Statistical Mechanics: A Set Of Lectures (Advanced Books Classics)

367 Pages·1998·5.97 MB·English

by  Richard P. Feynman

Frontiers in Physics
DAVID PINES, Editor
Volumes in Preparation:
E. R. Caianiello
COMBINATORICS AND RENORMALIZATION IN QUANTUM FIELD
THEORY
R. P. Feynman
PHOTON-HADRON INTERACTIONS
G. E. Pake and T. L. Estle
THE PHYSICAL PRINCIPLES OF ELECTRON PARAMAGNETIC
RESONANCE. Second Edition, completely revised, reset, and enlarged
STATISTICAL MECHANICS
A SET OF LECTURES
R. P. FEYNMAN California Institute of Technology
Notes taken by
R. Kikuchi and H. A. Feiveson
Hughes Aircraft Corporation
Edited by
Jacob Shaham
University of Illinois, Urbana
THE BENJAMIN/CUMMINGS PUBLISHING COMPANY, INC.
ADVANCED BOOK PROGRAM
Reading, Massachusetts
London Amsterdam Don Mills, Ontario Sydney Tokyo
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Statistical Mechanics: A Set of Lectures
First printing, 1972
Second printing, 1973
Third printing, 1974
Fourth printing, 1976
Fifth printing, 1979
Sixth printing, 1981
Seventh printing, 1982
International Standard Book Numbers
Clothbound: 0-8053-2508-5
Paperbound: 0-8053-2509-3
Library of Congress Catalog Card Number: 72-1769
Copyright C) 1972 by W. A. Benjamin, Inc.
Philippines copyright 1972 by W. A. Benjamin, Inc.
Published simultaneously in Canada.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photo-
copying, recording, or otherwise, without the prior written permission of the publisher,
W. A. Benjamin, Inc., Advanced Book Program, Reading, Massachusetts 01867,
U .S.A.
Printed in the United States of America
ISBN: 0-8053-2509-3
IJKLMNOP-MA-89876
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▪ (cid:9)
CONTENTS
Editor's Foreword (cid:9) xi
Acknowledgments (cid:9) . xiii
Chapter 1 Introduction to Statistical Mechanics
1.1 The Partition Function . (cid:9) 1
Chapter 2 Density Matrices
2.1 Introduction to Density Matrices . . (cid:9) . (cid:9) 39
2.2 Additional Properties of the Density Matrix . (cid:9) . (cid:9) 44
2.3 Density Matrix in Statistical Mechanics . . (cid:9) . (cid:9) 47
2.4 Density Matrix for a One-Dimensional Free Particle (cid:9) . (cid:9) 48
2.5 Linear Harmonic Oscillator (cid:9) . (cid:9) 49
2.6 Anharmonic Oscillator . (cid:9) . (cid:9) 53
2.7 Wigner's Function . (cid:9) . (cid:9) . (cid:9) . (cid:9) 58
2.8 Symmetrized Density Matrix for N Particles (cid:9) . (cid:9) 60
2.9 Density Submatrix . . . (cid:9) . (cid:9) 64
2.10 Perturbation Expansion of the Density Matrix (cid:9) . (cid:9) 66
2.11 Proof that F Fo + (H — (cid:9) 67
Chapter 3 Path Integrals
3.1 Path Integral Formation of the Density Matrix (cid:9) 72
3.2 Calculation of Path Integrals . (cid:9) . (cid:9) . (cid:9) 78
3.3 Path Integrals by Perturbation Expansion (cid:9) 84
3.4 Variational Principle for the Path Integral (cid:9) 86
3.5 An Application of the Variation Theorem (cid:9) 88
Chapter 4 Classical System of N Particles
4.1 Introduction . (cid:9) . (cid:9) . (cid:9) 97
4.2 The Second Vinai Coefficient (cid:9) . 100
4.3 Mayer Cluster Expansion (cid:9) . 105
4.4 Radial Distribution Function (cid:9) . 111
VII
viii (cid:9) Contents
4.5 Thermodynamic Functions. (cid:9) . . 113
4.6 The Born-Green Equation for n2 . . 115
4.7 One-Dimensional Gas . (cid:9) . . 117
4.8 One-Dimensional Gas with Potential of the Form e-. (cid:9)1x1 . . 120
4.9 Brief Discussion of Condensation . (cid:9) . . 125
