Bound States for Pseudoharmonic Potential of the Dirac Equation with Spin and Pseudo-Spin Symmetry via Laplace Transform Approach

Abstract

In the field of relativistic quantum mechanics, the Dirac equation plays an important role for spin-1/2 particles. This equation has been used extensively to study the relativistic heavy ion collisions, heavy ion spectroscopy and more recently in laser–matter interaction [1] and condensed matter physics [2]. The most interesting feature for the Dirac equation is the concept of spin and pseudospin symmetries. Although these symmetries of the Dirac Hamiltonian were discovered long ago, there has been a renewed interest in obtaining the solutions of the Dirac equations for some typical potentials under spin symmetry and pseudo-spin symmetry cases. The idea about spin symmetry and pseudo-spin symmetry with the nuclear shell model has been introduced in [3]. This idea has been widely used in explaining a number of phenomena in nuclear physics and related areas. Spin and pseudo-spin symmetric concepts have been used in the studies of certain aspects of deformed and exotic nuclei. Spin symmetry is relevant to meson with one heavy quark, which is being used to explain the absence of quark spin–orbit splitting (spin doublets) observed in heavy-light quark mesons [4] and pseudo-spin symmetry concept has been successfully used to explain different phenomena in nuclear structure including deformation, superdeformation, identical bands, exotic nuclei, and degeneracies of some shell model orbitals in nuclei (pseudospin doublets) [5, 6]. In recent times, many works have been done to solve the Dirac equation to obtain the bound states energy spectra and the corresponding eigenfunctions. Ginocchio [7–12] deduced that a Dirac Hamiltonian with scalar S(r) and vector V (r) harmonic oscillator potentials when V (r) = S(r) possesses a spin symmetry as well as a U(3) symmetry, whereas a Dirac