Dunkl-Schrodinger Equation in Higher Dimension

TL;DR

This paper derives analytical solutions for the Dunkl-Schrodinger equation in higher dimensions, solving for eigenvalues and eigenfunctions of harmonic oscillator and Coulomb problems using Cartesian and polar coordinates.

Contribution

It introduces exact solutions for the Dunkl-Schrodinger equation in higher dimensions for key quantum systems, employing separation of variables in multiple coordinate systems.

Findings

Eigenvalues and eigenfunctions for Dunkl harmonic oscillator

Analytical solutions for Coulomb potential with Dunkl operators

Graphical illustrations of energy eigenvalues and probability densities

Abstract

This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schr\"odinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms: the d-dimensional harmonic oscillator and the Coulomb potential. In order to obtain analytical solutions to these problems, both Cartesian and polar coordinate systems were employed. Firstly, the Dunkl-Schr\"odinger equation is derived in d-dimensional Cartesian coordinates, and then for the isotropic harmonic potential interaction, its solutions are given. Subsequently, using polar coordinates the angular and radial parts of the Dunkl-Schr\"odinger equation are obtained. It is demonstrated that the system permits the separation of variables in both coordinate systems, with the resulting separated solutions expressed through Laguerre and Jacobi polynomials. Then, the radial Dunkl-Schr\"odinger equation is solved using the isotropic harmonic, pseudoharmonic, and Coulomb potentials. The eigenstates and eigenvalues are obtained for each case and the behavior of the energy eigenvalue functions are illustrated graphically with the reduced probability densities.