Exact solutions, critical parameters and accidental degeneracy for the hydrogen atom in a spherical box
Francisco M. Fernández
TL;DR
The paper analyzes a hydrogen atom confined inside a spherical box with an impenetrable boundary, using a scaling transformation to reveal how energies transform with box size and Coulomb strength. It shows that the radial equation is conditionally solvable, yielding exact polynomial energies $E_l^{(\nu,i)} = -\frac{[\beta_l^{(\nu,i)}]^2}{2(l+\nu+2)^2}$ with the radial quantum number identified by the number of zeros of the polynomial, and identifies dimensionless parameter $\beta = \frac{m_e r_0 K}{\hbar^2}$ governing the spectrum and the relation $E(1,\beta)=\beta^{2}E(\beta,1)$. The work reveals accidental degeneracies at particular box radii determined by truncation conditions, and shows through Rayleigh-Ritz numerics that these roots tend to the model’s critical parameters $\beta_{nl}^{c}$ as the polynomial order grows. Overall, the study provides a detailed view of the spectral structure of a confined Coulomb problem and demonstrates how polynomial solvability and RR methods yield precise insights for confined atomic systems.
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 Schrödinger 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.