Solution of the Radial Schr\"{o}dinger Equation for the Potential Family $V(r)=\frac{A}{r^{2}}-\frac{B}{r}+Cr^{\kappa}$ using the Asymptotic Iteration Method

TL;DR

This paper derives exact and numerical solutions for the radial Schrödinger equation with a specific potential family using the asymptotic iteration method, covering various quantum states and parameter values.

Contribution

It provides exact solutions for certain parameter values and numerical solutions for others, expanding the analytical understanding of this potential family.

Findings

Exact solutions for κ=0, -1, -2

Numerical solutions for κ=1, 2

Results agree with previous studies

Abstract

We present the exact and iterative solutions of the radial Schr\"{o}dinger equation for a class of potential, $V(r)=\frac{A}{r^{2}}-\frac{B}{r}+Cr^{\kappa}$, for various values of $\kappa$ from -2 to 2, for any $n$ and $l$ quantum states by applying the asymptotic iteration method. The global analysis of this potential family by using the asymptotic iteration method results in exact analytical solutions for the values of $\kappa=0 ,-1$ and -2. Nevertheless, there are no analytical solutions for the cases $\kappa=1$ and 2. Therefore, the energy eigenvalues are obtained numerically. Our results are in excellent agreement with the previous works.