) =
K GK (x)φK (t)
Θ(x, t) =
P
P
P
K GK (x)pK (t)
K GK (x)πkK (t)
(5.18)
K GK (x)θK (t),
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 =
P
KM
1
[(∂/∂amM )ȧmM − (∂/∂πmM )π̇mM ]SMax − HMax
2
LSch =
Lgauge =
with integrals
SMax =
R
P 1
K [(∂/∂qK )q̇K − (∂/∂pK )ṗK ]SSch − HSch
2
P
KM
Π · Ad3 x
V
1
[(∂/∂φK )φ̇K − (∂/∂θK )θ̇K ]Sgauge − Hgauge
2
SSch =
R
P Qd3 x
V
Sgauge =
R
V
ΘΦd3 x.
(5.19)
(5.20)
(5.21)
(5.22)
Applying the calculus of variations to the above Lagrangians leads to the equations of motion:
∂H
∂ 2 SMax
ȧnN =
∂πmM ∂anN
∂πmM
∂H
∂ 2 Sgauge
φ̇L =
∂θI ∂φL
∂θI
or MmM,nN ȧnN = ∇πmM H
(5.23)
or NIL φ̇L = ∇θI H
(5.24)
89
∂ 2 SSch
∂H
q̇L =
∂pI ∂qL
∂pI
−
∂ 2 SMax
∂H
π̇mM =
∂anN ∂πmM
∂anN
−
or CIL q̇L = ∇pI H
(5.25)
T
− MnN
,mM π̇mM = ∇anN H
or
(5.26)
∂H
∂ 2 Sgauge
θ̇K =
∂φJ ∂θK
∂φJ
or
− NJT K θ̇K = ∇φJ H
(5.27)
∂H
∂ 2 SSch
ṗK =
∂qJ ∂pK
∂qJ
or
− CJT K ṗK = ∇qJ H
(5.28)
−
which are of the Hamiltonian form ω η̇ = ∂H/∂η. These equations may be written
more compactly as
M ȧ = ∇π H
− M T π̇ = ∇a H
N φ̇ = ∇θ H
− N T θ̇ = ∇φ H
C q̇ = ∇p H
− C T ṗ = ∇q H
and can be cast into matrix form as

T
0
0
 0 0 0 −M

 0 0 0
0
−N T
0



0
0
−C T
 0 0 0


 M 0 0
0
0
0


 0 N 0
0
0
0


0 0 C
0
0
0


















ȧ  
 

φ̇ 
 
 
 
q̇  
=
 

π̇ 
 
 

θ̇ 
 
 
ṗ
(5.29)

∂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-Schrödinger equations of motion.
90
The real phase space that carries the associated dynamics is endowed with the
Poisson bracket

 ∂F/∂a

 ∂F/∂φ



 ∂F/∂q
{F, G} = 

 ∂F/∂π


 ∂F/∂θ


∂F/∂p
T 






























0
0
0 −M
0
0
0
0
0
M
T
0
0
0
−N T
0
0
0
0
−C T
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 hGI |GK i =
R
hGI |xihx|GK id3 x.
V
5.2.1
Overview of Computer Program
The equations of motion (5.30) are coded in Fortran 90. The computer program 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 η b . The forces ∂H/∂η a are constructed from
this data. The symplectic form ωab is then inverted with the LAPACK [91] subroutine DGESVX, which is the expert driver for the AX = B solver DGESV. This
−1
establishes a first order system of differential equations of the form η̇ b = ωab
∂H/∂η a
which may be solved, for example, with an Euler stepping method. That is
−1
∂H/∂η a ](t).
η b (t + ∆t) = η b (t) + (∆t)η̇ b (t) = η b (t) + (∆t)[ωab
(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 numerically 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 electromagnetic radiation is still quite small compared to the dynamics of the matter. An
overview of the ENRD program is presented in Figure 5–1.
MAXENERGY.f90
RHS EM.f90
RIRJ.f90
• Maxwell energy.
• Compute hηM |η(t)i.
• RHS Maxwell.
• Create library for vector
arguments.
∂H ∂H
,
∂Π ∂A
enrd.f90
• Input deck.
GAENERGY.f90
RHS GA.f90
DERIVS.f90
• Build vector arguments.
• Gauge energy.
• RHS Gauge.
• Build forces ∂H
∂η on RHS.
• Create library for four index
matrix elements.
• Build symplectic form ω.
• Call integrator.
• Compute hηM |η(t)i.
∂H ∂H
∂Θ , ∂Φ
SCHENERGY.f90
RHS SH.f90
• Schrödinger energy.
• Compute hηM |η(t)i.
• RHS
Schrödinger.
∂H ∂H
∂P , ∂Q
• Call LAPACK routine
DGESVX.
• Invert ω η̇ = ∂H/∂η to
get η̇ = ω −1 ∂H/∂η.
• Write η(t + ∆t) to PS.dat.
RK4.f90
METRIC.f90
• Symplectic
form ω.
• Runge-Kutta 4 ODE solver.
η(t + ∆t) = η(t) + (∆t)η̇(t)
92
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 corresponding 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 electromagR
netic radiation was produced. The total charge q = V ρ(x)d3 x 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 Coulombic 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 deciR
mal places. The total charge q = V ρ(x)d3 x 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 constant.
frag replacements
94
20
10
0
πkK (t)
-10
-20
PSfrag replacements
-0.5
0.5
0
akK (t)
Figure 5–2: Phase space contour for the coefficients of the vector potential A and
its momentum Π.
5
pK (t) 0
-5
-5
0
qK (t)
5
Figure 5–3: Phase space contour for the coefficients of the real-valued Schrödinger
field Q and its momentum P.
20
g replacements
95
0.0005
θK (t)
0
-0.0005
-30
-20
-10
φK (t)
0
10
Figure 5–4: Phase space contour for the coefficients of the scalar potential Φ and its
momentum 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 Schrödinger eigenstates are found by
diagonalizing the time-independent Schrödinger equation
HC = SC,
(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 is
the matrix of energy eigenvalues. Similarly, the Maxwell eigenstates are found by
diagonalizing the free wave equation ∇2 φ − φ̈/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 ∇2 φ̃ = ω 2 φ̃, where ω is the frequency. In a basis
this equation turns into the matrix equation
HC = ω 2 SC,
(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
ω 2 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 according to
|η(t)i =
P
M |ηM ihηM |η(t)i,
(5.35)
where CM (t) ≡ hηM |η(t)i are the basis expansion coefficients. The real and imaginary parts of the Schrödinger 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 excitations of the s-, px -, and dy2 −z 2 -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 Schrödinger
coefficients are presented in Figure 5–8.
rag replacements
97
s
px
py
pz
1
d x2
2
dy2 −z
RehΨM |Ψ(t)i 0
0
10
t
20
30
Figure 5–5: Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t)
is a superposition of s- and px -waves.
s
px
py
pz
1
d x2
2
dy2 −z
ImhΨM |Ψ(t)i 0
−1
0
10
t
20
30
Figure 5–6: Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves.
PSfrag replacements
−1
−10
−20
−30
ImhΨM |Ψ(t)i
pK (t)
1
0.001
−0.001
−1
RehΨN |Ψ(t)i
qK (t)
P
98
2
M |hΨM |Ψ(t)i|
s
px
0.8
0.6
|hΨM |Ψ(t)i|2
0.4
0.2
0
0
10
t
20
30
Figure 5–7: Probability for the electron to be in a particular basis eigenstate.
d x2 2
dy2 −z
pK (t)
0.5
ImhΨM |Ψ(t)i 0
−0.5
−1
−1
−0.5
0
RehΨM |Ψ(t)i
0.5
1
Figure 5–8: Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves.
30
1
−10
−20
−30
0.001
−0.001
qK (t)
s
px
py
pz
99
5.3
Symplectic Transformation to the Coulomb Reference
Recall the basis representation of the Maxwell-Schrödinger equations of motion
in (5.30). They are

