Exact wave functions of two-electron quantum rings

TL;DR

This paper presents exact solutions for the Schrödinger equation of two electrons on a ring, revealing polynomial and irrational solutions, and analyzing their geometric phases and nodal structures, advancing understanding of quantum ring systems.

Contribution

It provides closed-form solutions for two-electron quantum rings at specific radii, including analysis of their geometric phases and nodal structures, which was not previously available.

Findings

Exact polynomial and irrational solutions for specific radii.

Degenerate singlet and triplet states have different geometric phases.

Distinct nodal structures associated with two-electron states.

Abstract

We demonstrate that the Schr\"odinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.