Spin and Pseudospin symmetries in the Dirac equation with central Coulomb potentials

TL;DR

This paper provides a detailed analytical study of the Dirac equation with Coulomb potentials, focusing on spin and pseudospin symmetries, their perturbative aspects, and their relation to hydrogenic solutions.

Contribution

It offers a comprehensive analysis of spin and pseudospin symmetries in the Dirac equation with Coulomb potentials, including their perturbative nature and quantum number structures.

Findings

Reproduces known relations between spin and pseudospin symmetries for nuclear potentials.

Analyzes the node structure of radial functions under these symmetries.

Finds similarities between solutions with symmetries and hydrogenic atom solutions.

Abstract

We analyze in detail the analytical solutions of the Dirac equation with scalar S and vector V Coulomb radial potentials near the limit of spin and pseudospin symmetries, i.e., when those potentials have the same magnitude and either the same sign or opposite signs, respectively. By performing an expansion of the relevant coefficients we also assess the perturbative nature of both symmetries and their relations the (pseudo)spin-orbit coupling. The former analysis is made for both positive and negative energy solutions and we reproduce the relations between spin and pseudospin symmetries found before for nuclear mean-field potentials. We discuss the node structure of the radial functions and the quantum numbers of the solutions when there is spin or pseudospin symmetry, which we find to be similar to the well-known solutions of hydrogenic atoms.