Analytical Solutions Of The Schr\"{o}dinger Equation For The Hulth\'en Potential Within SUSY Quantum Mechanics

TL;DR

This paper derives analytical solutions for the Schrödinger equation with the Hulthén potential using SUSY quantum mechanics and the Nikiforov-Uvarov method, providing energy levels and eigenfunctions for arbitrary angular momentum states.

Contribution

It introduces an improved approximation scheme for the centrifugal potential and applies SUSY quantum mechanics to obtain solutions for all angular momentum states.

Findings

Explicit energy spectra for the Hulthén potential.

Normalized eigenfunctions expressed in orthogonal polynomials.

Validation of the approximation scheme for arbitrary l states.

Abstract

The analytical solution of the modified radial Schr\"{o}dinger equation for the Hulth\'en potential is obtained within ordinary quantum mechanics by applying the Nikiforov-Uvarov method and supersymmetric quantum mechanics by applying the shape invariance consept that was introduced by Gendenshtein method by using the improved approximation scheme to the centrifugal potential for arbitrary $l$ states. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary $l$ states.