T
0
0
 0 0 0 −M

 0 0 0
0
−N T
0



 0 0 0
0
0
−C T


 M 0 0
0
0
0


 0 N 0
0
0
0


0 0 C
0
0
0
where
MKM ≡
∂ 2 hΠ|Ai
∂πK ∂aM
NKM ≡
















∂ 2 hΘ|Φi
∂θK ∂φM


ȧ  
 

φ̇ 
 
 
 
q̇  
=
 

π̇ 
 
 

θ̇ 
 
 
ṗ

∂H/∂a 

∂H/∂φ 



∂H/∂q 
,

∂H/∂π 


∂H/∂θ 


∂H/∂p
CKM ≡
∂ 2 hP |Qi
∂pK ∂qM
(5.30)
(5.36)
and where the integrals h·|·i 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

 

0
0
0
0
0 
 ∂/∂ã   ∂a/∂ã

 


 ∂/∂ φ̃  
0
∂φ/∂ φ̃
0
0
0
0 

 


 


 

0
∂φ/∂ q̃ ∂q/∂ q̃ ∂π/∂ q̃
0
0 
 ∂/∂ q̃  

=


 


 ∂/∂ π̃  
0
0
0
∂π/∂
π̃
0
0

 


 


 ∂/∂ θ̃  
0
0
0
0
∂θ/∂ θ̃
0 

 


 

0
∂φ/∂ p̃
0
∂π/∂ p̃
0
∂p/∂ p̃
∂/∂ p̃

∂/∂a 

∂/∂φ 



∂/∂q 


∂/∂π 


∂/∂θ 


∂/∂p
(5.37)
100
so that the symplectic form in (5.30) transforms as


T
0
0 
 0 0 0 −M


 0 0 0
0
−N T
0 





T 
0
0
−C 
 0 0 0




 M 0 0

0
0
0




 0 N 0
0
0
0 




0 0 C
0
0
0

0
−MÃQ̃ −MÃΠ̃
0
−MÃP̃
 0

 0
0
0
0
−NΦ̃Θ̃
0



0
0
0
−NQ̃Θ̃ −CQ̃P̃
 MQ̃Ã
→ 

 MΠ̃Ã
0
0
0
0
0


 0
0
0
NΘ̃P̃
NΘ̃Φ̃ NΘ̃Q̃


MP̃ Ã
0
CP̃ Q̃
0
−NP̃ Θ̃
0
where the new matrix elements are:
MX̃ Ỹ
NX̃ Ỹ
CX̃ Ỹ
∂a
∂π ∂ 2 hΠ̃ + ΠC |Ãi ∂a
∂ 2 hΠ̃ + ΠC |Ãi
∂π
M
=
=
≡
∂π∂a
∂ X̃ ∂ Ỹ
∂ X̃
∂ Ỹ
∂ X̃∂ Ỹ
2
2
∂θ ∂ hΘ|Φ̃ + ΦC i ∂φ
∂ hΘ̃|Φ̃ + ΦC i
∂θ ∂φ
N
=
=
≡
∂θ∂φ
∂ X̃ ∂ Ỹ
∂ X̃
∂ Ỹ
∂ X̃∂ Ỹ
2
2
∂p ∂q
∂p ∂ hP |Qi ∂q
∂ hP̃ |Qi
≡
C
=
=
∂ X̃ ∂ Ỹ
∂ X̃ ∂p∂q ∂ Ỹ
∂ X̃∂ Ỹ

(5.38)







,







(5.39)
for X and Y an arbitrary dynamical variable. The remaining elements are determined by transposition. Again the extra terms in ω̃ add new time-dependent
couplings to the theory. These new terms can all be performed analytically. The
101
resulting equations

0
 0

 0
0



 MQ̃Ã
0


 MΠ̃Ã
0


 0
NΘ̃Φ̃


MP̃ Ã
0
5.3.1
of motion are
−MÃQ̃ −MÃΠ̃
0
0
0
−NΦ̃Θ̃
0
0
−NQ̃Θ̃
0
0
0
NΘ̃Q̃
0
0
CP̃ Q̃
0
−NP̃ Θ̃

−MÃP̃  


0



−CQ̃P̃  



0



NΘ̃P̃ 


0


ã˙  
 
˙

φ̃ 
 
 
 
q̃˙  
=
 

π̃˙ 
 
 

θ̃˙ 
 
 
p̃˙

∂H/∂ã 

∂H/∂ φ̃ 



∂H/∂ q̃ 
.

