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
\hbarℏ
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 \hbarℏ)
on the order of unity.
Joint quasiprobability distribution on the measurement outcomes of MUB-driven operators
