The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.

中文翻译：

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具有多项式势的量子系统的Wigner函数

Wigner函数的Moyal方程是在电势是一个解析函数的假设下获得的。势的多项式形式是具有任何必要精度的分析势的自然近似。具有二阶多项式电势的最简单的量子系统是量子谐波振荡器。在本文中，对于具有任意阶多项式势的量子系统，基于谐波振荡器的本征函数，获得了核算子矩阵元素的显式表达式。使用核算子矩阵元素的显式表示，我们构造了具有多项式势能的量子系统在相空间中维格纳函数的分布。显示了修改后的Vlasov方程与Wigner函数的Moyal方程的关系。给出了用于高效找到维格纳函数的有效数值算法的示例。