Numerical methods for the Wigner equation with unbounded potential

TL;DR

This paper develops a numerical method for solving the time-dependent Wigner equation with unbounded potentials by combining polynomial approximation and spectral collocation, enabling accurate simulations of quantum systems.

Contribution

It introduces a novel approach that handles unbounded potentials in the Wigner equation using polynomial modeling and the Moyal expansion, improving accuracy and efficiency.

Findings

High-accuracy simulations of quantum systems with unbounded potentials.

Conservation of mass and energy in numerical solutions.

Reliable estimation of measurable quantum quantities.

Abstract

Unbounded potentials are always utilized to strictly confine quantum dynamics and generate bound or stationary states due to the existence of quantum tunneling. However, the existed accurate Wigner solvers are often designed for either localized potentials or those of the polynomial type. This paper attempts to solve the time-dependent Wigner equation in the presence of a general class of unbounded potentials by exploiting two equivalent forms of the pseudo-differential operator: integral form and series form (i.e., the Moyal expansion). The unbounded parts at infinities are approximated or modeled by polynomials and then a remaining localized potential dominates the central area. The fact that the Moyal expansion reduces to a finite series for polynomial potentials is fully utilized. Using a spectral collocation discretization which conserves both mass and energy, several typical quantum systems are simulated with a high accuracy and reliable estimation of macroscopically measurable quantities is thus obtained.