Any l-state solutions of the Woods-Saxon potential in arbitrary dimensions within the new improved quantization rule

TL;DR

This paper derives approximate energy eigenvalues and eigenfunctions for the Woods-Saxon potential in arbitrary dimensions using an improved quantization rule, exploring inter-dimensional degeneracies and related potentials.

Contribution

It introduces a new approach to solving the Woods-Saxon potential in any dimension with a focus on all l-states, expanding the analytical understanding of such quantum systems.

Findings

Derived energy eigenvalues and eigenfunctions for all l-states in D dimensions.

Analyzed inter-dimensional degeneracies for various quantum numbers.

Discussed solutions for related potentials like Hulthén and special cases.

Abstract

The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulth\'{e}n potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.