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 $Σ=S+V$ and $% Δ=V-S$ 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 $Σ=0$, for which pseudospin symmetry is exact. We also show that the case $U=Δ=0$ 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.

18 pages, 24 figures, uses RevTeX4, subfigure and caption macros. Revised version according to referee's comments, with typos and some figures corrected