Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential

TL;DR

This paper explores the solutions of the Dunkl-Klein-Gordon equation in three dimensions, focusing on the oscillator and Coulomb potential, revealing how Dunkl deformation influences eigenvalues and eigenfunctions.

Contribution

It introduces the Dunkl quantum mechanics framework and provides exact solutions for the Dunkl-Klein-Gordon oscillator and Coulomb problems in three dimensions.

Findings

Eigenfunctions expressed in Laguerre and Jacobi polynomials.

Dunkl deformation modifies eigenvalues and eigenfunctions.

Derived eigenvalues and eigenfunctions for Coulomb potential.

Abstract

Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic quantum mechanical differential equation in the presence of Dunkl operator-based deformation and we investigate solutions for two important problems in three-dimensional spatial space. To this end, after introducing the Dunkl quantum mechanics, we examine the Dunkl-Klein-Gordon oscillator solutions with the Cartesian and spherical coordinates. In both coordinate systems, we find that the differential equations are separable and their eigenfunctions can be given in terms of the associate Laguerre and Jacobi polynomials. We observe how the Dunkl formalism is affecting the eigenvalues as well as the eigenfunctions. As a second problem, we examine the Dunkl-Klein-Gordon equation with the Coulomb potential. We obtain the eigenvalue, their corresponding eigenfunctions, and the Dunkl-fine structure terms.