Phase Space Approach to Solving the Time-independent Schr\"odinger Equation

TL;DR

This paper introduces a phase space-based method for solving the time-independent Schrödinger equation using a von Neumann lattice with periodic boundary conditions, improving convergence and efficiency especially in higher dimensions.

Contribution

The authors develop a phase space approach with periodic boundary conditions that addresses convergence issues of the von Neumann method, enabling more efficient quantum calculations aligned with classical phase space.

Findings

Achieves convergence of the vN method with periodic boundary conditions.

Demonstrates high efficiency in the classical limit with one basis function per eigenstate.

Successfully applied to a two-dimensional potential where traditional methods fail.

Abstract

We propose a method for solving the time independent Schr\"odinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)] we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method for a challenging two-dimensional potential where the FGH method breaks down.