2004/12/15 by C. Quesne, V. M. Tkachuk · 1 citation
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP #quant-ph
paper · pdf · doi:10.1088/0305-4470/38/8/011
published as J.Phys. A38 (2005) 1747-1766 · 28 pages, no figure, submitted to JPA
arxiv created 2004/12/15 · arxiv updated 2009/12/01
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 ω→ - ω 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.