More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum

TL;DR

This paper extends supersymmetric quantum mechanical methods to analyze harmonic oscillators with minimal uncertainties in position and momentum, deriving spectra and eigenstates under deformed commutation relations and electric fields, including multi-dimensional cases.

Contribution

It provides the first explicit spectrum and eigenstates for a harmonic oscillator with minimal uncertainties under electric fields using SUSYQM, and extends methods to multi-dimensional problems.

Findings

Eigenvalues depend on deformation parameters and electric field when minimal momentum uncertainty exists.

Energy spectrum shifts are uniform when minimal position uncertainty exists without minimal momentum uncertainty.

The methods are extended to solve multi-dimensional harmonic oscillator problems with minimal uncertainties.

Abstract

We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commutation 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 $\alpha$, $\beta$. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for $\alpha \ne 0$, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for $\alpha = 0$, 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 $\beta \ne 0$. 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 $\beta$, $\beta'$. 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.