Approximate bound state solutions of Dirac equation with Hulthén potential including Coulomb-like tensor potential

Abstract

We solve the Dirac equation approximately for the attractive scalar S(r) and repulsive vector V (r) Hulthén potentials including a Coulomb-like tensor potential with arbitrary spin-orbit coupling quantum number κ. In the framework of the spin and pseudospin symmetric concept, we obtain the analytic energy spectrum and the corresponding two-component upper-and lowerspinors of the two Dirac particles by means of the Nikiforov-Uvarov method in closed form. The limit of zero tensor coupling and the non-relativistic solution are obtained. The energy spectrum for various levels is presented for several κ values under the condition of exact spin symmetry in the presence or absence of tensor coupling. Within the framework of the Dirac equation the spin symmetry arises if the magnitude of the attractive Lorentz scalar potential S(r) and the time-component repulsive vector potential are nearly equal, S(r) ∼ V (r) in nuclei (i.e., when the difference potential ∆(r) = V (r) − S(r) = C s = constant). However, the pseudospin symmetry occurs when S(r) ∼ −V (r) are nearly equal (i.e., when the sum potential Σ(r) = V (r) + S(r) = C ps = constant) [1-3]. The bound states of nucleons seem to be sensitive to some mixtures of these potentials. The cases ∆(r) = 0 and Σ(r) = 0 correspond to SU(2) symmetries of the Dirac Hamiltonian [3]. The spin symmetry is relevant for mesons [4]. The pseudospin symmetry concept has been applied to many systems in nuclear physics and related areas [2-7]. Further, it is used to explain features of deformed nuclei [8], the super-deformation [9] and to establish an effective nuclear shell-model scheme [5,6,10]. The pseudospin symmetry appeared in nuclear physics refers to a quasi-degeneracy of the single-nucleon doublets and can be characterizedwith the non-relativistic quantum numbers (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2), where n, l and j are the single-nucleon radial, orbital and total angular momentum quantum numbers for a single particle, respectively [5,6]. The total angular momentum is given as j = l + s, where l = l + 1 is a pseudo-angular momentum and s = 1/2 is a pseudospin angular momentum. In real nuclei, the pseudospin symmetry is only an approximation and the quality of approximation depends on the pseudo-centrifugal potential and pseudospin orbital potential [11]. The Dirac Hamiltonian with vector and scalar potentials quadratic in space coordinates has been studied [12]. It was shown that the the Dirac equation can be solved exactly for the cases ∆(r) = 0 and Σ(r) = 0. In addition, the linear tensor potential and mixture of quadratic scalar and vector potentials are studied [13]. It is shown that a linear tensor potential with quadratic ∆(r) or Σ(r) generates a harmonic-oscillator-like second order differential equation which can be solved analytically. Recently, Akcay [14,15] has shown that the Dirac equation for scalar and vector quadratic potentials including the Coulomb-like tensor potential with the spin and pseudospin symmetries can be solved exactly. These results in ...

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Solutions of the Dirac Equation With the Shifted Deng–fan Potential Including Yukawa-Like Tensor Interaction

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