Analytical bound-state solutions of the Klein-Fock-Gordon equation for the sum of Hulth\'en and Yukawa potential within SUSY quantum mechanics

TL;DR

This paper derives analytical bound-state solutions for the Klein-Fock-Gordon equation with combined Hulthén and Yukawa potentials using supersymmetric quantum mechanics, providing explicit energy levels and wave functions.

Contribution

It introduces a novel scheme to solve the Klein-Fock-Gordon equation with combined potentials using SUSY QM and shape invariance, including closed-form normalization constants.

Findings

Energy eigenvalues are highly sensitive to potential parameters.

Relativistic and non-relativistic results coincide with traditional quantum mechanics.

Wave functions are expressed via Jacobi polynomials with recursion relations.

Abstract

The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound-state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulth\'en and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and non-equal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary $l$ states. Beyond that, a closed-form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The non-relativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.