Dynamical symmetries of two-dimensional systems in relativistic quantum mechanics

TL;DR

This paper explores the symmetries of two-dimensional relativistic quantum systems described by the Dirac equation, revealing SO(3) and SU(2) symmetries for Coulomb and harmonic oscillator potentials, respectively.

Contribution

It identifies and derives the dynamical symmetries and their generators for these systems, connecting them to their energy spectra and non-relativistic limits.

Findings

Coulomb potential exhibits SO(3) symmetry with derived generators.

Harmonic oscillator potential exhibits SU(2) symmetry with derived generators.

Energy spectra obtained from Casimir operators.

Abstract

The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum $L$. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the harmonic oscillator potential possesses an SU(2) symmetry. The generators of the symmetric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits are also discussed.