Dirac oscillator with nonzero minimal uncertainty in position

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.

28 pages, no figure, submitted to JPA

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