Two electrons on a hypersphere: a quasi-exactly solvable model

TL;DR

This paper introduces a quasi-exactly solvable model of two electrons confined to a hypersphere, revealing polynomial wave functions at specific radii and energies, with implications for understanding electron interactions in constrained geometries.

Contribution

It demonstrates that the wave function for two Coulomb-interacting electrons on a hypersphere is polynomial at certain radii, providing a new exactly solvable model in quantum physics.

Findings

Polynomial wave functions occur at specific radii.

The model's $ ext{D}=3$ case closely resembles real physical systems.

A set of energies and radii for ground and excited states are reported.

Abstract

We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a $\mathcal{D}$-sphere ($\mathcal{D} \ge 1$), is a polynomial in the interelectronic distance $u$ for a countably infinite set of values of the radius $R$. A selection of these radii, and the associated energies, are reported for ground and excited states on the singlet and triplet manifolds. We conclude that the $\mathcal{D}=3$ model bears the greatest similarity to normal physical systems.