Numerical Methods for the Wigner Equation with Unbounded Potential

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. In order to accurately resolve both the pseudo-differential operator and the linear differential operator, a spectral collocation scheme for the phase space and an explicit fourth-order Runge–Kutta time discretization are adopted. We are able to prove that the resulting full discrete spectral scheme 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.