Two-electron quantum dot model revisited: bound states and other analytical and numerical solutions

TL;DR

This paper revisits the two-electron quantum dot model, correcting previous solutions, comparing them with numerical methods, and exploring new Coulomb-like potentials to identify bound states and eigenfunctions.

Contribution

It provides corrected analytical solutions, introduces new numerical solutions for Coulomb-like potentials, and predicts bound states in a planar two-electron quantum dot system.

Findings

Corrected previous theoretical solutions with polynomial solutions.

Numerically validated solutions using the Numerov method.

Predicted bound states and calculated eigenvalues for new potentials.

Abstract

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. In particular, some corrections are also made in previous theoretical calculations. The corrected polynomial solutions are confronted with numerical calculations based on the Numerov method, in a good agreement between both. Then, new solutions considering the $1/r$ and $\ln r$ Coulombian-like potentials in (1+2)D, not yet obtained, are discussed numerically. In particular, we are able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound states for such planar system in the case $l=0$ is predicted and the respective eigenvalues are determined.