Table Of ContentNON-RELATIVISTIC
QUANTUM ELECTRODYNAMICS
NON-RELATIVISTIC
QUANTUM ELECTRODYNAMICS
W. P. Healy
Department of Mathematics
Royal Melbourne Institute of Technology
Melbourne, Victoria
Australia
1982
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Copyright © 1982 by ACADEMIC PRESS INC. (LONDON) LTD
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Healy, W. P.
Non-relativistic quantum electrodynamics.
1. Quantum electrodynamics
I. Title
537.6 QC680 LCCCN 82-71002
ISBN 0-12-335720-9
Typeset and printed by The Universities Press (Belfast) Ltd.
Preface
Non-relativistic quantum electrodynamics is the theory of the interaction
of photons and slowly moving charged particles, such as those found in
atoms and molecules under normal laboratory conditions. In recent years
there has been a resurgence of interest in this subject, due in part to the
development of lasers and their use as spectroscopic tools for investigat
ing a variety of physical and chemical phenomena. This book presents a
systematic account of non-relativistic quantum electrodynamics and is
aimed at theoretical physicists, applied mathematicians and physical
chemists who have some knowledge of classical electrodynamics and of
Dirac’s formulation of ordinary quantum mechanics.
The theory is derived by the method of canonical quantization and is
thus related to classical electrodynamics through Bohr’s correspondence
principle. This method seems the most natural and easiest to use in a
non-relativistic context and, historically, is the one by which the electro
magnetic field was first quantized. It is true that the description of the
spin of the charged particles does not emerge automatically from the
resulting formalism, but this can, if desired, be added on in an ad hoc
fashion, as in non-relativistic quantum mechanics. The canonical formula
tion of classical electrodynamics in the Coulomb gauge is dealt with at
some length, as are the transformations of the energy and momentum
densities of the system. Particular attention is drawn to the inversion of
the relations defining the electromagnetic potentials that may be achieved
by taking certain line integrals over the fields. These path-dependent
potentials are interesting in their own right, but do not appear to be
widely known. Moreover, they are closely associated with the path-
dependent polarization and magnetization fields that occur in the multi
polar form of the Hamiltonian, which is related to the more standard
minimal-coupling form through the unitary transformation that was first
carried out by E. A. Power and S. Zienau. The interpretation of this
transformation is still, even now, a subject of debate in the literature. It is
hoped that the treatment given here will show (i) that the correct
canonical variables and consistent equations of motion are obtained by
interpreting the transformation as the quantum analogue of a classical
VI PREFACE
canonical transformation and (ii) that the partitioning (for the purposes of
perturbation theory) of the multipolar Hamiltonian is intrinsically path-
dependent, due to the path-dependence of the polarization and magne
tization fields. This can be regarded as a kind of gauge dependence. The
gauge invariance of physical results, however, is demonstrated for some
cases.
The summation convention is used, so that a repeated Latin suffix
always implies a sum. References are listed alphabetically according to
the surname of the author at the end of each chapter. They are cited in
the text by author name and year of publication. A list of physical
constants, some theorems in vector analysis, a note on longitudinal and
transverse vector fields and two operator identities appear in Appendices
A, B, C and D, respectively. The reader is referred to these where
necessary in the body of the book. The problems that are given at the end
of many sections are meant to illustrate the theory, and in some cases to
develop it further.
The work leading to this book was mostly carried out in the Research
School of Chemistry, Institute of Advanced Studies, and the Department
of Applied Mathematics, Faculty of Science, at the Australian National
University. Some of the last section of Chapter 7 is based on work done
in collaboration with Dr. R. G. Woolley. I am very grateful to Professor
D. P. Craig of the Research School of Chemistry for suggesting and
encouraging the project and to Mrs B. Hawkins of the Department of
Applied Mathematics for typing the manuscript.
W. P. Healy
Melbourne
March 1982
Contents
Preface v
Chapter 1 Introduction
1.1 Non-relativistic theory 1
1.2 Planck’s radiation law 2
1.3 Einstein’s A and B coefficients 5
1.4 Uncertainty relations 8
Chapter 2 The Classical Equations of Motion
2.1 Maxwell-Lorentz theory 14
2.2 C, P and T symmetries 16
2.3 Vector and scalar potentials 20
2.4 Gauge transformations 23
2.5 The Coulomb gauge 27
2.6 Energy and momentum balance 31
Chapter 3 Canonical Formalism
3.1 Discrete systems—particles 38
3.2 Continuous systems—fields 43
3.3 Transverse fields 47
3.4 Canonical formulation of electrodynamics in the Coulomb gauge 50
3.5 Conservation of energy and momentum; Noether’s principle 55
Chapter 4 Canonical Quantization
4.1 Introduction 61
4.2 Equations of motion 63
4.3 Photons 68
4.4 Product space for the coupled systems 83
4.5 The free field 86
Chapter 5 Symmetries and Conservation Laws
5.1 Relations between observers 96
5.2 Linear and antilinear operators 100
5.3 Continuous symmetries 102
5.4 Discrete symmetries 108
vüi CONTENTS
Chapter 6 Interaction of Photons and Atoms
6.1 Approximations 119
6.2 Emission, absorption and scattering of radiation 123
6.3 Line width and level shift 138
Chapter 7 Path-dependent Electrodynamics
7.1 Introduction 147
7.2 Polarization and magnetization fields 149
7.3 Line integral Lagrangians 156
7.4 The multipolar Hamiltonian 162
7.5 Applications 169
Appendices
A Values of Physical Constants 177
B Theorems in Vector Analysis 178
C Longitudinal and Transverse Vector Fields 183
D Operator Identities 185
Subject Index 187
Chapter 1
Introduction
1.1 Non-relativistic theory
Quantum electrodynamics is the fundamental theory of the interaction of
radiation and charged particles. It is founded on the hypothesis that the
behaviour of the electromagnetic field under the influence of its micros
copic sources is governed by the laws of quantum mechanics as well as by
the Maxwell-Lorentz equations. The theory was first established by Dirac
(1927) and its modern version, due mainly to Tomonaga, Schwinger,
Feynman and Dyson, complies fully with the requirements of special
relativity. Indeed any theory of the pure radiation field based on the
source-free Maxwell-Lorentz equations must be relativistically covariant,
even though it might not be expressed in a form that makes this evident.
Quantum electrodynamics has, however, a well defined non-relativistic
limit in so far as the motion of the sources is concerned. The non-
relativistic theory is of an approximate character but involves a much
simpler formalism than its relativistic counterpart. Moreover, it is applica
ble to a wide range of problems in physics and chemistry, particularly in
the areas of spectroscopy, laser physics and intermolecular forces.
The conditions under which non-relativistic quantum electrodynamics
provides an accurate description of phenomena will first be determined. It
is assumed that the charged particles move at such low speeds (in the
inertial frame of a fixed observer) that their masses can be considered as
constant and equal to their rest masses. More specifically, since the relativistic
mass of a particle with speed v and rest mass m0 is m0(l — v2/c2)~112, we
require that vie« 1, c being the speed of light in vacuo.t This inequality
generally holds for the motion of atomic particles. Thus the root mean
square speed v of the electron of a hydrogen-like ion in a state with
principal quantum number n is (Pauling and Wilson 1935) 27rZe2/nh,
where Ze is the nuclear charge and h is Planck’s constant. If Z = 1 and
t
For the values of physical constants, see Appendix A.