∂H/∂ π̃ 


∂H/∂ θ̃ 


∂H/∂ p̃
(5.40)
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


0
−MÃQ̃ −MÃΠ̃
0
−MÃP̃ 
 0



 0
0
0
0
−NΦ̃Θ̃
0






 0
0
0
0
0
−CQ̃P̃ 
.

(5.41)
ω̃ → 


 MΠ̃Ã
0
0
0
0
0




 0
0
0
NΘ̃P̃ 
NΘ̃Φ̃ NΘ̃Q̃




0
0
CP̃ Q̃
0
0
0
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 η̃ b . The forces
∂H/∂ η̃ a 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 stationary state of the hydrogen atom. In a basis of six gaussians an s-wave was
constructed. The corresponding basis representation of the Coulombic scalar potential 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
R
produced. The total charge q = V ρ(x)d3 x 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
R
q = V ρ(x)d3 x 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 constant.
5.3.5
Analysis of Solutions in Coulomb Basis
As done previously, the evolving state vector η̃(t) in the Coulomb basis is expanded in terms of the stationary eigenbasis η̃M ≡ ηM according to
|η̃(t)i =
P
M |η̃M ihη̃M |η̃(t)i.
(5.35)
As before, the real and imaginary parts of the Schrödinger coefficients for a superposition 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 dynamics, which correspond to excitations of the s-, px -, and dy2 −z 2 -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 sand px -states. Lastly, the phase space contour of the Schrödinger 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 behavior is due to the fact the electromagnetic radiation produced by the electron
cannot escape to infinity and carry away energy, momentum, and angular momentum. Rather, the radiation reflects off of the artificial boundaries of the finite spatial
basis and reexcites the electron.
rag replacements
104
s
px
py
pz
1
d x2
2
dy2 −z
RehΨM |Ψ(t)i 0
0
10
t
20
30
Figure 5–9: Real part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where η(t)
is a superposition of s- and px -waves.
s
px
py
pz
1
d x2
2
dy2 −z
ImhΨM |Ψ(t)i 0
−1
0
10
t
20
30
Figure 5–10: Imaginary part of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i,
where η(t) is a superposition of s- and px -waves.
PSfrag replacements
−1
−10
−20
−30
ImhΨM |Ψ(t)i
pK (t)
1
0.001
−0.001
−1
RehΨN |Ψ(t)i
qK (t)
P
105
2
M |hΨM |Ψ(t)i|
s
px
0.8
0.6
|hΨM |Ψ(t)i|2
0.4
0.2
0
0
10
t
20
30
Figure 5–11: Probability for the electron to be in a particular basis eigenstate.
pK (t)
0.5
ImhΨM |Ψ(t)i 0
−0.5
−1
−1
−0.5
0
RehΨM |Ψ(t)i
0.5
1
Figure 5–12: Phase space of the Schrödinger coefficients CM (t) ≡ hηM |η(t)i, where
η(t) is a superposition of s- and px -waves.
30
d x2 2
dy2 −z
−10
−20
−30
0.001
−0.001
qK (t)
s
px
py
pz
1
106
The asymptotic problem, be it electromagnetic radiation or free (ionized) electrons, 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 Schrödinger
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 problems. 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 functions, repetitive projection and complex rotation methods, and Siegert pseudostates
are common theoretical tools. These techniques are discussed by Yoshida, Watanabe, Reinhold, and Burgdörfer 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 formulation of scattering theory. The exterior complex scaling method maps all coordinates 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.
g replacements
107
Asymptotic basis
Local basis
φ(t)
t
Figure 5–13: Schematic picture of the local and asymptotic basis proposed for the
description of electromagnetic radiation and electron ionization. The amplitude
from the asymptotic basis is dumped into the free field φ, which acts as a storage
tank 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-Schrödinger 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 ).
Further away, a set of complex basis functions of the form
(5.4)
108
Gk (x) =
P
lm Clm Ylm (x̂)
eikr
exp(−ar 2 )
r
(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 ∗ ψ̇ − {i∇ψ ∗ · −i∇ψ}/2m
or the free field Lagrangian ∂α φ∗ ∂ α φ. 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 Schrödinger 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
− c∇Φ · Π − cΘ∇ · A
8π
Q2 + P 2
[∇Q + qAP/c]2 + [−∇P + qAQ/c]2
+ qΦ
+
4mq
2
2
2
2
[∇U + q̄AW/c] + [−∇W + q̄AU/c]
U + W2
+
+ q̄Φ
4mq̄
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) =
P (x, t) =
P
P
K GK (x)qK (t)
U (x, t) =
K GK (x)pK (t)
W (x, t) =
P
P
K GK (x)uK (t)
(5.45)
K GK (x)wK (t).
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 −M
T
0
0
0
0
0
−N
0
0
0
0
0
0
0
M
0
0
0
T

0
0
0
0
−C T
0
0
0
0
−K T
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





















ȧ
φ̇
q̇
u̇
π̇
θ̇
ṗ
ẇ


 
 
