Justification for the group-theoretical method as the right way to solve the infinite spherical well in quantum mechanics

TL;DR

This paper advocates for the group-theoretical method as the most appropriate approach to solving the infinite spherical well in quantum mechanics, supported by classical-quantum correspondence evidence.

Contribution

It provides a classical mechanics perspective and compares classical and quantum predictions, justifying the group-theoretical method for this quantum problem.

Findings

Quantum and classical predictions converge at high energies.

Group-theoretical method resolves peculiarities in standard solutions.

Classical analysis supports the quantum solution approach.

Abstract

Recently, the problem of the infinite spherical well was solved by the group-theoretical method to resolve all the peculiarities in the currently accepted solution [DOI: 10.13140/RG.2.2.18172.44162 (Researchgate, 2017)]. With a view to further justifying the group-theoretical method, the problem is first studied from the viewpoint of classical mechanics. Then the radial probability densities predicted by classical mechanics are compared with those predicted from solutions of the problem obtained by the group-theoretical method. The comparisons clearly indicate the convergence of predictions of quantum mechanics and classical mechanics in the limit of large eigen-energies. Therefore, the group-theoretical method is justified as the right way to solve the problem of the infinite spherical well.