Improved analytical approximation to arbitrary l-state solutions of the Schrodinger equation for the hyperbolical potentials

TL;DR

This paper introduces a new approximation method for solving the Schrödinger equation with hyperbolic potentials for arbitrary angular momentum states, providing accurate energy eigenvalues and wave functions.

Contribution

A novel approximation scheme for the centrifugal term in the Schrödinger equation, enabling analytical solutions for l≠0 states with hypergeometric functions.

Findings

Numerical energy eigenvalues show high accuracy compared to existing methods.

Wave functions are expressed in terms of Jacobi polynomials.

Method effectively handles short- and long-range potentials.

Abstract

A new approximation scheme to the centrifugal term is proposed to obtain the $l\neq 0$ bound-state solutions of the Schr\"{o}dinger equation for an exponential-type potential in the framework of the hypergeometric method. The corresponding normalized wave functions are also found in terms of the Jacobi polynomials. To show the accuracy of the new proposed approximation scheme, we calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $\sigma_{\text{0}}.$ Our numerical results are of high accuracy like the other numerical results obtained by using program based on a numerical integration procedure for short-range and long-range potentials. The energy bound-state solutions for the s-wave ($l=0$) and $\sigma_{0}=1$ cases are given.