 
 
 
 
 
 
 
 
=
 
 
 
 
 
 
 
 
 
 
∂H/∂a
∂H/∂φ
∂H/∂q
∂H/∂u
∂H/∂π
∂H/∂θ
∂H/∂p
∂H/∂w











,










(5.30)
where
MKM ≡
∂ 2 hΠ|Ai
∂πK ∂aM
NKM ≡
∂ 2 hΘ|Φi
∂θK ∂φM
CKM ≡
∂ 2 hP |Qi
∂pK ∂qM
and where the integrals h·|·i involve only gaussian functions.
KKM ≡
∂ 2 hW |U i
∂wK ∂uM
(5.46)
CHAPTER 6
CONCLUSION
Nonperturbative analytical and numerical methods for the solution of the nonlinear Maxwell-Schrödinger equations have been presented including the complete
coupling of both systems. The theory begins by applying the calculus of variations
to the Maxwell and Schrödinger 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
ωab η̇ b = ∂H/∂η a .
(6.1)
These Maxwell-Schrödinger equations are closed when the Schrödinger 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 welldefined 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-Schrödinger equations are represented in a finite basis
of gaussian functions GK (x) and solved numerically. That is, each dynamical variable
is expanded in this basis according to
η(x, t) =
P
K GK (x)ηK (t),
(6.2)
where the time-dependent coefficients ηK (t) carry the dynamics. As a result, a hierarchy of approximate equations of motion are generated that basis-represent the exact Maxwell-Schrödinger 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-Schödinger 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 atomfield 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-Schrödinger equations to isolate the Coulombic or electrostatic 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 ∇2 ΦC = −4πρ and then calculating EC = −∇ΦC . By removing 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 situations 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-Schrödinger theory lie
in photon-electron-phonon dynamics in semiconductor quantum wells [10], spontaneous 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 A
GAUGE TRANSFORMATIONS
A.1
Gauge Symmetry of Electrodynamics
The basic equations of electrodynamics [9] are:
4π
Ė
J+
c
c
∇ · E = 4πρ
∇×B=
∇·B=0
Ḃ
= 0,
∇×E+
c
(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
∇ · Ȧ
= −4πρ
c
h
Ä
Φ̇ i
4π
2
∇ A− 2 −∇ ∇·A+
= − J
c
c
c
∇2 Φ +
(A.2a)
(A.2b)
by defining the electric field E = −Ȧ/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 → Φ0 = Φ −
Ġ
c
(A.3)
A → A0 = A + ∇G,
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 = ∇ · A0 +
Φ̇ G̈
Φ̇0
= ∇ · A + ∇2 G + −
c
c
c
113
(A.4)
114
implies that the Lorenz gauge function satisfies the equation G = −[∇ · A + Φ̇/c].
This choice of gauge function leads to the manifestly Lorentz invariant equations of
motion:
Φ̈
= −4πρ
c2
Ä
4π
∇2 A − 2 = − J.
c
c
∇2 Φ −
(A.5a)
(A.5b)
Another choice is the Coulomb (radiation) gauge, which leads to the equations of
motion:
∇2 Φ = −4πρ
∇2 A T −
ÄT
4π
= − JT
2
c
c
(A.6a)
(A.6b)
with the gauge function satisfying ∇2 G = −∇ · 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)
R d3 k ik·x h ab k a k b i b
∂a∂b i b
δ − 2 Ã (k, t),
e
= δ − 2 A (x, t) = V
∇
(2π)3
k
h
ab
(A.7)
where PTab = δ ab − ∂ a ∂ b /∇2 is the transverse projection operator (see Appendix
R
C) and 1/∇2 is shorthand for the Green’s function V d3 x0 [−4π|x − x0 |]−1 of the
Laplacian operator (see Appendix B), where (∇2 /∇2 )g = g for g a well-behaved
function. More precisely
R 3 0
−1
1 2
∇
g(x)
=
d x g(x0 )∇02
= g(x).
V
2
∇
4π|x − x0 |
(A.8)
The dynamical radiation fields associated with AT are almost separated from the instantaneous or static fields associated with Φ, as seen in (A.6). A closer examination
of the transverse projection operator PTab will show that even AT contains instantaneous 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 ∇2 G = 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 situation 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 mechanics exhibits the additional symmetry
Ψ → Ψ0 = 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 invariant coupling schemes, the minimal coupling prescription embodied in (3.20) is
the simplest.
APPENDIX B
GREEN’S FUNCTIONS
The Green’s function or fundamental solution G associated with the inhomogeneous (partial) differential equation Lu(k) = h(k) satisfies the equation
LG(k, k0 ) = δ (n) (k − k0 )
(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, k0 ) = G(k0 , 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
R
u(k) = d(n) k 0 h(k0 )G(k0 , 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
R
h(k) = L d(n) k 0 h(k0 )G(k0 , k)
R
= d(n) k 0 h(k0 )δ (n) (k0 − 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
R d(n) k ik·x
1e
= δ (n) (x),
n
(2π)
(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
R
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


