Analytical solution of the Klein Gordon equation with a Multi-parameter q-Deformed Woods-Saxon Type Potential

TL;DR

This paper provides an analytical solution to the Klein-Gordon equation with a multi-parameter q-deformed Woods-Saxon potential, analyzing scattering and bound states, and confirming results numerically.

Contribution

It introduces a new analytical approach to solve the Klein-Gordon equation with a complex potential, including scattering and bound state analysis, under the spin symmetric limit.

Findings

Derived reflection and transmission probabilities confirming probability conservation.

Established a quantization scheme for bound states.

Numerical validation of analytical results using Newton Raphson method.

Abstract

In this manuscript, we present analytical solution of the Klein-Gordon equation with the multi-parameter q-deformed Woods-Saxon type potential energy under the spin symmetric limit in $(1+1)$ dimension. In the scattering case, we obtain the reflection and transmission probabilities and prove the conservation of the total probability. Moreover, we analyze the correlation between the potential parameters with the reflection and transmission probabilities. In the bound state case, we use the continuity conditions and derive a quantization scheme. To confirm our results numerically, in both cases we randomly assign values to the potential parameters and find numerical results by using the Newton Raphson method.