Chapter (cid:9) 5 Order-Disorder Theory
5.1 Introduction (cid:9) . (cid:9) . . 127
5.2 Order-Disorder in One-Dimension. (cid:9) . (cid:127) 130
5.3 Approximate Methods for Two Dimensions . (cid:127) 131
5.4 The Onsager Problem (cid:9) . (cid:127) 136
5.5 Miscellaneous Comments (cid:127) 149
Chapter (cid:9) 6 Creation and Annihilation Operators
6.1 A Simple Mathematical Problem . 151
6.2 The Linear Harmonic Oscillator . 154
6.3 An Anharmonic Oscillator . (cid:9) . . 156
6.4 Systems of Harmonic Oscillators . 157
6.5 Phonons (cid:9) (cid:127) (cid:9) . . 159
6.6 Field Quantization (cid:9) . (cid:127) 162
6.7 Systems of Indistinguishable Particles. . 167
6.8 The Hamiltonian and Other Operators . 176
6.9 Ground State for a Fermion System . . 183
6.10 Hamiltonian for a Phonon-Electron System . 185
6.11 Photon-Electron Interactions (cid:9) . . 190
6.12 Feynman Diagrams . (cid:127) 192
Chapter (cid:9) 7 Spin Waves
7.1 Spin-Spin Interactions (cid:127) 198
7.2 The Pauli Spin Algebra . . 200
7.3 Spin Wave in a Lattice . (cid:9) . . 202
7.4 Semiclassical Interpretation of Spin Wave . 206
7.5 Two Spin Waves. (cid:9) (cid:127) (cid:9) . (cid:9) (cid:127) (cid:9) (cid:127) . 207
7.6 Two Spin Waves (Rigorous Treatment) . 209
7.7 Scattering of Two Spin Waves . . 212
7.8 Non-Orthogonality (cid:9) (cid:127) (cid:9) . . 215
7.9 Operator Method (cid:9) . (cid:9) . 217
7.10 Scattering of Spin Waves-Oscillator Analog . 218
Chapter (cid:9) 8 Polaron Problem
8.1 Introduction (cid:9) . (cid:9) (cid:127) . 221
8.2 Perturbation Treatment of the Polaron Problem. . 225
8.3 Formulation for the Variational Treatment (cid:9) . (cid:9) . . 231
8.4 The Variational Treatment . . 234
8.5 Effective Mass (cid:9) . (cid:9) . (cid:127) 241
Contents (cid:9) ix
Chapter (cid:9) 9 Electron Gas in a Metal
9.1 Introduction: The State Function q7 242
9.2 Sound Waves. (cid:9) . (cid:9) . 244
9.3 Calculation of P(R) . 246
9.4 Correlation Energy (cid:9) . 248
9.5 Plasma Oscillation (cid:9) . 249
9.6 Random Phase Approximation. 252
9.7 Variational Approach (cid:9) . (cid:9) . 254
9.8 Correlation Energy and Feynman Diagrams 255
9.9 Higher-Order Perturbation . 262
Chapter 10 Superconductivity
10.1 Experimental Results and Early Theory (cid:9) . 265
10.2 Setting Up the Hamiltonian 269
10.3 A Helpful Theorem . (cid:9) . 273
10.4 Ground State of a Superconductor (cid:9) . 274
10.5 Ground State of a Superconductor (continued) 277
1C.6 Excitations (cid:9) . 279
10.7 Finite Temperatures . 281
10.8 Real Test of Existence of Pair States and Energy Gap 285
10.9 Superconductor with Current (cid:9) . 290
10.10 Current Versus Field (cid:9) . (cid:9) . 293
10.11 Current at a Finite Temperature 298
10.12 Another Point of View . (cid:9) . 303
Chapter 11 Superfluidity
11.1 Introduction: Nature of Transition 312
11.2 Superfluidity-An Early Approach (cid:9) . 319
11.3 Intuitive Derivation of Wave Functions: Ground State. 321
11.4 Phonons and Rotons 326
11.5 Rotons (cid:9) . 330
11.6 Critical Velocity . (cid:9) . 334
11.7 Irrotational Superfluid Flow 335
11.8 Rotational of the Superfluid (cid:9) . 337
11.9 A Reasoning Leading to Vortex Lines 339
11.10 The A. Transition in Liquid Helium (cid:9) . 343
Index (cid:9) . . (cid:9) 351

EDITOR'S FOREWORD
The problem of communicating in a coherent fashion the recent developments
in the most exciting and active fields of physics seems particularly pressing today.
