Exact Solutions of the Schr\"odinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well

TL;DR

This paper derives exact analytical solutions for the Schrödinger-Dunkl equation describing a free particle in cylindrical wells, revealing how reflection operators influence wavefunctions and energy spectra.

Contribution

It provides the first explicit solutions of the Schrödinger-Dunkl equation in cylindrical wells, incorporating reflection symmetries and classifying solutions by Dunkl parameters.

Findings

Exact wavefunctions expressed via Bessel functions

Energy spectra classified by reflection operator eigenvalues

Conditions for definite parity wavefunctions derived

Abstract

In this paper, we study the Schr\"odinger equation with Dunkl derivative for a free particle confined in a cylindrical potential well. We consider both the finite and infinite height cases. The Dunkl formalism introduces reflection operators that modify the structure of the Hamiltonian and affect the parity of the solutions. By working in cylindrical coordinates, we obtain exact analytical expressions for the radial and axial wavefunctions in terms of Bessel functions. The energy spectrum and the solutions are classified according to the eigenvalues of the reflection operators in the three coordinates. We analyze in detail the conditions under which the wavefunctions acquire definite parity and discuss the resulting constraints on the Dunkl parameters.