 0, if x lies outside Ω
R (n) (n)
d
y
δ
(y
−
x)
=
,
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
R
d(n) y g(y)
Ω
∂ (n)
∂g(y) δ (y − x) = −
∂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
The ∇2 Operator
B.2
Consider the Poisson equation from electrodynamics
∇2 Φ = −4πρ,
(B.8)
where Φ is the scalar potential and ρ is the charge density. The corresponding
Green’s function satisfies the equation
∇2 G(x, x0 ) = δ(x − x0 ),
(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)2 G̃ = exp(−ik · x0 ). By another Fourier
inversion
0
G(x, x0 ) = −
R
−1
d3 k eik·(x−x )
=
.
(2π)3
k2
4π|x − x0 |
V
(B.10)
With (B.10) the Poisson equation (B.8) is recast as the integral equation
Φ(x, t) =
=
R
R
V
d3 x0 [−4πρ(x0 , t)]G(x0 , x)
d 3 x0
V
ρ(x0 , t)
.
|x − x0 |
(B.11)
Together with the homogeneous solutions of the Laplace equation ∇2 Φ = 0, the
total solution is obtained.
B.3
The ∂ 2 Operator
Consider the wave equation from electrodynamics
−∂ 2 Φ = Φ = ∇2 Φ −
Φ̈
= −4πρ,
c2
(B.12)
where Φ is the scalar potential and ρ is the charge density as before. The corresponding Green’s function satisfies
h
1 ∂2 i
∇ − 2 2 G(x, t; x0 , t0 ) = δ(x − x0 )δ(t − t0 ).
c ∂t
2
(B.13)
119
The solution of (B.13) will require the Fourier inversion in both x and t to obtain
˜ = − exp(−i[k · x0 − ωt0 ]). Division by [k2 − ω 2 /c2 ] followed by another
[k2 − ω 2 /c2 ]G̃
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
¨ + (ck)2 Φ̃ = 4πc2 ρ̃,
Φ̃
(B.14)
with Green’s function satisfying
h
i
(d2 /dt2 ) + (ck)2 Dk (t, t0 ) = δ(t − t0 ).
(B.15)
Fourier inversion in t results in [ω 2 − c2 k 2 ]D̃k = − exp(iωt0 ) from which the retarded
Green’s function becomes
(+)
Dk (t, t0 )
0
R ∞ dω e−iω(t−t0 )
0 sin ck(t − t )
= − −∞
=
Θ(t
−
t
)
,
2π ω 2 − c2 k 2
ck
(B.16)
where the integration was performed in lower half complex plane. Similarly the
(−)
advanced solution is Dk (t, t0 ) = Θ(t0 − t) sin ck(t − t0 )/ck by integrating in the
upper half plane.
The solution to (B.13) is more clear, now that this first step has been accom˜ = − exp(−i[k · x0 − ωt0 ]) must
plished. It has already been shown that [k2 − ω 2 /c2 ]G̃
be Fourier inverted twice in order to obtain G. The resulting Green’s function is
0
d3 k ik·(x−x0 ) R ∞ dω e−iω(t−t )
G(x, t; x , t ) = c V
e
−∞ 2π ω 2 − c2 k 2
(2π)3
0 R
cΘ(t − t ) ∞
(B.17)
= 2
dk sin ck(t − t0 ) sin(k|x − x0 |)
0
0
2π |x − x |
i
cΘ(t − t0 ) h
0
0
0
0
δ |x − x | − c(t − t ) − δ |x − x | + c(t − t ) ,
=
4π|x − x0 |
0
0
2
R
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; x0 , t0 ) =
cΘ(t − t0 )
0
0
δ
|x
−
x
|
−
c(t
−
t
)
4π|x − x0 |
(B.18)
while the second term is just zero since both |x − x0 | and c(t − t0 ) are positive.
An analogous computation with the advanced Green’s function D (−) leads to the
advanced solution
PSfrag replacements
G(−) (x, t; x0 , t0 ) =
cΘ(t0 − t)
0
0
δ
|x
−
x
|
+
c(t
−
t
)
.
4π|x − x0 |
(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-time
ct
r(t)
future
elsewhere
x
past
Figure B–1: The trajectory or world line r(t) of a massive particle moves from past
to future within the light cone. A massless particle such as a photon propagates on
the light cone.
originated from a point that is a distance |x − x0 | = c(t − t0 ) away and at a time
t0 = t − |x − x0 |/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
R
3 0 ∞
d
x
dt0 [−4πρ(x0 , t)]G(±) (x0 , t0 ; x, t)
V
−∞
R 3 0R ∞ 0 ρ(x0 , t0 ) 0 h
|x − x0 | i
= V d x −∞ dt
δ t − t∓
.
|x − x0 |
c
Φ(x, t) =
R
(B.20)
121
Together with the homogeneous solutions of the wave equation Φ = 0, the total
solution is obtained.
APPENDIX C
THE 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
PSfrag replacements
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
k
vk ∼ k(k · v)/k 2
v
v⊥
−k(k · v)/k 2
r(t)
Figure C–1: Since à = h̃v, the transverse vector potential Ã⊥ = [v − k(k · v)/k 2 ]h̃
and the longitudinal vector potential Ãk = [k(k·v)/k 2 ]h̃, where h̃ is a scalar function.
is easily visualized by going to the Fourier space. There ik · Ã⊥ = 0 and ik × Ãk =
0, where Ã⊥ is the Fourier inverse of AT and Ãk 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 PTij = δ ij − ∂ i ∂ j /∇2
according to
h
n −h(x, t) o
∂i∂j i
PTij h(x, t)vj = δ ij − 2 h(x, t)vj = [∂ i ∂ j − ∇2 δ ij ]
vj = T ij (x, t)vj .
2
| {z }
∇
∇
{z
}
|
Aj (x,t)
g(x,t)
(C.2)
Note that PTik = PTij PTjk and P† = P. In Fourier space T ij is of the form
T̃ ij (k, t) = [k i k j − k 2 δ ij ]g̃(k, t)
= [3k i k j − k 2 δ ij ]g̃(k, t) + [−2k i k j ]g̃(k, t)
|
{z
} |
{z
}
Q̃ij (k,t)
(C.3)
M̃ ij (k,t)
where g̃ 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 J̃ = qvΘ(t − t1 )Θ(t2 − t) exp(−ik · [r + vt] − k 2 /8`2 ). The
first instantaneous or Coulomb-like piece is
√
erf
2`|x
−
(r
+
vt)|
qv
v
A(0) (x, t) = Φ(0) (x, t) =
c
c
|x − (r + vt)|
while the remaining three pairs of radiating terms are of the form
√
qv n erf 2`[c(t − tk ) − |x − (r + vtk )|]
v (k)
(k)
A (x, t) = Φ (x, t) =
c
2c
|x − (r + vtk )|
√
erf 2`[c(t − tk ) + |x − (r + vtk )|] o
−
|x − (r + vtk )|
(C.5)
(C.6)
124
for k = 1, 2. The first term in (C.3) resembles the traceless quadrupole moment
tensor and is labeled Q̃ij , while the second term M̃ ij is the remainder. It will be
seen that the M̃ ij tensor can be completely determined from Q̃ij .
Solving for each component of T ij first and then contracting with the velocity
eliminates the Fourier inversion of
Ã⊥ (k, t) = [k(k · v) − k 2 v]g̃(k, t)
(C.7)
in favor of the angularly inferior
T ij (x, t) =
R
V
d3 k i j
[k k − k 2 δ ij ]g̃(k, t)eik·x ,
(2π)3
(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 Q̃ij (although the same is true for both M̃ ij and T̃ ij ), it can
be shown that while Q̃ij 6= 0 for all i and j, its Fourier inverse Qij is diagonal in a
certain frame. For example (for A(0) )
(0)
(0)
−4πq −ik·(r+vt)−k2 /8`2
e
6= 0,
ck 4
(C.10)
−4πq ik·[x−(r+vt)]−k2 /8`2
d3 k
3kx ky
e
=0
3
(2π)
ck 4
(C.11)
Q̃12 (k, t) = 3kx ky g̃C (k, t) = 3kx ky
while
(0)
Q12 (x, t) =
R
V
in the boosted frame of origin r + vt where x − (r + vt) is rotated about the origin
to lie along the êz -axis. As expected, if x − (r + vt) is placed in general along the
constant vector n̂0 = êx sin θ0 cos φ0 + êy sin θ0 sin φ0 + êz cos θ0 , then the angular
(0)
part of Q12 becomes
R
dΩ sin θ cos φ sin θ sin φ 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 x0 = r(0, 0, 1) which lies along the êz -axis.
This spatial rotation is performed via the rotation matrix


