Path integral solution for a Klein-Gordon particle in vector and scalar deformed radial Rosen-Morse-type potentials

TL;DR

This paper derives a path integral solution for a Klein-Gordon particle in Rosen-Morse-type potentials, providing explicit Green's functions, energy equations, and wave functions, with numerical solutions for bound states and analysis of special cases.

Contribution

It introduces a path integral approach to solve the Klein-Gordon equation with deformed Rosen-Morse potentials, deriving explicit Green's functions and energy quantization conditions.

Findings

Explicit radial Green's functions obtained

Energy levels for different potential shapes derived

Numerical solutions for bound states provided

Abstract

The problem of a Klein-Gordon particle moving in equal vector and scalar Rosen-Morse-type potentials is solved in the framework of Feynman's path integral approach. Explicit path integration leads to a closed form for the radial Green's function associated with different shapes of the potentials. For $q\leq -1$, and $\frac{1}{2\alpha }\ln \left\vert q\right\vert <r<+\infty $, the energy equation and the corresponding wave functions are deduced for the $l$ states using an appropriate approximation to the centrifugal potential term. When $-1<q<0$ or $q>0$, it is shown that the quantization conditions for the bound state energy levels $E_{n_{r}}$ are transcendental equations which can be solved numerically. Three special cases such as the standard radial Manning-Rosen potential $(\left\vert q\right\vert =1)$, the standard radial Rosen-Morse potential $% (V_{2}\rightarrow -V_{2},q=1)$ and the radial Eckart potential $% (V_{1}\rightarrow -V_{1},q=1)$ are also briefly discussed.