Pseudospin symmetry and the relativistic harmonic oscillator

Abstract

A generalized relativistic harmonic oscillator for spin 1∕2 particles is studied. The Dirac Hamiltonian contains a scalar $S$ and a vector $V$ quadratic potentials in the radial coordinate, as well as a tensor potential $U$ linear in $r$. Setting either or both combinations $\ensuremath{\Sigma}=S+V$ and $\ensuremath{\Delta}=V\ensuremath{-}S$ to zero, analytical solutions for bound states of the corresponding Dirac equations are found. The eigenenergies and wave functions are presented and particular cases are discussed, devoting a special attention to the nonrelativistic limit and the case $\ensuremath{\Sigma}=0$, for which pseudospin symmetry is exact. We also show that the case $U=\ensuremath{\Delta}=0$ is the most natural generalization of the nonrelativistic harmonic oscillator. The radial node structure of the Dirac spinor is studied for several combinations of harmonic-oscillator potentials, and that study allows us to explain why nuclear intruder levels cannot be described in the framework of the relativistic harmonic oscillator in the pseudospin limit.