On the phase-integral method for the radial Dirac equation

TL;DR

This paper applies the phase-integral method to the radial Dirac equation in potential models, deriving formulae for different potentials, analyzing Stokes lines, and providing a criterion for choosing base functions, with numerical testing.

Contribution

It extends the phase-integral technique to the Dirac equation with central potentials, offering new formulae and a criterion for base function selection, including numerical validation.

Findings

Phase-integral formulae derived for various potentials

A criterion for choosing the base function established

Numerical tests confirm the approach's effectiveness

Abstract

In the application of potential models, the use of the Dirac equation in central potentials remains of phenomenological interest. The associated set of decoupled second-order ordinary differential equations is here studied by exploiting the phase-integral technique, following the work of Froman and Froman that provides a powerful tool in ordinary quantum mechanics. For various choices of the scalar and vector parts of the potential, the phase-integral formulae are derived and discussed, jointly with formulae for the evaluation of Stokes and anti-Stokes lines. A criterion for choosing the base function in the phase-integral method is also obtained, and tested numerically. The case of scalar confinement is then found to be more tractable.