Solution of the Dirac equation with position-dependent mass in the Coulomb field

TL;DR

This paper derives exact solutions for the Dirac equation with a position-dependent mass in a Coulomb field, revealing a relativistic mass component and potential applications in quantum field theory renormalization.

Contribution

It introduces a novel exact solution to the Dirac equation with a specific position-dependent mass, expanding theoretical understanding of relativistic quantum systems.

Findings

Explicit discrete energy spectrum derived

Wave-functions explicitly obtained

Relativistic mass component proportional to 1/r

Abstract

We obtain exact solution of the Dirac equation for a charged particle with position-dependent mass in the Coulomb field. The effective mass of the spinor has a relativistic component which is proportional to the square of the Compton wavelength and varies as 1/r. It is suggested that this model could be used as a tool in the renormalization of ultraviolet divergences in field theory. The discrete energy spectrum and spinor wave-function are obtained explicitly.