Narrow Results

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.

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.