EC 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