Exact solutions, critical parameters and accidental degeneracy for the hydrogen atom in a spherical box

TL;DR

This paper investigates the quantum properties of a hydrogen atom confined in a spherical box, deriving exact solutions and analyzing degeneracies and critical parameters using analytical and numerical methods.

Contribution

It presents new exact polynomial solutions for the hydrogen atom in a spherical box and analyzes accidental degeneracies and critical parameters.

Findings

Exact polynomial solutions for the radial Schrödinger equation.

Identification of accidental degeneracies at specific box radii.

Accurate eigenvalues obtained via Rayleigh-Ritz method.

Abstract

We derive some properties of the hydrogen atom inside a box with an impenetrable wall that have not been discussed before. Suitable scaling of the Hamiltonian operator proves to be useful for the derivation of some general properties of the eigenvalues. The radial part of the Schr\"{o}dinger equation is conditionally solvable and the exact polynomial solutions provide useful information. There are accidental degeneracies that take place at particular values of the box radius, some of which can be determined from the conditionally-solvable condition. Some of the roots stemming from the conditionally-solvable condition appear to converge towards the critical values of the model parameter. This analysis is facilitated by the Rayleigh-Ritz method that provides accurate eigenvalues.