Quantum eigenstates from classical Gibbs distributions

02/08/2021 - 2:00pm
Speaker(s) / Presenter(s): 
Anatoli Polkovnikov, Boston U.


I will 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 ℏ 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. 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. 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. At the end I will mention some unpublished results on emergence of negative probabilities and associated doubling of the Hilbert space, which is similar to emergence of spin degrees of freedom.


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