The bound state solutions of the $D$-dimensional Schr\"{o}dinger equation for the Woods-Saxon potential

TL;DR

This paper derives analytical bound state solutions of the D-dimensional Schrödinger equation with the Woods-Saxon potential using advanced approximation and solution methods, providing explicit energy spectra and wave functions.

Contribution

It introduces a novel improved scheme for the centrifugal term and applies both NU and SUSYQM methods to obtain energy eigenvalues and wave functions for arbitrary angular momentum.

Findings

Explicit energy eigenvalues for various quantum numbers

Radial wave functions expressed in closed form

Finite energy spectrum depending on potential parameters

Abstract

In this work, the analytical solutions of the $D$-dimensional Schr\"odinger equation are studied in great detail for the Wood-Saxon potential by taking advantage of the Pekeris approximation. Within a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any angular momentum case within the context of the Nikiforov-Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformed each other is demonstrated. In addition, a finite number energy spectrum depending on the depth of the potential $V_{0}$, the radial $n_{r}$ and orbital $l$ quantum numbers and parameters $D, a, R_{0}$ are defined as well.