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Quantum Mechanics and Statistics

J. D. Fast

Published Jan 1, 1968• 0 citations
Publisher

Abstract

In the previous chapters it was stated that the entropy of a system can be determined by statistical means by counting the number of micro-states in which the system may occur under the given conditions. By failing to enlarge upon this point, we have falsely given the impression that the way in which the count should be made was definitely established. Even in the simple case of the various configurations of 50 white and 50 red billiard balls (Fig. 11), counting is more ambiguous than was suggested in Chapter 2. If two white balls are interchanged, a state is reached which, to all appearances, is indistinguishable from the original state and which was thus not counted separately. However, a more thorough investigation with the help of scales and measuring instruments would certainly make it possible to distinguish two similarly coloured balls from one another. The number of visually-distinguishable configurations is thus much smaller than the number — 100! — which can be distinguished on closer investigation.

DOI
10.1007/978-1-349-15380-0_5

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In the previous chapters it was stated that the entropy of a system can be determined by statistical means by counting the number of micro-states in which the system may occur under the given conditions. By failing to enlarge upon this point, we have falsely given the impression that the way in which the count should be made was definitely established. Even in the simple case of the various configurations of 50 white and 50 red billiard balls (Fig. 11), counting is more ambiguous than was suggested in Chapter 2. If two white balls are interchanged, a state is reached which, to all appearances, is indistinguishable from the original state and which was thus not counted separately. However, a more thorough investigation with the help of scales and measuring instruments would certainly make it possible to distinguish two similarly coloured balls from one another. The number of visually-distinguishable configurations is thus much smaller than the number — 100! — which can be distinguished on closer investigation.
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Differences of opinion concerning fundamental issues in statistical mechanics directly related to the thermodynamic entropy have persisted through more than a century of debate. One reason is the lack of consensus on the definitions of key terms, especially the terms “distinguishable,” “indistinguishable,” and “identical.” Several definitions occur in the literature, but are not always made explicit. The multiplicity of definitions has created confusion about the basic conditions under which entropy is to be defined. In this paper, I present an overview of definitions in current use for terms associated with distinguishability and relate them to various definitions that have been suggested for entropy. My hope is that consensus will be achievable if the definitions are clarified and agreed upon.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
If a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged … A satisfactory theory ought, of course, to count any two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. ( Dirac, 1958, p. 207 ) Introduction In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.