An Investigation of the Bound State Solutions of the Klein-Gordon Equation for the Generalized Woods-Saxon Potential in Spin Symmetry and Pseudo-spin Symmetry Limits

TL;DR

This paper analytically investigates the bound state solutions of the Klein-Gordon equation with a generalized Woods-Saxon potential, revealing bound states only in the spin symmetric limit and confirming this with numerical calculations for a Kaon particle.

Contribution

It provides an analytical proof of the existence of bound states only in the spin symmetric limit and employs numerical methods to calculate the spectra for a specific particle.

Findings

Bound states exist only in the spin symmetric limit.

Numerical calculations confirm the non-existence of bound states in the pseudo-spin symmetric limit.

Theoretical and numerical analysis are consistent regarding bound state spectra.

Abstract

Recently, scattering of a Klein-Gordon particle in the presence of mixed scalar-vector generalized symmetric Woods-Saxon potential was investigated for the spin symmetric and the pseudo-spin symmetric limits in one spatial dimension. In this manuscript, the bound state solutions of the Klein-Gordon equation with mixed scalar-vector generalized symmetric Woods-Saxon potential are examined analytically within the framework of spin and pseudo-spin symmetry limits. We prove that the occurrence of bound state energy spectrum exists only in the spin symmetric limit, while in the pseudo-spin symmetric limit, the bound state spectrum does not exist. Besides the theoretical proof, the Newton-Raphson numerical methods are used to calculate the bound state energy spectra of a neutral Kaon particle, confined in a generalized symmetric Woods-Saxon potential, energy well constituted with repulsive or attractive surface interactions, for the spin and pseudo-spin symmetric limits, respectively. Numerical results are consistent with the non-existence of the bound state energy spectrum in the pseudo-spin symmetric limit.