Approximate path integral solution for a Dirac particle in a deformed Hulth\'en potential

TL;DR

This paper develops an approximate path integral method to solve the Dirac equation in a deformed Hulthen potential, deriving the Green's function and energy spectrum for bound states.

Contribution

It introduces a novel approximation approach within the path integral formalism to analyze Dirac particles in deformed Hulthen potentials, including special cases.

Findings

Derived a closed-form Green's function for the Dirac equation in the potential.

Calculated the energy spectrum from the Green's function poles.

Analyzed special cases like the standard Hulthen potential and hydrogen-like ions.

Abstract

The problem of a Dirac particle moving in a deformed Hulthen potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green's function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green's function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green's function. A Dirac particle in the standard Hulthen potential for q=1 and a Dirac hydrogen-like ion when q = 1 and a that tends to infinity are considered as particular cases.