Semi-Spectral Method for the Wigner equation

TL;DR

This paper introduces a spectral numerical method for solving the Wigner equation in quantum systems, combining techniques from quantum mechanics and fluid dynamics, validated through convergence tests and behavior analysis.

Contribution

It presents a novel spectral decomposition approach for the Wigner equation, enabling efficient numerical solutions by splitting reaction and advection steps.

Findings

Second order convergence verified

Method captures non-classical Wigner function behavior

Validated on 1D harmonic potential

Abstract

We propose a numerical method to solve the Wigner equation in quantum systems of spinless, non-relativistic particles. The method uses a spectral decomposition into $L^2(\mathbb{R}^d)$ basis functions in momentum-space to obtain a system of first-order advection-reaction equations. The resulting equations are solved by splitting the reaction and advection steps so as to allow the combination of numerical techniques from quantum mechanics and computational fluid dynamics by identifying the skew-hermitian reaction matrix as a generator of unitary rotations. The method is validated for the case of particles subject to a one-dimensional (an-)harmonic potential using finite-differences for the advection part. Thereby, we verify the second order of convergence and observe non-classical behavior in the evolution of the Wigner function.