Exact solutions of the Schr\"odinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

TL;DR

This paper provides exact analytical solutions to the Schrödinger equation with Dunkl derivatives for specific potentials, extending traditional quantum mechanics solutions to include Dunkl parameter effects.

Contribution

It introduces exact solutions for the Schrödinger equation with Dunkl derivatives for free particles, pseudo-harmonic oscillators, and Mie-type potentials in three dimensions.

Findings

Wave functions derived analytically

Energy spectra obtained explicitly

Results reduce to known cases without Dunkl parameters

Abstract

We solve exactly the Schr\"odinger equation for the free-particle, the pseudo-harmonic oscillator and the Mie-type potential in three dimensions with the Dunkl derivative. The equations for the radial and angular parts are obtained by using spherical coordinates and separation of variables. The wave functions and the energy spectrum for these potentials are derived in an analytical way and it is shown that our results are adequately reduced to those previously reported when we remove the Dunkl derivative parameters.