Wigner function of a quantum system with polynomial potential

Abstract

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.

Parallel Algorithm for Calculating the Wigner Function for a Quantum System With a Polynomial Potential

“…Используя выражения (i.2)-(i.4), в работе [22] был описан метод построения функции Вигнера для квантовой системы с полиномиальным потенциалом…”

The Wigner function negative value domains and energy function poles of the harmonic oscillator

“…The problems regarding the search for positive quasi-distribution functions as well as possible interpretations of negative quasi-probability values remain to be challenging today. A.A. Vlasov obtained an infinite self-linked chain of equations for the distribution density functions of higher kinematic quantities [16]. Let us consider the first two equations from the infinite self-linked Vlasov equations chain for the probability…”

“…The vector fields Nt determines the number of particles in the system, which can be non-integer [16]. For a constant number of particles ( N const  ) the value N is used as a normalizing factor when calculating the total probability.…”

“…U is an analytical function. It was shown [16] that the Moyal equation can be regarded as a special case of the second Vlasov equation (i.4) with the Vlasov-Moyal approximation [16] of the vector acceleration field…”

“…Averaging over the velocity space of the Vlasov-Moyal approximation (i.12) will give the classical Vlasov approximation [18] for equation (i.10): 1 1 .…”