Analytical solutions of the $D$-dimensional Schr\"{o}dinger equation with the Woods-Saxon potential for arbitrary $l$ state

TL;DR

This paper derives analytical solutions for the D-dimensional Schrödinger equation with the Woods-Saxon potential, providing explicit energy levels and wavefunctions for arbitrary angular momentum states.

Contribution

It introduces a method to solve the hyper-radial Schrödinger equation with the Woods-Saxon potential in D dimensions for any l state using the Nikiforov-Uvarov approach and Pekeris approximation.

Findings

Explicit energy eigenvalues for various quantum numbers

Analytical wavefunctions derived for arbitrary l states

Extension of solutions to D-dimensional space

Abstract

In this work, the analytical solution of the hyper-radial Schr\"{o}dinger equation for the spherical Woods-Saxon potential in D dimensions is presented. In our calculations, we have applied the Nikiforov-Uvarov method by using the Pekeris approximation to the centrifugal potential for arbitrary $l$ states. The bound state energy eigenvalues and corresponding eigenfunctions are obtained for various values of $n$ and $l$ quantum numbers.