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Modern Physics Letters A

Exact solution of the Dunkl–Schrödinger equation for an angular-dependent potential and application of the Hellman–Feynman theorem

A. Arabsaghari, HASSAN HASSANABADI, Won Sang Chung

Published Dec 7, 2024• 3 citations
Publisher

Abstract

We determine the closed-form solution of a three-dimensional stationary Schrödinger equation for an angular-dependent potential in spherical coordinates within the Dunkl formalism. Furthermore, we calculate the change of the system’s total energy with respect to coupling constants in the potential by means of the Hellman–Feynman theorem.

DOI
10.1142/s0217732324501177

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We determine the closed-form solution of a three-dimensional stationary Schrödinger equation for an angular-dependent potential in spherical coordinates within the Dunkl formalism. Furthermore, we calculate the change of the system’s total energy with respect to coupling constants in the potential by means of the Hellman–Feynman theorem.
In this paper, we solve exactly the Schrödinger 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.
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Abstract A central coordinate system is investigated within the framework of the Wigner-Dunkl deformation of quantum mechanics, treated using Feynman’s path integral approach. This approach involves performing path integration of a Lagrangian in polar coordinates, with an effective potential term that depends on both radial and angular coordinates. The success in separating variables within the two-dimensional Dunkl system facilitated obtaining exact expressions for the propagators associated with the free particle, inverse square potential, harmonic oscillator, and Coulomb potential. The energy eigenvalues and their corresponding wave functions are also determined.