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

The problem of a relativistic spin 1/2 particle confined to a one-dimensional box is solved in a way that resembles closely the solution of the well known quantum-mechanical textbook problem of a non-relativistic particle in a box. The energy levels and probability density are computed and compared with the non-relativistic case. Resumo. O problema de uma partícula de spin 1/2 confinada por uma caixa a uma dimensãoé resolvido de uma maneira muito semelhanteà da resolução do problema de uma partícula no-relativista numa caixa referido em muitos livros introdutórios de Mecânica Quntâica. Os níveis de energia e a densidade de probabilidade são calculados e comparados com os valores não-relativistas.

Pseudospin symmetry in nuclei is investigated considering the Dirac equation with a Lorentz structured Woods-Saxon potential. The isospin correlation of the energy splittings of pseudospin partners with the nuclear potential parameters is studied. We show that, in an isotopic chain, the pseudospin symmetry is better realized for neutrons than for protons. This behavior comes from balance effects among the central nuclear potential parameters. In general, we found an isospin asymmetry of the nuclear pseudospin interaction, opposed to the nuclear spin-orbit interaction which is quasi isospin symmetric.PACS numbers: 21.10. Hw, 21.30.Fe, 21.60.Cs In some heavy nuclei a quasi-degeneracy is observed between single-nucleon states with quantum numbers (n, ℓ, j = ℓ + 1/2) and (n − 1, ℓ + 2, j = ℓ + 3/2) where n, ℓ, and j are the radial, the orbital, and the total angular momentum quantum numbers, respectively. This doublet structure is better expressed using a "pseudo" orbital angular momentum quantum number,l = ℓ + 1, and a "pseudo" spin quantum number,s = 1/2. For example, for [ns 1/2 , (n − 1)d 3/2 ] one hasl = 1, for [np 3/2 , (n − 1)f 5/2 ] one hasl = 2, etc. Exact pseudospin symmetry means degeneracy of doublets whose angular momentum quantum numbers are j =l ±s. This symmetry in nuclei was first reported about 30 years ago [1], but only recently has its origin become a topic of intense theoretical research.In recent papers [2,3,4,5,6] possible underlying mechanisms to generate such symmetry have been discussed. We briefly review the main points of these studies.Blokhin et al.[2] performed a helicity unitary transformation in a nonrelativistic single-particle Hamiltonian. They showed that the transformed radial wave functions have a different asymptotic behavior, implying that the helicity transformed mean field acquires a more diffuse surface. Application of the helicity operator to the nonrelativistic single-particle wave function maps the normal state (l, s) onto the "pseudo" state (l,s), while keeping all other global symmetries [2]. The same kind of unitary transformation was also considered earlier by Bahri et al.[3] to discuss the pseudospin symmetry in the nonrelativistic harmonic oscillator. They showed that a particular condition between the coefficients of spin-orbit and orbit-orbit terms, required to have a pseudospin symmetry in that non-relativistic single particle Hamiltonian, was consistent with relativistic mean-field (RMF) estimates.Ginocchio [4], for the first time, identified the pseudospin symmetry as a symmetry of the Dirac Hamiltonian. He pointed out that the pseudo-orbital angular momentum is just the orbital angular momentum of the lower component of the Dirac spinor. Thus, the pseudospin symmetry started to be regarded and understood in a relativistic way. He also showed that the pseudospin symmetry would be exact if the attractive scalar, S, and the repulsive vector, V , components of a Lorentz structured potential were equal in magnitude: S + V = 0. Under this condition, the pseudospi...

Pseudospin symmetry in nuclei is investigated by solving the Dirac equation with Woods-Saxon scalar and vector radial potentials, and studying the correlation of the energy splittings of pseudospin partners with the nuclear potential parameters. The pseudospin interaction is related to a pseudospin-orbit term that arises in a Schroedinger-like equation for the lower component of the Dirac spinor. We show that the contribution from this term to the energy splittings of pseudospin partners is large. The near pseudospin degeneracy results from a significant cancelation among the different terms in that equation, manifesting the dynamical character of this symmetry in the nucleus. We analyze the isospin dependence of the pseudospin symmetry and find that its dynamical character is behind the different pseudospin splittings observed in neutron and proton spectra of nuclei.Comment: 13 pages, 9 figures, uses REVTeX4 macro

In this work we study the contribution of the isoscalar tensor coupling to the realization of pseudospin symmetry in nuclei. Using realistic values for the tensor coupling strength, we show that this coupling reduces noticeably the pseudospin splittings, especially for single-particle levels near the Fermi surface. By using an energy decomposition of the pseudospin energy splittings, we show that the changes in these splittings come mainly through the changes induced in the lower radial wave function for the low-lying pseudospin partners and through changes in the expectation value of the pseudospin-orbit coupling term for surface partners. This allows us to confirm the conclusion already reached in previous studies, namely that the pseudospin symmetry in nuclei is of a dynamical nature.