Quantum eigenstates from classical Gibbs distributions

TL;DR

This paper demonstrates how classical Gibbs distributions can be mapped to quantum eigenstates and Schrödinger-like equations, revealing a deep classical-quantum correspondence without relying on traditional quantum postulates.

Contribution

It introduces a novel framework connecting classical Gibbs ensembles to quantum eigenstates and Schrödinger equations through the inverse Wigner-Weyl transform.

Findings

Classical Gibbs distributions can reproduce quantum phenomena like tunneling and band structures.

Negative probabilities in classical distributions correspond to quantum quasiprobabilities, disappearing with increased uncertainty.

The classical eigenstates satisfy an integral equation reducing to Schrödinger's equation in a saddle-point approximation.

Abstract

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.