The Wigner Function of Ground State and One-Dimensional Numerics

TL;DR

This paper develops a theoretical and numerical framework for computing the ground state Wigner function of many-body quantum systems, linking it to the Schrödinger equation and demonstrating its effectiveness through 1D numerical experiments.

Contribution

It introduces a novel eigenvalue problem for the Wigner function, along with a numerical method based on quantum hydrodynamics and imaginary time propagation for 1D systems.

Findings

The eigenvalue problem accurately finds ground states consistent with Schrödinger solutions.

The numerical method successfully computes Wigner functions for 1D systems.

Potential application to large-scale systems demonstrated with density functional theory examples.

Abstract

In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schr\"odinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum hydrodynamic model [Z. Cai et al, J. Math. Chem., 2013] to reduce the dimension of the problem, ii) an imaginary time propagation method is designed for solving the model, and numerical techniques such as solution reconstruction are proposed for the feasibility of the method. Results of several numerical experiments verify our method, in which the potential application of the method for large scale system is demonstrated by examples with density functional theory.