Dirac oscillator with nonzero minimal uncertainty in position. (English) Zbl 1061.81023

Summary: 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 the one 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 \(l= j-{1\over 2}\) and \(l= j +{1\over 2}\) cases, both features being connected with supersymmetry or, equivalently, the \(\omega\to\omega\) transformation. In contrast to the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from the one associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the \(l= j-{1\over 2}\) 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 exists.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q60 Supersymmetry and quantum mechanics
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
81R15 Operator algebra methods applied to problems in quantum theory
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