Pseudospin symmetry and the relativistic harmonic oscillator

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 Sigma = S + V and Delta = 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 Sigma = 0, for which pseudospin symmetry is exact. We also show that the case U=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.