cos θ cos φ cos θ sin φ − sin θ 


Λkl (Ω) = 
cos φ
0 
 − sin φ
,


sin θ cos φ sin θ sin φ cos θ
(C.13)
where x0k = Λkl xl or in matrix notation x0 = Λx. With Λkl , the transverse projection
tensor in (C.8) may be rotated to the diagonal frame by
T kl (x) → T 0kl (x0 ) =
=
∂x0k ∂x0l ij −1 0
T (Λ x )
∂xi ∂xj
(C.14)
Λki Λl j T ij (x)
where Λki Λl j T ij = Λki T ij [Λ−1 ]j l or T0 = 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 0kl (x0 ) = Λki T ij (x)[Λ−1 ]j l =
R
V
R d3 k 0 ik0 ·x0 0kl 0
d3 k ik·x k ij
−1 l
e
Λ
T̃
(k)[Λ
]
=
e
T̃ (k ),
i
j
V
(2π)3
(2π)3
(C.15)
where d3 k 0 = det Λ d3 k = d3 k and k0 · x0 = k · x. In this frame the three diagonal
elements of T 0kl are known. They are




011
0
011
0
012
0
013
0
0
0 
T (x )
T̃ (k ) T̃ (k ) T̃ (k )
 R d3 k 0 ik0 ·x0 


=
 0
T̃ 021 (k0 ) T̃ 022 (k0 ) T̃ 023 (k0 ) .
022
0
e
T
(x
)
0
V




(2π)3




0
0
T 033 (x0 )
T̃ 031 (k0 ) T̃ 032 (k0 ) T̃ 033 (k0 )
(C.16)
Each of the elements T 0ii (x0 ) will be computed in the diagonal frame. However the physics is not correct until x0 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


2
2
2
cos θ sin φ cos φ − sin θ cos θ cos φ
 cos θ cos φ

 011 0
2
T kl (x) = 
cos2 θ sin2 φ
− sin θ cos θ sin φ 
 cos θ sin φ cos φ
 T (x )


2
− sin θ cos θ cos φ − sin θ cos θ sin φ
sin θ


2
− sin φ cos φ 0
 sin φ

 022 0
+
cos2 φ
0
− sin φ cos φ
 T (x )


0
0
0


2
2
2
sin θ sin φ cos φ sin θ cos θ cos φ
 sin θ cos φ

 033 0
 T (x )
2
2
2
+
sin
θ
sin
φ
cos
φ
sin
θ
sin
φ
sin
θ
cos
θ
sin
φ




sin θ cos θ cos φ sin θ cos θ sin φ
cos2 θ
(C.17)
which may be written in matrix form as T = Λ−1 T0 Λ. The transverse vector potential AkT (x) is now obtained by the simple contraction T kl (x)vl . Note that the T 0kk (x0 )
appearing in (C.17) are scalar functions that only depend upon the norm of x0 .
A proper treatment of the tensor calculus reveals that ∇0 · A0T (x0 , t) = ∇ ·
AT (x, t). Componentwise that is
∂k0 T 0kl (x0 , t)vl0 = ∂i T ij (x, t)vj .
(C.18)
It must still be verified that AT is divergenceless in either frame. It is sufficient to
show that ∂i T ij vj = 0. This result is most easily shown by working in the (r, θ, φ)
basis with unit vectors
êx = êr sin θ cos φ + êθ cos θ cos φ − êφ sin φ
êy = êr sin θ sin φ + êθ cos θ sin φ + êφ cos φ
êz = êr cos θ − êθ sin θ.
(C.19)
127
In these spherical coordinates the the velocity becomes






