Table Of ContentEC A ICS
TH TEGRALS
Edition
DOVER PUBLICATIONS, INC.
MINEOLA, NEW YORK
Copyright
Copyright© 1965 by Richard P. Feynman and Albert R. Hibbs
Emended Edition © 2005 by Daniel F. Styer.
All rights reserved
Bibliographical Note
This Dover edition, first published in 2010, is an unabridged,
emended republication of the work originally published in 1965 by
McGraw-Hill Companies, Inc., New York.
Library of Congress Cataloging-in-Publication Data
Feynman, Richard Phillips.
Quantum mechanics and path integrals I Richard P. Feynman,
Albert R. Hibbs, and Daniel F. Styer.- Emended ed.
p . em.
Originally published: Emended edition. New York : McGraw-
Hill, 2005.
Includes bibliographical references and index.
ISBN-13: 978-0-486-47722-0
ISBN-10: 0-486-47722-3
1. Quantum Theory. I. Hibbs, Albert R. II. Styer, Daniel F. III.
Title.
QC174.12.F484 2010
530.12-dc22
2010004550
Manufactured in the United States by Courier Corporation
47722303
www.doverpublications.com
Preface
The fundamental physical and mathematical concepts which underlie
the path integral approach to quantum mechanics were first developed
by R.P. Feynman in the course of his graduate studies at Princeton,
although more fully developed ideas, such as those described in this
volume, were not worked out until a few years later. These early in
quiries were involved with the problem of the infinite self-energy of the
electron. In working on that problem, a "least-action" principle using
half advanced and half retarded potentials was discovered. The princi
ple could deal successfully with the infinity arising in the application of
classical electrodynamics.
v
Preface
Vl
The problem then became one of applying this action principle to
quantum mechanics in such a way that classical mechanics could arise
naturally as a special case of quantum mechanics when 1i was allowed
to go to zero.
Feynman searched for any ideas which might have been previously
worked out in connecting quantum-mechanical behavior with such clas
sical ideas as the lagrangian or, in particular, Hamilton's principle func
tion S, the indefinite integral of the lagrangian. During some conversa
tions with a visiting European physicist, Feynman learned of a paper in
which Dirac had suggested that the exponential function of iE times the
lagrangian was analogous to a transformation function for the quantum
mechanical wave function in that the wave function at one moment could
be related to the wave function at the next moment (a time interval
E
later) by multiplying with such an exponential function.
The question that then arose was what Dirac had meant by the
phrase "analogous to," and Feynman determined to find out whether
or not it would be possible to substitute the phrase "equal to." A brief
analysis showed that indeed this exponential function could be used in
this manner directly.
Further analysis then led to the use of the exponent of the time
integral of the lagrangian, S (in this volume referred to as the action),
as the transformation function for finite time intervals. However, in the
application of this function it is necessary to carry out integrals over all
space variables at every instant of time.
In preparing an article1 describing this idea, the idea of "integral
over all paths" was developed as a way of both describing and evalu
ating the required integrations over space coordinates. By this time a
number of mathematical devices had been developed for applying the
path integral technique and a number of special applications had been
worked out, although the primary direction of work at this time was
toward quantum electrodynamics. Actually, the path integral did not
then provide, nor has it since provided, a truly satisfactory method of
avoiding the divergence difficulties of quantum electrodynamics, but it
has been found to be most useful in solving other problems in that field.
In particular, it provides an expression for quantum-electrodynamic laws
in a form that makes their relativistic invariance obvious. In addition,
useful applications to other problems of quantum mechanics have been
found.
The most dramatic early application of the path integral method to
an intractable quantum-mechanical problem followed shortly after the
1 R.P. Feynman, Space-Time Approach to Non-relativistic Quantum Mechanics,
Rev. Mod. Phys., vol. 20, pp. 367-387, 1948.
Preface vii
discovery of the Lamb shift and the subsequent theoretical difficulties
in explaining this shift without obviously artificial means of getting rid
of divergent integrals. The path integral approach provided one way of
handling these awkward infinities in a logical and consistent manner.
The path integral approach was used as a technique for teaching
quantum mechanics for a few years at the California Institute of Tech
nology. It was during this period that A.R. Hibbs, a student of Feyn
man's, began to develop a set of notes suitable for converting a lecture
course on the path integral approach to quantum mechanics into a book
on the same subject.
Over the succeeding years, as the book itself was elaborated, other
subjects were brought into both the lectures of Dr. Feynman and the
book; examples are statistical mechanics and the variational principle.
At the same time, Dr. Feynman's approach to teaching the subject of
quantum mechanics evolved somewhat away from the initial path in
tegral approach. At the present time, it appears that the operator
technique is both deeper and more powerful for the solution of more
general quantum-mechanical problems. Nevertheless, the path integral
approach provides an intuitive appreciation of quantum-mechanical be
havior which is extremely valuable in gaining an intuitive appreciation
of quantum-mechanical laws. For this reason, in those fields of quantum
mechanics where the path integral approach turns out to be particularly
useful, most of which are described in this book, the physics student is
provided with an excellent grasp of basic quantum-mechanical princi
ples which will permit him to be more effective in solving problems in
broader areas of theoretical physics.
