Approximate k-state solutions to the Dirac-Yukawa problem based on the spin and pseudospin symmetry

TL;DR

This paper derives approximate analytical solutions for the Dirac equation with Yukawa potential under spin and pseudospin symmetry, revealing how potential parameters influence energy spectra and degeneracies, and extends to special cases including Coulomb and non-relativistic limits.

Contribution

It provides new approximate analytical formulas for Dirac bound states with Yukawa potential considering spin and pseudospin symmetry, including special cases and non-relativistic limits.

Findings

Energy levels depend on screening parameter and symmetry constants.

Degeneracies are lifted by adding centrifugal-like terms.

Nonrelativistic solutions align with existing methods.

Abstract

Using an approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation with the screened Coulomb (Yukawa) potential for any arbitrary spin-orbit quantum number {\kappa}. Based on the spin and pseudospin symmetry, analytic bound state energy spectrum formulas and their corresponding upper- and lower-spinor components of two Dirac particles are obtained using a shortcut of the Nikiforov-Uvarov method. We find a wide range of permissible values for the spin symmetry constant C_{s} from the valence energy spectrum of particle and also for pseudospin symmetry constant C_{ps} from the hole energy spectrum of antiparticle. Further, we show that the present potential interaction becomes less (more) attractive for a long (short) range screening parameter {\alpha}. To remove the degeneracies in energy levels we consider the spin and pseudospin solution of Dirac equation for Yukawa potential plus a centrifugal-like term. A few special cases such as the exact spin (pseudospin) symmetry Dirac-Yukawa, the Yukawa plus centrifugal-like potentials, the limit when {\alpha} becomes zero (Coulomb potential field) and the non-relativistic limit of our solution are studied. The nonrelativistic solutions are compared with those obtained by other methods.