 
 cos θ 
 sin θ sin φ 
 sin θ cos φ 
v x 






 
 = v1 cos θ cos φ + v2 cos θ sin φ + v3 − sin θ 
v=
v






 y






 
0
cos φ
− sin φ
vz
and the transverse vector potential is just






33
33
33
 T cos θ 
 T sin θ sin φ 
 T sin θ cos φ 







 11

 11

11
AT = v 1 
T cos θ cos φ + v2 T cos θ sin φ + v3 −T sin θ  .






22
22
0
T cos φ
−T sin φ
(C.20)
(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 0ii as
seen in (C.21). In the following section the three T 0ii 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 0kk (x0 , 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
d3 k
[Q̃ij (k, t) + M̃ ij (k, t)]eik·x vj
V
3
(2π)
R d3 k
R d3 k
i j
2 ij
ik·x
= V
[3k
k
−
k
δ
]g̃(k,
t)e
v
+
[−2k i k j ]g̃(k, t)eik·x vj ,
j
V
(2π)3
(2π)3
AiT (x, t) =
R
(C.22)
128
where it was noticed that Q̃ij is analogous to the traceless quadrupole moment tensor
from electrodynamics [9]. In the diagonal frame Q̃0ij looks like


011
0
0
0
Q̃ (x , t)



0ij
0


Q̃ (x , t) = 
0
Q̃022 (x0 , t)
0



033
0
0
0
Q̃ (x , t)
(C.23)
with Q̃011 + Q̃022 + Q̃033 = 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
√
v (0)
qv erf 2`|x − (r + vt)|
A (x, t) = Φ (x, t) =
c
c
|x − (r + vt)|
(0)
(C.25)
and three pairs of radiating terms
n erf √2`[c(t − t ) − |x − (r + vt )|]
qv
v
k
k
A(k) (x, t) = Φ(k) (x, t) =
c
2c
|x − (r + vtk )|
√
erf 2`[c(t − tk ) + |x − (r + vtk )|] o
−
|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
d3 k
(0)
(1)
[3k i k j − k 2 δ ij ]{g̃C (k, t) + g̃R (k, t)}eik·x
V (2π)3

−4πq −ik·(r+vt)−k2 /8`2


 ck 4 e
R d3 k
[3k i k j − k 2 δ ij ]
= V

(2π)3

 + 4πq e−ik·(r+vt1 )−k2 /8`2 cos ck(t − t1 )
ck 4
Qij (x, t) =
R




eik·x



(C.27)
and
M ij (x, t) =
(0)
R
V
d3 k
(0)
(1)
[−2k i k j ]{g̃C (k, t) + g̃R (k, t)}eik·x
3
(2π)
(C.28)
(k)
where −k 2 g̃C v is the Fourier inverse of A(0) and −k 2 g̃R v is the Fourier inverse of
A(k) . Higher order terms in v/c are omitted from (C.27).
(0)
(0)
Taking g̃ in (C.22) as g̃C , the Qij tensor becomes
(0)
Qij
d3 k
−4πq ik·[x−(r+vt)]−k2 /8`2
2
= V
[3k
k
−
k
δ
]
e
.
i
j
ij
(2π)3
ck 4
R
(C.29)
(0)
Using cylindrical symmetry it is found that all off-diagonal elements of Qij are zero
(0)
(0)
and Q11 = Q22 when the vector x − (r + vt) is rotated to the frame where it lies
(0)
(0)
(0)
(0)
along the z-axis. But since Qij is traceless, Q33 = −2Q11 . Thus, solving for Q33
determines Q(0) entirely. That is
(0)
Q33
d3 k
−4πq ik·[x−(r+vt)]−k2 /8`2
= V
[3kz2 − k 2 ]
e
3
(2π)
ck 4
z 2 Φ(0)
3q R ∞
−z 2 /16`2
dz
cos(|x
−
(r
+
vt)|z)e
W
=−
+
−1,−1/2
πc 0
8`2
c
(0)
2q`
Φ
1 3 1 5
, ; , ; −2`2 |x − (r + vt)|2 +
= −√
2 F2
2 2 2 2
c
2πc
√
2 [x−(r+vt)]2
−2`
3q erf 2`|x − (r + vt)|
3q e
Φ(0)
√
=− 2
,
+
+
8` c
|x − (r + vt)|3
c
8π`c |x − (r + vt)|2
R
(C.30)
130
where the primes which denote this diagonal frame have been temporarily omitted
(0)
and where Gradshteyn and Ryzhik [87] was used. Notice that Q33 not only de(0)
(0)
(0)
(0)
(0)
(0)
termines Q11 and Q22 , it also determines M11 , M22 , and M33 . Taking M̃ij as
(0)
M̃ij = [−2ki kj ]
(0)
(0)
−4πq −ik·(r+vt)−k2 /8`2
e
ck 4
(0)
(C.31)
(0)
(0)
it is found that M11 = M22 = [cQ33 + 2Φ(0) ]/3c and M33 = −2[cQ33 − Φ(0) ]/3c
(0)
with Mij = 0 for i 6= j. With this, the vector potential transverse to A(0) becomes
(0)
AT = [Q(0) + M(0) ]v.
(1)
(C.32)
(1)
Taking g̃ in (C.22) as g̃R , the Qij tensor becomes
(1)
Qij =
R
V
4πq
d3 k
2
2
[3ki kj − k 2 δij ] 4 eik·[x−(r+vt1 )]−k /8` cos ck(t − t1 ).
3
(2π)
ck
(C.33)
(1)
Using spherical symmetry it is again found that all off-diagonal elements of Qij are
(1)
(1)
zero and Q11 = Q22 when the vector x − (r + vt1 ) is chosen to lie along the z-axis.
(1)
(1)
(1)
(1)
Again since Qij is traceless, Q33 = −2Q11 . Thus, solving for Q33 determines Q(1)
entirely. That is
(1)
d3 k
4πq
2
2
[3kz2 − k 2 ] 4 eik·[x−(r+vt1 )]−k /8` cos ck(t − t1 )
V
3
(2π)
ck
12q` −2`2 c2 (t−t1 )2 R 1
2
2 2
du u2e−2` [x−(r+vt1 )] u
e
=√
0
2πc
Q33 =
R
Φ(1)
· cosh 4`2 c(t − t1 )|x − (r + vt1 )|u +
c
n
−2`2 [c(t−t1 )+|x−(r+vt1 )|]2
3q
[c(t − t1 ) − |x − (r + vt1 )|]e
=√
|x − (r + vt1 )|3
8π`c
2
2
[c(t − t1 ) + |x − (r + vt1 )|]e−2` [c(t−t1 )−|x−(r+vt1 )|] o
−
|x − (r + vt1 )|3
√
3q 1 + 4`2 c2 (t − t1 )2 n
erf
2`[c(t
−
t
)
+
|x
−
(r
+
vt
)|]
+ 2
1
1
8` c |x − (r + vt1 )|3
√
o Φ(1)
,
− erf 2`[c(t − t1 ) − |x − (r + vt1 )|] +
c
(C.34)
131
(1)
(1)
(1)
where Gradshteyn and Ryzhik [87] was used. Similarly Q33 determines M11 , M22 ,
(1)
(1)
and M33 . Taking M̃ij as
(1)
M̃ij = [−2ki kj ]
(1)
4πq −ik·(r+vt1 )−k2 /8`2
e
cos ck(t − t1 )
ck 4
(1)
(1)
(1)
(C.35)
(1)
it is found that M11 = M22 = [cQ33 + 2Φ(1) ]/3c and M33 = −2[cQ33 − Φ(1) ]/3c
(1)
with Mij = 0 for i 6= j. With this, the vector potential transverse to A(1) becomes
(1)
AT = [Q(1) + M(1) ]v
(0)
(C.36)
(1)
so that within the step ATin = AT + AT .
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
d3 k
(1)
(2)
[3k i k j − k 2 δ ij ]{g̃R (k, t) − g̃R (k, t)}eik·x
V
3
(2π)

