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arXiv:quant-ph/9906035

The Generalized Quantum Statistics

Won-Young Hwang, Jeong‐Young Ji, Jongbae Hong

Published Jun 11, 1999• 0 citations
arXivHTMLPDF

Abstract

The concept of wavefunction reduction should be introduced to standard quantum mechanics in any physical processes where effective reduction of wavefunction occurs, as well as in the measurement processes. When the overlap is negligible, each particle obey Maxwell-Boltzmann statistics even if the particles are in principle described by totally symmetrized wavefunction [P.R.Holland, The Quantum Theory of Motion, Cambridge Unversity Press, 1993, p293]. We generalize the conjecture. That is, particles obey some generalized statistics that contains the quantum and classical statistics as special cases, where the degree of overlapping determines the statistics that particles should obey among continuous generalized statistics. We present an example consistent with the conjecture.

DOI
10.48550/arxiv.quant-ph/9906035

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The concept of wavefunction reduction should be introduced to standard quantum mechanics in any physical processes where effective reduction of wavefunction occurs, as well as in the measurement processes. When the overlap is negligible, each particle obey Maxwell-Boltzmann statistics even if the particles are in principle described by totally symmetrized wavefunction [P.R.Holland, The Quantum Theory of Motion, Cambridge Unversity Press, 1993, p293]. We generalize the conjecture. That is, particles obey some generalized statistics that contains the quantum and classical statistics as special cases, where the degree of overlapping determines the statistics that particles should obey among continuous generalized statistics. We present an example consistent with the conjecture.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
When identical particles are not localized they can no longer be distinguished . This situation occurs in the case of a gas since the particles of a gas are unconstrained in their motion except by the walls of the container. It also occurs when the particles are so close together that their wave functions overlap, that is when they are free to exchange positions. It should be stressed that this is a typical quantum-mechanical phenomenon, formulated as QM 3. Classical particles are always distinguishable because one can in principle follow each particle along its trajectory. According to quantum mechanics this is impossible even in principle. According to principle QM 3, there are in fact two different kinds of statistics depending on the type of particle: Bose-Einstein statistics for bosons and Fermi-Dirac statistics for fermions . This is related to the fact that fermions satisfy the Pauli exclusion principle ; there can be no more than one particle in each eigenstate.
The possibility of consistency between the basic quantum principles and reduction (wave function reduction) is reexamined. The mathematical description of an organized macroscopic device is constructed explicitly as a convenient tool for this investigation. A derivation of reduction from quantum mechanics is proposed on a specific example, using standard methods of statistical physics. Although these methods are valid only "for all practical purposes", arguments are given to ascribe an emerging status to reduction (or the uniqueness of physical reality), similar to the status of classical physics. Examination of measurements of the particles in an EPR pair by two space-like separated apparatuses shows that the model is consistent with the non-separable character of quantum mechanics, although both measurements are local. Quantitative estimates are given.
The basic postulates The statistical interpretation of the wavefunction is in accord with experimental facts. An interference pattern on a screen is built up by a series of apparently random events, and the wavefunction correctly predicts where the particle is most likely to land over an ensemble of trials (cf. the frontispiece ). Yet the interpretation of the wavefunction which ascribes to it a purely statistical significance is not forced on us by the experimental results – the latter show only that it is consistent to regard the wavefunction as containing information on probability; they do not exclude the possibility that it may possess other properties. On the contrary, one may take the view that the characteristic distribution of spots on a screen which build up an interference pattern is evidence that the wavefunction indeed has a more potent physical role than a mere repository of information on probabilities, for how are the particles guided so that statistically they fall into such a pattern? Such a question is naturally ruled out by the purely probabilistic interpretation. But the latter is appropriate only if we wish to reduce physics to a kind of algorithm which is efficient at correlating the statistical results of experiments. If we wish to do more, and attempt to understand the experimental results as the outcome of a causally connected series of individual processes, then we are free to enquire as to the possible further significance of the wavefunction (beyond its probabilistic aspect), and to introduce other concepts in addition to the wavefunction.
We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were Gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\Delta} is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a sound foundation for quantum statistical mechanics.