Walter Thirring A Course in Mathematical Physics 4 Quantum Mechanics of Large Systems Translated by Evans M. Harrell Springer-Verlag New York Wien Dr. Walter Thirring Dr. Harrell Institute for Theoretical Physics The Johns Hopkins University University of Vienna Baltimore, Maryland Austria U.
Translation of Lehrbuch der Mathematischen Physik Band 4: Quantenmechanik grosser Systeme Wien—New York: Springer-Verlag 1980 © 1980 by Springer-Verlag! Wien ISBN 3-21 1-81604-6 Springer-Verlag Wien New York ISBN 0-387-81604-6 Springer-Verlag New York Wien Library of Congress Cataloging in Publication Data Thirring, Walter E., 1927— Quantum mechanics of large systems. (A course in mathematical physics; 4) Translation of: Quantenmechanik grosser Systeme. Series: Thirring, Walter E. Lehrbuch der mathematischen Physik.l'Ss 82-19159 With 39 Figures © 1983 by Springer-Verlag New York Inc.
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. Typeset by Composition House Ltd Salisbury England Printed and bound by R. Printed in the United States of America 987654321 ISBN 0-387-81701-8 Springer-Verlag York ISBN 3-211-81701-8 Springer-Verlag Wien New York www.com Preface In this final volume I have tried to present the subject of statistical mechanics in accordance with the basic principles of the series.
The effort again entailed following Gustav Mahler's maxim, "Tradition = Schlamperei" (i., filth) and clearing away a large portion of this tradition-laden area. The result is a book with little in common with most other books on the subject. The ordinary perturbation—theoretic calculations are not very useful in this field. Those methods have never led to propositions of much substance.
Even when perturbation series, which for the most part never converge, can be given some asymptotic meaning, it cannot be determined how close the nth order approximation comes to the exact result. Since analytic solutions of nontrivial problems are beyond human capabilities, for better or worse we must settle for sharp bounds on the quantities of interest, and can at most strive to make the degree of accuracy satisfactory. The last two decades have seen successful and beautiful treatments of many fundamental issues—I have in mind the ordering of the states (2. 1), properties of the entropy (2.2), noncommutative ergodic theory (3.1), the proof of the existence of the thermodynamic functions (4.3), and the mathematical analysis of Thomas-Fermi theory (4.2), which provides an understanding of the stability of matter.
The day is surely not far off when most of the remaining holes in the conceptual structure of quantum statistical mechanics will have been filled in and the questions that are not satisfactorily answered today will be added to the list of achievements. The successful completion of this course of mathematical physics in a. reasonable time required the fortunate conjunction of several circumstances. As with volume III, I had active support from several collaborators, and in particular I am greatly obliged to B.
Countless other colleagues have helped indirectly by coping www.com Vi Prefac. duties for me. The English rdition has again gT 1mm the cntical reading of B. The working con- .ii the University of\ tenna were invaluable for thc completion of hut not least, the fricLionless collaboration of Springer-Verlag ii Vienna and my secretary and calligrapher F.
Wagner enabled the books to appear quickly and at a reasonable price. I am aware that the uncompromising way of mathematical physics is not the easiest. Yet I feel that it has been one of the greatest intellectual accomplishments of our era to cast the laws of Nature in a clear mathematical form with rigorously deducible consequences. No amount of labor is too high a price to have paid for this.
Let me conclude by also acknowledging and expressing my thanks to the reader who has borne with me to the end of the course. Walter Thirring www.com Contents Systems with Many Particles 1.1 Equilibrium and Irreversibility 1 1.2 The Limit of an Infinite Number of Particles 11 1.3 Arbitrary Numbers of Particles in Fock Space 20 1.4 Representations with N = 29 2 Thermostatics 45 2.1 The Ordering of the States 45 2.2 The PropertIes of Entropy 57 2.3 The Microcanonical Ensemble 73 2.4 The Canonical Ensemble 103 2.5 The Grand Canonical Ensemble 115 3 Thennodynamics 144 3.2 The Equilibrium State 173 3.3 Stability and Passivity 191 4 Physical Systems 209 4.1 Thomas-Fermi Theory 209 4.3 Normal Matter 256 Bibliography 279 Index VII www.com Systems with Many Particles 1.1 Equilibrium and Irreversibility Macroscopic bodies cci in wi irreversible and deterministic manner in con frost with rhe rerersible and indetermmisti character of the of quantum physics. How can the apparent contra- diction We have learned Lo describe systems of finitely many partic!es algebra. ,nforma(ion about the systems a SU.e it on the algebra (ci.32)'; As our main goal is the study of e'.eryday matter, our framework wilt oe that of nonrelativistic quantuu theory.
For the purposes of contrast, or of aiding intuition, we shall also have oecasion to call upon classicai where states measures on phase space, and extremal states are measures. In either framework time-evolution an automorphism a a, for a E d in the Heisenberg picture. If desired, time can alternatively, in the SchrOdinger picture, he put upon the state: w. the algebra is Abetian (classical mechanics), then the point of an extremal state moves along a classical trajectory in phase-space.
