On the physics of the $so_q(4)$ hydrogen atom

TL;DR

This paper explores the q-deformation of the $so(4)$ symmetry in the hydrogen atom, deriving the energy spectrum and highlighting issues with previous approaches that used deformed Casimir operators.

Contribution

It introduces a consistent method for deriving the energy spectrum of the q-deformed hydrogen atom and clarifies the physical implications of the deformation.

Findings

Deformation causes degeneracy breaking in the energy levels.

The Hilbert space of the system is reduced due to deformation.

Using deformed Casimir operators leads to inconsistencies.

Abstract

In this work we investigate the $q$-deformation of the $so(4)$ dynamical symmetry of the hydrogen atom using the theory of the quantum group $su_q(2)$. We derive the energy spectrum in a physically consistent manner and find a degeneracy breaking as well as a smaller Hilbert space. We point out that using the deformed Casimir as was done before leads to inconsistencies in the physical interpretation of the theory.