Select SciPost Phys. 10, 014 (2021) Quantum eigenstates from classical Gibbs distributions Pieter W. Claeys1? and Anatoli Polkovnikov2 1 TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK 2 Department of Physics, Boston University, Boston, MA 02215, USA ?
[email protected] Abstract We discuss how the language of wave functions (state vectors) and associated non- commuting Hermitian operators naturally emerges from classical mechanics by apply- ing the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schrödinger equation follows from the Liouville equa- tion, with ~h 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 nega- tive probabilities in Wigner’s quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distribu- tions. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that re- duces to the Schrödinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradig- matic 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 ~h) on the order of unity. Copyright P.W.