Narrow Results

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.

By a simple algebraic approach we study the exact solution to the Dirac equation with scalar and vector trigonometric Scarf potentials in the case of spin symmetry. The transcendental energy equation and spinor wave functions are presented. It is found that there exist only positive energy bound states in the case of spin symmetry. Also, the energy eigenvalue approaches a constant when the potential parameter α goes to zero. The equally scalar and vector case is studied briefly.

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.

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by $d{s}^2={(1+{\alpha }^{2}U(r))}^2(d{t}^2-d{r}^2)-r^{2}d{𝜃}^2-r^2{sin}^2𝜃d{\varphi }^2$, where $U(r)$ is a scalar potential, coupled to vector $V(r)$ and tensor $A(r)$ potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

In this work, we analyzed the anharmonic oscillator with spin and pseudo-spin symmetries in deformed nuclei. For that, we consider the Dirac equation in curved space–time which has a line element given by $d{s}^2={(1+{\alpha }^{2}U(r))}^2(d{t}^2-d{r}^2)-r^{2}d{𝜃}^2-r^2{sin}^2𝜃d{\varphi }^2$ with electromagnetic field $A_{\mu }=(V(r),cA(r),0,0)$. We consider two forms of coupling of the spin 1/2 particle with the electromagnetic field and curved space–time: $V(r)=U(r)$ and $V(r)=-U(r)$, where the spin and pseudo-spin symmetries were manifested, respectively. We calculate the spinorial wave function and the energy spectrum of the anharmonic oscillator quasi-exactly, and from this result it is obtained that the symmetries are broken due to the coupling of the electromagnetic field with the curvature of space–time. We analyzed the densities of radial probabilities and energy spectra in both symmetries.

Several aspects about Λ-hypernuclei in the relativistic mean field theory, including the effective Λ-nucleon coupling strengths based on the successful effective nucleon-nucleon interaction PK1, hypernuclear magnetic moment and -hypernuclei, have been presented. The effect of tensor coupling in Λ-hypernuclei and the impurity effect of to nuclear structure have been discussed in detail.

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.

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum $E{3}^{\prime }$, anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a $SU(2)$ non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

The spin and pseudo-spin symmetries are analytically investigated by solving the three-dimensional Dirac equation for the Kratzer potential plus a ring-shaped potential. Relativistic Schrödinger-like wave equations coupled in energy are derived from Dirac equation. The energy eigenvalues and eigenfunctions are calculated by solving the coupled relativistic radial, and angular wave equations in the framework of asymptotic iteration method. Our numerical results revealed that the spin and pseudo-spin symmetries are relativistic symmetries of the Dirac Hamiltonian. Effects of the angle-dependent potential on the relativistic energy spectra are also investigated. In addition, we include illustrative tables to examine the solutions in detail.

Several aspects about Λ-hypernuclei in the relativistic mean field theory, including the effective Λ-nucleon coupling strengths based on the successful effective nucleon-nucleon interaction PK1, hypernuclear magnetic moment and -hypernuclei, have been presented. The effect of tensor coupling in Λ-hypernuclei and the impurity effect of to nuclear structure have been discussed in detail.