Narrow Results

In this work, we study the Dirac equation with scalar, vector, and tensor interactions. The Dirac Hamiltonian contains quadratic scalar and vector potentials, as well as a tensor potential. The tensor potential is taken as a sum of a linear term and a Coulomb-like term. It is shown that the tensor potential preserves the form of the harmonic oscillator potential and generates spin-orbit terms. The energy eigenvalues and the corresponding eigenfunctions are obtained for different alternatives.

We examine the pseudospin symmetry in deformed nuclei with axially-symmetry and obtain the eigenfunctions and energy equation for the case of axially-symmetric harmonic oscillator potential in the Dirac equation.

Based on the significant role of spin and pseudospin symmetries in hadron and nuclear spectroscopy, we have investigated Dirac equation under scalar and vector potentials of cotangent hyperbolic form besides a Coulomb tensor interaction via an approximate analytical scheme. The considered potential for small potential parameter resembles the well-established Kratzer potential. In addition, we see how the tensor term removes the degeneracy of doublets. After an acceptable approximation, namely a Pekeris-type one, we see that the problem is simply solved via the quantum mechanical idea of supersymmetry without having to deal with the cumbersome, complicated and time-consuming numerical programming.

Under the condition of pseudospin symmetry, the exact solution of Dirac equation is studied and that no bound solutions are observed for generalized asymmetrical Hartmann potential, which is in agreement with that for Coulomb potential. With the analytic continuation method, the unbound solutions are presented by mapping the wave functions of bound states in the complex momentum plane. Furthermore, the scattering phase shifts are obtained from the radial wave function by analyzing the asymptotic behavior of the confluent hypergeometric functions.

We obtain the bound state energy eigenvalues and the corresponding wave functions of the Dirac particle for the generalized Hulthén potential plus a ring-shaped potential with pseudospin and spin symmetry. The Nikiforov–Uvarov method is used in the calculations. Contribution of the angle-dependent part of the potential to the relativistic energy spectra are investigated. In addition, it is shown that the obtained results coincide with those available in the literature.

There is now motivating experimental evidence for relativistic symmetries in nuclei and hadrons, namely pseudospin and spin symmetry limits of the Dirac equation besides the old theoretical backgrounds. The most fundamental ingredients in such studies are definitely the wave functions and energy eigenvalues. Here, having in mind the importance of the Coulomb term as well as the degeneracy-removing role of tensor interaction, we obtain the exact solutions to the problem for Coulomb scalar, vector and tensor terms in both spin and pseudospin symmetry limits. We see that, contrary to many other common cumbersome techniques, the problem is simply solved via the methodology of supersymmetric quantum mechanics.

The spin and pseudospin symmetries in the Dirac Hamiltonian are investigated in the presence of the Hartmann and the Higgs oscillator superintegrable potentials. The Pauli-Dirac representation is used in the Dirac equation with scalar and vector potentials of equal magnitude. Then, the Dirac equation is reduced to a Schrödinger-like equation. The symmetry algebras of the Schrödinger-like equation corresponding to the superintegrable potentials are represented. Also, the associated irreducible representations are shown by means of the quadratic algebras. Finally, the relativistic energy spectra of the Hartmann and the Higgs oscillator superintegrable potentials are calculated.

We show that relativistic mean fields theories with scalar S, and vector V, quadratic radial potentials can generate a harmonic oscillator with exact pseudospin symmetry and positive energy bound states when S=-V. The eigenenergies are quite different from those of the non-relativistic harmonic oscillator. We also discuss a mechanism for perturbatively breaking this symmetry by introducing a tensor potential. Our results shed light into the intrinsic relativistic nature of the pseudospin symmetry, which might be important in high density systems such as neutron stars.

Pseudospin symmetry is an approximate relativistic symmetry of the nucleus as demonstrated by experimental data. This symmetry follows from the fact that the vector and scalar potentials of nucleons moving in a relativistic mean field are approximately equal in magnitude and opposite in sign. QCD sum rules in nuclear matter support this conclusion. Such an observation suggests a fundamental reason for pseudospin symmetry. We review the status of pseudospin symmetry conservation in the nucleon–nucleon interaction.

We investigated the analytical -wave solutions of Dirac equation for trigonometric Pöschl–Teller (PT) potential under the pseudospin symmetry condition. The energy eigenvalues equation and corresponding wave functions are obtained by using the Nikiforov–Uvarov (NU) method. The energy bound states are also calculated numerically.