In our earlier experience systems of N particle are so complex for large N that it becomes impossible to reach precise, quantitative conclusions. It turns out. however, that the theoretical analysis again simplifies in the limit N —+ Many properties become independent of the exact numbt'r of particles and other detailed characteristics of the physical system, somewhat in analogy to what happens in the central limit theorem of probability theory. This may seem peculiar at first: we have always had d = www.com 2 1 Systems with Many Particles separable Hubert space, and time-evolution was given by a unitary group on.
What, then, appears so special about a many-particle system? Just that the information contained in a pure state about a many-particle system is so overwhelming that it would be too ambitious to employ the whole of for the observables. Actual measurements could never be made on more than a few observables, so has to be cut down to size. For instance, suppose that a device is only equipped to observe one particle at a time, and is unable to detect correlations between particles. Then, rather than taking the entire tensor product of the individual particles as the algebra of observ- ables, it is reasonable to regard d as a single factor.
Accordingly, many states differing on reduce to the same state when restricted to d. (The classical situation is similar; the restriction of fd3qi. w(x1, PN), so whole cylindrical regions of phase-space reduce to a single restricted state.) As a consequence large portions of the space of states on are quite similar from the point of view of the reduced algebra. If, in the Schrodinger picture, the state W, travels throughout the space of states, then its restriction takes on a certain value with a very high probability, unless prevented by some constants of the motion.
This most probable state is called the equilibrium state over d. The irreversible tendency toward equilThrium has always aroused wonder, especially as the basic equations of dynamics are invariant under reversal of the motion (III: 3. We have even seen in classical mechanics that the trajectory of any point on a compact energy surface returns arbitrarily close to its initial position (1: 2. In quantum theory the Hamiltonian H of a system confined to a finite volume has purely discrete spectrum.
If and denote the eigenvalues and eigenvectors of H, then the time-dependence of an observable a is given by w(a) = — 3. k where the state w is represented by the vector L if>. The state is now an almost-periodic function oft; if the sum is finite, and the are rationally dependent, then it is actually strictly periodic. At any rate, to arbitrarily good accuracy, w,(a) again becomes nearly w(a) after some sufficiently long delay.
The trouble is that the recurrence times are so unimaginably long that they have no physical relevance. Suppose, for instance, that there are N distinct energy differences The recurrence time can then be estimated as follows. The factors exp(iw3t) can be pictured as N clocks with hands moving at N different rates. The question is how long it takes for a certain configuration www.1 Equilibrium and Irreversibility 3 of clock faces to reappear to within some angular accuracy The con- figuration in the space of angles has measure so the recurrence time is on the order of (&p/2irY where the reciprocal angular velocity 1/co is an average of the 1/wi.
Even for just N = 10, 1/w = sec. and 1 = 1/100, so that w, returns to w to within 1 oo accuracy, the recurrence time is 1020 sec., which is much longer than the age of the universe. The approach to equilibrium is connected to a loss of information; to be more precise, information does not get lost, but only less accessible. We have seen that when the wave-packet of a free particle spreads (III: 3.3), grows linearly with time, although the state remains pure and thus has maximal information content.
The observable with least deviation from the mean is, however, not x(r) but x(0) x(t) — pi. This behavior can be seen even in classical motion if a minimal spread of the support of the probability distribution function in phase space is hypo- thesized to account for quantum effects. If, say, the initial probability density p(p, q) is concentrated on a part of the energy shell {(q; p)1p1 p P2) and is not pointlike. and it moves freely on a torus, then it eventually fills the energy shell densely with a "fuzzy" distribution.
Faster particles overtake the slower ones, as bicycles racing in a stadium start packed closely together but later draw apart and eventually spread around the whole track (see Figure 1). The ergodic hypothesis has figured importantly in the history of statistical mechanics; it is the assumption that the trajectory of almost every point winds densely around the energyshell in phase space, so that the time average can be replaced with the average over the energy shell. On the one hand this requires more than is necessary, since it suffices to fill a sufficiently typical part of the energy shell, the average on which equals the average on the whole shell for the reduced algebra of observables. On the other hand, although macroscopic measurements last much longer than the collision time, they last much less than the recurrence time, so one does not wait for the whole energy shell to be sampled.
We shall discuss examples in which the equilibrium state is actually attained by the state in a reasonable time after reduction to one particle. A pictorial description of the situation is as follows. The information about a subsystem (i., the opposite of the entropy, to be defined later) as a function on the space of states of the total system Consists mainly of a plain with few hills and still fewer mountains. The larger the total system, the further apart the prominences.
Even if a path begins on a peak. it soon descends to the plain, and there is only the slightest probability that it will ascend another mountain in any conceivable time. The time of descent to the plain and the recurrence time are of completely different orders of magnitude. It takes only the time corresponding physically to a few collisions to descepd to a level near that of the plain, whereas the other mountains lie in the un- fathomable distance.
This means that equilibrium is reached long before the immense recurrence time required to wind throughout the space of states; www.com Systems with Many Particles Figure 1 The motion of the density in phase space for a free particle on a torus.