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.

中文翻译：

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经典吉布斯分布的量子本征态

我们讨论了通过将逆Wigner-Weyl变换应用于相空间概率分布和可观测量，波函数（状态向量）和相关的非交换Hermitian运算符的语言如何自然地从经典力学中出现。用这种语言，Schrrodinger方程式来自Liouville方程式，其中$ \ hbar $现在是一个自由参数。经典平稳分布可以表示为具有离散（量化）能量的平稳状态的总和，其中这些状态直接对应于有趣的是，现在是经典力学允许表面上的负概率占据本征态，这是维格纳拟概率分布中负概率的两倍。当在经典分布中允许足够的不确定性时，这些负概率将消失。我们证明了这种对应关系对于典型的吉布斯合奏尤为明显，经典的本征态满足一个积分本征值方程，该方程在逆温度控制的鞍点近似中简化为Schr \“ odinger方程。可以从经典的吉布斯系综中以惊人的精度再现诸如隧道效应，能带结构，贝里相，朗道能级，能级统计量和量子本征态之类的范例示例，而无需任何量子力学参考，并且具有所有参数（包括$ \ hbar $）按统一顺序排列。逆温度控制的鞍点近似中的odinger方程。我们通过显示可以从经典吉布斯合奏以惊人的精确度重现出混沌势能中的一些范式示例（例如隧穿，能带结构，贝里相，朗道能级，能级统计量和量子本征态）来说明这种对应关系，而无需参考量子力学且所有参数（包括$ \ hbar $）均按统一顺序排列。逆温度控制的鞍点近似中的odinger方程。我们通过显示可以从经典吉布斯合奏以惊人的精确度重现出混沌势能中的一些范式示例（例如隧穿，能带结构，贝里相，朗道能级，能级统计量和量子本征态）来说明这种对应关系，而无需参考量子力学且所有参数（包括$ \ hbar $）均按统一顺序排列。