Spin breaking for the resonant states in ²⁰⁸Pb is investigated by solving the Dirac equation with Woods–Saxon vector and scalar potentials in combination with an analytic continuation in the coupling constant method, where the stable and convergent energies and widths are obtained. Spin breaking are shown in correlation with the nuclear mean field shaped by the central depth Σ₀, a radius (range) R and a diffusivity a, which play an important role in the splittings of energy and width. The energy-level crossings appear in several spin partners of resonant states, where the decay time is found to be different for the spin doublets even when their energies are fully degenerate.

We explore the origins and breaking mechanisms of the spin and pseudospin symmetries for the antinucleon spectrum by the use of the similarity renormalization group, which transforms the Dirac Hamiltonian into a diagonal form and decomposes it into several independent components. By comparing the contribution of every component to the spin and pseudospin splittings, it is found that the energy splitting of spin doublets is dominated by the spin-orbit coupling while the pseudospin breaking comes mainly from these contributions of the nonrelativistic term and the dynamical term. In addition, the dependencies of the spin and pseudospin symmetries on the shape of the potential and the quantum numbers of the doublets are clarified.

Motivated by the prominent role of tensor interactions in nuclear spectroscopy and many applications of spin and pseudospin symmetry in hadronic and nuclear physics, we solve the Dirac equation with a CPRS potential and a Cornell tensor interaction, in the spin and pseudospin symmetry limits, by using the quasi-exactly solvable method. We obtain explicitly the wave functions for the two lowest energy levels, both for spin and pseudospin symmetry. We also discuss the degeneracy of the system.

Based on the similarity renormalization group (SRG) method, proton and neutron asymmetry in the nuclear pseudospin is investigated for the deformed system. We have checked influences of every term on the proton pseudospin splitting and their correlation with nuclear deformation. Further, we have found that dynamical term always made a similar effect on the proton pseudospin symmetry (PSS), which is different from the neutron case. By exerting its influence on the dynamical term, the vector–isovector $V_{\rho }$ potential gives the main contribution to the isospin asymmetry in the PSS except for several deeply bound states. The phenomenon that isospin asymmetry in the PSS becomes worse for the levels closer to the continuum and its reasons is disclosed.

It is known that pseudospin symmetry plays a crucial role in formation of many physical phenomena. By combining the relativistic mean field theory with the complex momentum representation method, the pseudospin symmetry in the single particle resonant states in the deformed nucleus ${}^{154}$Dy is investigated through the energy and width splittings, the quadrupole deformation parameter, the radial density distributions and occupation probabilities of the pseudospin doublets. Near the continuum threshold, the pseudospin symmetry is well reserved in both bound and resonant states. The energy and width splittings of pseudospin doublets in resonant states exhibit correlations with the deformation and quantum numbers. The good pseudospin symmetry is expected with lower pseudo-orbital angular momentum projection $\tilde{\mathrm{\Lambda }}$ and the main quantum number N. In general, an increase in deformation tends to weaken the quality of the pseudospin symmetry. The understanding of the evolution of the pseudospin doublets in the resonant states has been deepened by studying the pseudospin symmetry in the deformed nuclei.

Following a paper [R. Lisboa, Phys. Rev. C 67 054305 (2003)], we reinvestigate the role of the Coulomb and ρ potentials in the isospin asymmetry of nuclear pseudospin. By comparing the contributions of these potentials to the splitting of pseudospin doublets, we found the effect of the ρ potential is small and the Coulomb potential gives the main contribution to the observed isospin asymmetry of the psuedospin wavefunction splittings.

Within the density-dependent relativistic Hartree-Fock-Bogoliubov theory, the structure properties of superheavy nuclei are systematically investigated and Z = 120 (N = 184) is predicted as the next proton (neutron) magic number. The emergences of proton and neutron magic shells are found to be essentially related with the broken and restoration of pseudo-spin symmetry, respectively. It is also found that the nodal effects play a substantial role in determining the systematics of shell quenching and enhancement.

We show that pseudospin symmetry is a symmetry of the Dirac Hamiltonian for which the sum of the scalar and vector potentials are a constant. In this paper we discuss some of the implications of this relativistic symmetry and the experimental data that support these predictions. We show that pseudo-U(3) symmetry is a symmetry of the Dirac Hamiltonian for which the sum of harmonic oscillator vector and scalar potentials are equal to a constant, and we give the generators of pseudo-U(3) symmetry. We also show that pseudospin for nuclei implies spin symmetry for anti-nucleons moving in a nuclear environment.