R.P. Feynman
A.R. Hibbs
Preface to Emended Edition
In the forty years since the first publication of Quantum Mechanics and
Path Integrals, the physics and the mathematics introduced here has
grown both rich and deep. Nevertheless this founding book - full of
the verve and insight of Feynman remains the best source for learning
about the field. Unfortunately, the 1965 edition was flawed by extensive
typographical errors as well as numerous infelicities and inconsistencies.
This edition corrects more than 879 errors, and many more equations
are recast to make them easier to understand and interpret. Notation is
made uniform throughout the book, and grammatical errors have been
corrected. On the other hand, the book is stamped with the rough and
tumble spirit of a creative mind facing a great challenge. The objective
throughout has been to retain that spirit by correcting, but not polish
ing. This edition does not attempt to add new topics to the book or to
bring the treatment up to date. However, some comments are added in
an appendix of notes. (The existence of a relevant comment is signaled
in the text through the symbol0 Equation numbers are the same here
.)
as in the 1965 edition, except that equations (10.63) and (10.64) are
swapped.
I thank Edwin Tayor for encouragement and Daniel Keren, J ozef
Hanc, and especially Tim Hatamian for bringing errors to my attention.
A research status leave from Oberlin College made this project possible.
I can well remember the day thirty years ago when I opened the
pages of Feynman-Hibbs, and for the first time saw quantum mechanics
as a living piece of nature rather than as a flood of arcane algorithms
that, while lovely and mysterious and satisfying, ultimately defy under
standing or intuition. It is my hope and my belief that this emended
edition will open similar doors for generations to come.
Daniel F. Styer
viii
Contents
Preface v
Preface to Emended Edition
VIn
chapter 1 The Fundamental Concepts of Quantum Mechanics 1
1-1 Probability in quantum mechanics 2
1-2 The uncertainty principle 9
1-3 Interfering alternatives 13
1-4 Summary of probability concepts 19
1-5 Some remaining thoughts 22
1-6 The purpose of this book 23
ix
Contents
X
chapter 2 The Quantum-mechanical Law of Motion 25
2-1 The classical action 26
2-2 The quantum-mechanical amplitude 28
2-3 The classical limit 29
2-4 The sum over paths 31
2-5 Events occurring in succession 36
2-6 Some remarks 39
chapter 3 Developing the Concepts with Special Examples 41
3-1 The free particle 42
3-2 Diffraction through a slit 4 7
3-3 Results for a sharp-edged slit 55
3-4 The wave function 57
3-5 Gaussian integrals 58
3-6 Motion in a potential field 62
3-7 Systems with many variables 65
3-8 Separable systems 66
3-9 The path integral as a functional 68
3-10 Interaction of a particle and a harmonic oscillator 69
3-11 Evaluation of path integrals by Fourier series 71
i
chapter 4 The Schrodinger Description of Quantum Mechanics 75
4-1 The Schrodinger equation 76
4-2 The time-independent hamiltonian 84
4-3 Normalizing the free-particle wave functions 89
chapter 5 Measurements and Operators 95
5-1 The momentum representation 96
5-2 Measurement of quantum-mechanical variables 106
5-3 Operators 112
chapter 6 The Perturbation Method in Quantum Mechanics 119
6-1 The perturbation expansion 120
6-2 An integral equation for K v 126
6-3 An expansion for the wave function 127
6-4 The scattering of an electron by an atom 129
6-5 Time-dependent perturbations and transition amplitudes 144
Contents
Xl
chapter 7 Transition Elements 163
7-1 Definition of the transition element 164
7-2 Functional derivatives 170
7-3 Transition elements of some special functionals 174
7-4 General results for quadratic actions 182
7-5 Transition elements and the operator notation 184
7-6 The perturbation series for a vector potential 189
7-7 The hamiltonian 192
chapter 8 Harmonic Oscillators 197
8-1 The simple harmonic oscillator 198
8-2 The polyatomic molecule 203
8-3 Normal coordinates 208
8-4 The one-dimensional crystal 212
8-5 The approximation of continuity 218
8-6 Quantum mechanics of a line of atoms 222
8-7 The three-dimensional crystal 224
8-8 Quantum field theory 229
8-9 The forced harmonic oscillator 232
chapter 9 Quantum Electrodynamics 235
9-1 Classical electrodynamics 237
9-2 The quantum mechanics of the radiation field 242
9-3 The ground state 244
9-4 Interaction of field and matter 24 7
9-5 A single electron in a radiative field 253
9-6 The Lamb shift 256
9-7 The emission of light 260
9-8 Summary 262
chapter· 10 Statistical Mechanics 267
10-1 The partition function 269
10-2 The path integral evaluation 273
10-3 Quantum-mechanical effects 279
10-4 Systems of several variables 287
10-5 Remarks on methods of derivation 296
