# Bound State Solution of the Klein-Fock-Gordon equation with the Hulth\'en plus a Ring-Shaped like potential within SUSY quantum mechanics

## TL;DR

This paper derives bound state solutions for the Klein-Fock-Gordon equation with a combined Hulthén and ring-shaped potential using SUSY quantum mechanics and NU methods, providing explicit energy levels and wave functions for arbitrary angular momentum.

## Contribution

It introduces a novel approach to solve the Klein-Fock-Gordon equation with complex potentials using SUSYQM and NU methods, including explicit formulas for energy eigenvalues and wave functions.

## Key findings

- Energy eigenvalues depend on quantum numbers and potential parameters.
- Wave functions are expressed in terms of Jacobi polynomials.
- Normalization constants for wave functions are explicitly derived.

## Abstract

In this paper, the bound state solution of the modified Klein-Fock-Gordon equation is obtained for the Hulth\'en plus ring-shaped lake potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any $l\neq0$ angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein-Fock-Gordon equation of the Hulth\'en plus ring-shaped like potential are obtained by Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary $l$ states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to $n_r$ radial and $l$ orbital quantum numbers.