Interacting particles in two dimensions: numerical solution of the four-dimensional Schr\"odinger equation in a hypercube

TL;DR

This paper introduces a high-precision finite difference numerical method to solve the four-dimensional Schr"odinger equation for two interacting particles in a confined two-dimensional domain, achieving sixth-order accuracy.

Contribution

It develops a fully discretized Numerov-type finite difference scheme with 89-point stencil for solving 4D Schr"odinger equations with high accuracy.

Findings

Eigenvalue errors are consistent with theoretical predictions.

The method achieves an $ ext{O}(h^6)$ local truncation error.

Numerical results confirm the method's effectiveness for two-particle quantum systems.

Abstract

We study numerically the Coulomb interacting two-particle stationary states of the Schr\"odinger equation, where the particles are confined in a two-dimensional infinite square well. Inside the domain the particles are subjected to a steeply increasing isotropic harmonic potential, resembling that in a nucleus. For these circumstances we have developed a fully discretized finite difference method of the Numerov-type that approximates the four-dimensional Laplace operator, and thus the whole Schr\"odinger equation, with a local truncation error of $\mathcal{O}(h^6)$, with $h$ being the uniform step size. The method is built on a 89-point central difference scheme in the four-dimensional grid. As expected from the general theorem by Keller [Num.\ Math. \textbf{7}, 412 (1965)], the error of eigenvalues so obtained are found to be the same order of magnitude which we have proved analytically as well.