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Quantum eigenstates from classical Gibbs distributions

by Pieter W. Claeys, Anatoli Polkovnikov

Submission summary

As Contributors: Pieter W. Claeys · Anatoli Polkovnikov
Arxiv Link: (pdf)
Date submitted: 2020-07-28 14:32
Submitted by: Claeys, Pieter W.
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
Approaches: Theoretical, Computational


We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schrödinger equation follows from the Liouville equation, with $\hbar$ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schrödinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including $\hbar$) on the order of unity. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner's quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions.

Current status:
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Submission & Refereeing History

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Submission 2007.07264v2 on 28 July 2020

Reports on this Submission

Anonymous Report 1 on 2020-8-28 Invited Report


Mini review-styled paper with many details.


Originality, but compensated by its style, see report.


This paper reviews in principle well known results, but in a style which brings together in a clear language various aspects connected here. Many detailed examples are given, a strengths of this mini review styled paper.

I suggest the authors consider the following minor issues:

(1) ref. 26 spells wrongly the author's name! Its JK Moser! Details of the ref. must be given also.

(2) The introductions links to complementary approaches to the main idea here, both are indeed well known in the literature. hence, I suggest to stress this better, also by citing relevant literature:

- on the first approach (semiclassical expansion a la Weyl-Moyal): Hamiltonian Systems: Chaos and Quantization (Cambridge Monographs on Mathematical Physics) from de Almeida, Alfredo M. Ozorio
- on the second one (truncated Wigner expansion): quantum optics literature by the New Zealand groups, Peter Drummond, H. J. Carmichael, CW Gardinger and others, see e.g. M. J. Werner and P. D. Drummond, J. Comput. Phys. 132, 312 (1997) or C.W. Gardiner, StochasticMethods: A Handbook for the Natural and Social Sciences, Springer Series in Synergetics (Springer-Verlag, Berlin, 2009).

  • validity: top
  • significance: high
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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