Dirac oscillator with nonzero minimal uncertainty in position

TL;DR

This paper exactly solves the Dirac oscillator under deformed commutation relations with minimal position uncertainty, revealing unique spectral properties and supersymmetry behavior differing from the conventional case.

Contribution

It provides the first exact solution of the Dirac oscillator with nonzero minimal position uncertainty, using supersymmetric quantum mechanics and shape invariance methods.

Findings

Energy spectrum lacks degeneracy except rotational symmetry.

Supersymmetry remains unbroken for small and intermediate angular momentum states.

Large angular momentum states do not have bound states.

Abstract

In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for $E=-1$, nor symmetry between the $l = j - {1/2}$ and $l = j + {1/2}$ cases, both features being connected with supersymmetry or, equivalently, the $\omega \to - \omega$ transformation. In contrast with the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from that associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the $l = j - {1/2}$ states corresponding to small, intermediate and very large $j$ values in the sense that only for the first ones supersymmetry remains unbroken, while for the second ones no bound state does exist.