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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
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 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:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 3 on 2020-11-2 Invited Report


1- A different/fresh perspective on classical quantum correspondence
2- Very pedagogical and intuitive presentation


Very few,
perhaps that the basic idea in the paper is not original.


The paper elaborates on a beautiful correspondence between classical and quantum mechanics in phase space. Given a classical phase space density P(x,p), one can define a quasi-density matrix W(x1,x2), with respect to a "formal" resolution parameter \epsilon, such that W(x1,x2) corresponds to a proper density matrix when P(x,p) is a Wigner function of a quantum state and \eps is the Planck's constant \hbar. One can now derive quantum formulation of classical mechanics, such as the eigenvectors and the spectrum of the density matrix, writing the corresponding classical Schroedinger equation (which becomes an integral equation, reducing to the usual Schroedinger equation when \eps or \beta are small enough), etc. One can even extend this intriguing correspondence to dynamics and define the Liouvillian propagator for classical quasi-density matrices. Particularly intriguing is the connection between the classical Gibbs phase space distribution and the corresponding quantum eigenfunctions, obtained as eigenvectors of the classical quasi-density matrix. Another intriguing but well discussed issue is the emergence of "negative classical probabilities" for small parameter \eps (smaller than the scale of variation (or uncertainties) of classical probability density).

It is true that most of results discussed here appeared in the literature before, but this paper gives an overarching discussion and a clear physical picture. Even though such a formulation of classical dynamics may not be practical, it can stimulate further interesting studies into quantum classical correspondence, in particular from the viewpoint of quantum chaos. I thus recommend the paper for publication in SciPost Physics.

Requested changes

- The precise mathematical meaning of the spectrum and eigenfunctions of the the classical quasi-density matrix seems unclear. For example, there exist a rigorous formulation of classical mechanics within the Hilbert space, the so-called Koopman-Von Neumann picture. Perhaps the results of the present paper could be phrased or linked to this broad picture. Maybe just a sentence or two would be helpful to give a broad picture.

- How is the classical spectrum related to chaos in two or more dimensions? Is there an extension of Bohigas-Giannoni-Schmidt conjecture to that case? It would be an interesting question for a followup study.

- Top of page 7: typo: particule -> particular

- Text after Eq. (44): The authors discuss that classical probabilities are "oscillatory", in n? Is this true as the formula in the text does not say that, it just says they are negative, if \tilde{\beta}_q is real (as it is said in the text after Eq. (43)).

- Text after Eq. (7): "is" is missing after the first word.

- Caption title of Fig.4: To unformise the style with other figures, add the information that the figure refers to a "quartic potential"

- Caption title of Fig. 7: .. add information that the figure referes to "double well potential".

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

Anonymous Report 2 on 2020-10-6 Invited Report


Report on 'Quantum eigenstates from classical Gibbs distributions'

The density operator corresponding to the quantum canonical ensemble can be represented by its Wigner function as a real function in the corresponding classical phase space. In the limit of high temperatures (small \beta), this function is accurately approximated by the classical canonical probability density. Thus, the inverse Wigner transform of this classical density can be equated approximately to a spectral decomposition of the projectors onto the eigenstates |n> of the Hamiltonian, weighed by w_n = \exp -\beta E_n. Of course, this is just the spectral decomposition of the evolution operator with imaginary time and temperature replacing Planck's constant. Actually, \hbar is treated as a mobile parameter in semiclassical methods, while here it is replaced by \epsilon. Thus, \epsilon (or \hbar) should also be small for the classical approximation to hold, without the spectral graininess being perceptible. A further step taken here is to transfer, through the inverse Wigner transform, the classical Liouville evolution to the canonical density matrix in equation (19). This is an approximation to the quantum evolution for small \epsilon.

The obvious direct course is to investigate the quantum canonical Wigner function at lower temperatures, but, surprisingly, this paper proceeds in the opposite direction. Not only are the spectral weights obtained for the continuation of the classical density to lower temperatures, but also the eigenstates themselves. No justification is given for the physical relevance of these hybrid classical-quantum constructions, except for the possible implicit assertion that the classical high temperature regime of the Wigner function
provides a competitive method for the evaluation of individual eigenstates. Unfortunately, the presentation is not sufficiently clear to confirm this unequivocally, so as to distinguish an important new metod from a theoretical curiosity. Indeed, the detailed explanation, as to how the stationary wave functions are actually calculated, is missing.

The general equation for the matrix elements of the density matrix in an arbitrary basis of orthogonal eigenstates is given by (50), but there is no special role for Wigner functions in this. Then in section 5.2 the inverse Wigner transform of the classical canonical distribution (71) is treated as a bona fide density matrix, so that its 'exact' eigenvalue equation is (72). Since the distribution is stationary by construction, there is no need to consider explicitly the transform of the Liouville equation. In the rest of this section, it is shown that for sufficiently small \epsilon and \beta, this is approximately equivalent to the stationary Schrödinger equation (with exponential eigenvalues).

Is there any advantage to solve this equation instead of the Schrödinger equation? Are the surprisingly accurate 'classical eigenfunctions', which are compared in the examples,
calculated in this way? One should note that it is fairly standard to compute numerically low lying eigenstates; is there any advantage to compute them in this new way? Could one use this method for high excited states, which are difficult computationally? Even further, could one get at eigenstates of classically chaotic systems in this way?

The recommendation for this paper to be published in a first class journal depends on these issues. Careful consideration should be given to the presentation in a revised version of the paper. The exact results for the linear potential and the harmonic oscillator do not require the solution of a new eigenvalue equation, so they do not prepare the reader for this novel use of the canonical density. It is certainly not true that the Hamilton-Jacobi equation (75) is 'exactly equivalent' to the Schrödinger equation. The formulae for 'General potentials' in section 5.1 are not identical to the ones for the linear potential and the harmonic oscillator in those special cases...

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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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