  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.
