Corrected Analytical Solution of the Generalized Woods-Saxon Potential for Arbitrary $\ell$ States

TL;DR

This paper derives an analytical solution for the generalized Woods-Saxon potential in quantum mechanics, providing explicit energy levels for arbitrary angular momentum states and validating results against numerical methods.

Contribution

It presents a new analytical approach to solve the Schrödinger equation with the generalized Woods-Saxon potential for any angular momentum quantum number.

Findings

Analytical energy eigenvalues match numerical results for =0.

Closed-form solutions are obtained for arbitrary nd igenfunctions.

The method improves understanding of single-particle energy levels in nuclear physics.

Abstract

The bound state solution of the radial Schr\"{o}dinger equation with the generalized Woods-Saxon potential is carefully examined by using the Pekeris approximation for arbitrary $\ell$ states. The energy eigenvalues and the corresponding eigenfunctions are analytically obtained for different $n$ and $\ell$ quantum numbers. The obtained closed forms are applied to calculate the single particle energy levels of neutron orbiting around $^{56}$Fe nucleus in order to check consistency between the analytical and Gamow code results. The analytical results are in good agreement with the results obtained by Gamow code for $\ell=0$.