The enormous growth in the number of physicists has tended to make the
familiar channels of communication considerably less effective. It has become
increasingly difficult for experts in a given field to keep up with the current
literature; the novice can only be confused. What is needed is both a consistent
account of a field and the presentation of a definite "point of view" concerning
it. Formal monographs cannot meet such a need in a rapidly developing field,
and, perhaps more important, the review article seems to have fallen into
disfavor. Indeed, it would seem that the people most actively engaged in devel-
oping a given field are the people least likely to write at length about it.
FRONTIERS IN PHYSICS has been conceived in an effort to improve the
situation in several ways. First, to take advantage of the fact that the leading
physicists today frequently give a series of lectures, a graduate seminar, or a
graduate course in their special fields of interest. Such lectures serve to sum-
marize the present status of a rapidly developing field and may well constitute
the only coherent account available at the time. Often, notes on lectures exist
(prepared by the lecturer himself, by graduate students, or by postdoctoral
fellows) and have been distributed in mimeographed form on a limited basis.
One of the principal purposes of the FRONTIERS IN PHYSICS Series is to make
such notes available to a wider audience of physicists.
It should be emphasized that lecture notes are necessarily rough and
informal, both in style and content, and those in the series will prove no excep-
tion. This is as it should be. The point of the series is to offer new, rapid,
more informal, and it is hoped, more effective ways for physicists to teach
one another. The point is lost if only elegant notes qualify.
A second way to improve communication in very active fields of physics
is by the publication of collections of reprints of recent articles. Such collections
are themselves useful to people working in the field. The value of the reprints
would, however, seem much enhanced if the collection would be accompanied
by an introduction of moderate length, which would serve to tie the collection
together and, necessarily, constitute a brief survey of the present status of the
xii (cid:9) Editor's Foreword
field. Again, it is appropriate that such an introduction be informal, in keeping
with the active character of the field.
A third possibility for the series might be called an informal monograph,
to connote the fact that it represents an intermediate step between lecture
notes and formal monographs. It would offer the author an opportunity to
present his views of a field that has developed to the point at which a sum-
mation might prove extraordinarily fruitful, but for which a formal monograph
might not be feasible or desirable.
Fourth, there are the contemporary classics—papers or lectures which
constitute a particularly valuable approach to the teaching and learning of
physics today. Here one thinks of fields that lie at the heart of much of present-
day research, but whose essentials are by now well understood, such as quantum
electrodynamics or magnetic resonance. In such fields some of the best pedago-
gical material is not readily available, either because it consists of papers long
out of print or lectures that have never been published.
The above words, written in August, 1961, seem equally applicable today
(which may tell us something about developments in communication in physics
during the past decade). Richard Feynman contributed two lecture note vol-
umes ("Quantum Electrodynamics" and "The Theory of Fundamental Pro-
cesses") to the first group of books published in this series, and, with the publica-
tion of the present volume and the forthcoming publication of "Photon-Hadron
Interactions," it gives me special pleasure to welcome him back as a major
contributor to FRONTIERS IN PHYSICS.
"Statistical Mechanics: A Set of Lectures" will be of interest to everyone
concerned with teaching and learning statistical mechanics. In addition to
providing an elegant introduction to the basic concepts of statistical physics,
the notes contain a description of some of the many original and profound
contributions, (ranging from polaron theory to the theory of liquid helium)
which Professor Feynman has made in this field.
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Urbana, Illinois
DAVID PINES
June, 1972