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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 accepted: 2021-01-13
Date submitted: 2020-12-22 13:56
Submitted by: Claeys, Pieter W.
Submitted to: SciPost Physics
Academic field: Physics
  • 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 Schr\"odinger 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. 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. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr\"odinger 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, Berry phases, Landau levels, level statistics 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.

Published as SciPost Phys. 10, 014 (2021)

Author comments upon resubmission

We thank the referees for providing valuable feedback. Following their suggestions we rearranged the material to highlight the new results related to the classical Gibbs ensemble. We rearranged the material such that original derivations and the main results of the paper appear sooner, moving some of the introductory material and the results on microcanonical ensembles to the Appendices. Next to some minor textual changes, we also added some additional topics suggested by the referees and also by our other colleagues, in particular by Sir. Michael Berry. The key changes can be found below.

List of changes

- We have rearranged the structure of the paper such that the new results on the Gibbs distribution appear earlier. Introductory sections on the dynamics and microcanonical ensemble have been moved to Appendix.

- We have added a derivation of the two leading-order corrections in $\beta$ to the Gibbs Hamiltonian (Sec. 4.2). These corrections highlight that this is {\emph not} a semiclassical expansion: different orders in $\beta$ have the same order in $\epsilon$. We tested these corrections in the example of tunneling and even for $\beta=1$, i.e. when the small parameter is not that small, we observed that the relative mistake in the tunneling gap can be reduced by orders of magnitude by taking into account higher-order corrections (see the inset in Fig. 6).

- We have added a derivation of the Gibbs Hamiltonian for a particle in the presence of a vector potential and showed how, to leading order in $\beta$, the resulting Gibbs Hamiltonian again agrees with the quantum Hamiltonian. For this reason, all Berry-type phases encoded in the quantum states also appear to be contained in the classical eigenstates, extending the applicability of this formalism. We illustrated the agreement between quantum and classical eigenstates for a confined particle particle in a 2D potential in the presence of a magnetic field and found excellent correspondence both in terms of absolute values and the phases of the wave functions (Fig. 9). Furthermore, we also added an exact analytic derivation of Landau Levels from the classical Gibbs ensemble in the presence of a constant magnetic field (Sec. 5.5.).

- We have included a discussion of chaotic and integrable two-dimensional potentials, which was independently suggested by several people including the third referee. We found that the classical spectrum indeed satisfies the BGS and Berry-Tabor conjectures (Sec. 6). It is remarkable that having the Gibbs distribution alone in hand and without analyzing any dynamics, one can determine whether the classical system is chaotic or not.

- Minor textual changes following the Referees' comments (including corrected typos, an explanation of how all numerical results were obtained, and a short comment on the Koopman-von Neumann construction).

Reports on this Submission

Anonymous Report 2 on 2021-1-7 Invited Report

  • Cite as: Anonymous, Report on arXiv:2007.07264v3, delivered 2021-01-07, doi: 10.21468/SciPost.Report.2376


Very original work, which increases our understanding of the complex relation between classical and quantum mechanics.


The authors have substantially clarified the text and provided extentions that are of great value. Publication of this new veresion is fully recommended.

The single fault that seems to have slipped through is that equation (53) is not "exactly equivalent to the Schrödinger equation".

Another improvement could be to substitute the notation for the normalization integral Z_x in (14), since it tends to mislead the reader into attributing an
x-dependence to it.

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

Anonymous Report 1 on 2021-1-5 Invited Report


I believe that the authors have adequately taken care of all the points raised, hence I recommend the manuscript for publication.

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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