4πq −ik·(r+vt1 )−k2 /8`2


cos ck(t − t1 )
e

R d3 k
ck 4
i j
2 ij
= V
[3k k − k δ ]

(2π)3

 − 4πq e−ik·(r+vt2 )−k2 /8`2 cos ck(t − t2 )
ck 4
Qij (x, t) =
R




eik·x



(C.37)
and
M ij (x, t) =
(k)
R
V
d3 k
(1)
(2)
[−2k i k j ]{g̃R (k, t) − g̃R (k, t)}eik·x ,
3
(2π)
where −k 2 g̃R v is the Fourier inverse of A(k) .
(C.38)
132
(1)
(1)
Taking g̃ in (C.22) as g̃R , the zz-component of the tensor Qij is
2
(1)
Q33
2
3q n [c(t − t1 ) − |x − (r + vt1 )|]e−2` [c(t−t1 )+|x−(r+vt1 )|]
=√
|x − (r + vt1 )|3
8π`c
2
2
[c(t − t1 ) + |x − (r + vt1 )|]e−2` [c(t−t1 )−|x−(r+vt1 )|] o
−
|x − (r + vt1 )|3
√
3q 1 + 4`2 c2 (t − t1 )2 n
+ 2
2`[c(t
−
t
)
+
|x
−
(r
+
vt
)|]
erf
1
1
8` c |x − (r + vt1 )|3
√
o Φ(1)
− erf 2`[c(t − t1 ) − |x − (r + vt1 )|] +
c
(C.39)
(1)
(1)
(1)
(1)
(1)
and as before M11 = M22 = [cQ33 + 2Φ(1) ]/3c and M33 = −2[cQ33 − Φ(1) ]/3c with
(1)
Mij = 0 for i 6= j.
(2)
(2)
Lastly, taking g̃ in (C.22) as g̃R , the zz-component of Qij is found to be
2
2
3q n [c(t − t2 ) − |x − (r + vt2 )|]e−2` [c(t−t2 )+|x−(r+vt2 )|]
|x − (r + vt2 )|3
8π`c
2
2
[c(t − t2 ) + |x − (r + vt2 )|]e−2` [c(t−t2 )−|x−(r+vt2 )|] o
−
|x − (r + vt2 )|3
√
3q 1 + 4`2 c2 (t − t2 )2 n
erf 2`[c(t − t2 ) + |x − (r + vt2 )|]
+ 2
3
8` c |x − (r + vt2 )|
√
o Φ(2)
.
− erf 2`[c(t − t2 ) − |x − (r + vt2 )|] +
c
(C.40)
(2)
Q33 = √
(2)
Taking M̃ij as
(2)
M̃ij = [−2ki kj ]
(2)
4πq −ik·(r+vt1 )−k2 /8`2
cos ck(t − t1 )
e
ck 4
(2)
(2)
(2)
(C.41)
(2)
it is found that M11 = M22 = [cQ33 + 2Φ(2) ]/3c and M33 = −2[cQ33 − Φ(2) ]/3c
(2)
with Mij = 0 for i 6= j. With this, the vector potential transverse to A(2) becomes
(2)
AT = [Q(2) + M(2) ]v
(1)
(2)
so that outside the step ATout = AT − AT .
(C.42)
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 −
(C.43)
(0)
t)[AT
These results are presented in [88].
+
(1)
AT ]
+ Θ(t −
(1)
t2 )[AT
−
(2)
AT ].
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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 University 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 Öhrn and Dr. Erik Deumens at the Quantum Theory Project. During
his third year of graduate school David married his college sweetheart, Kathryn
Allida Masiello.
142