Solutions of the Generalized Dunkl-Schrödinger Equation for Harmonic and Coulomb Potentials in two Dimensions

Abstract

The recent generalization of the Dunkl operator, incorporating six parameters, offers a refined approach to bridging theoretical models and experimental observations. In this study, we apply the fully generalized Dunkl derivatives to solve two cornerstone quantum mechanical problems-the harmonic oscillator and the Coulomb potential-in the non-relativistic context. Our analysis begins with the systems formulated in two-dimensional Cartesian coordinates, followed by a transition to polar coordinates to achieve angular solutions. For the radial component, we identify a required constraint that reduces the set of Wigner parameters by one. This leads to the determination of the radial eigenfunctions and the corresponding energy spectra for both systems, all within the non-relativistic context. Our analysis reveals that the Wigner parameters significantly influence the probability densities, altering the localization of the particle within the potential and highlighting the role of parity in shaping the radial distribution.