X iv : m at h - ph / 0 31 20 29 v 2 2 1 Ju n 20 04 More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and / or momentum

We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commuta-tion relations leading to nonzero minimal uncertainties in position and/or momentum. Here we determine for the first time the spectrum and the eigenvectors of a one-dimensional harmonic oscillator in the presence of a uniform electric field in terms of the deforming parameters α, β. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for α = 0, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for α = 0, we recover the conventional quantum mechanical picture of an overall energy-spectrum shift even when there is a nonzero minimum uncertainty in position, i.e., for β = 0. Then we consider the problem of a D-dimensional harmonic oscillator in the case of isotropic nonzero minimal uncertainties in the position coordinates, depending on two parameters β, β ′. We extend our methods to deal with the corresponding radial equation in the momentum representation and rederive in a simple way both the spectrum and the momentum radial wave functions previously found by solving the differential equation. This opens the way to solving new D-dimensional problems.