AnnotationA mathematically rigorous derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.