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We discuss how the language of wave functions (state vectors) andassociated non-commuting Hermitian operators naturally emerges fromclassical mechanics by applying the inverse Wigner-Weyl transform to thephase space probability distribution and observables. In this language,the Schr"odinger equation follows from the Liouville equation, with \hbar ℏ now a free parameter. Classical stationary distributions can berepresented 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 apparentnegative probabilities to occupy eigenstates, dual to the negativeprobabilities in Wigner’s quasiprobability distribution. These negativeprobabilities are shown to disappear when allowing sufficientuncertainty in the classical distributions. We show that thiscorrespondence is particularly pronounced for canonical Gibbs ensembles,where classical eigenstates satisfy an integral eigenvalue equation thatreduces to the Schr"odinger equation in a saddle-pointapproximation controlled by the inverse temperature. We illustrate thiscorrespondence by showing that some paradigmatic examples such astunneling, band structures, Berry phases, Landau levels, levelstatistics and quantum eigenstates in chaotic potentials can bereproduced to a surprising precision from a classical Gibbs ensemble,without any reference to quantum mechanics and with all parameters(including \hbar ℏ )on the order of unity.