Approximate solutions of the Dirac equation for the Rosen–Morse potential including the spin-orbit centrifugal term

Abstract

We give the approximate analytic solutions of the Dirac equations for the Rosen-Morse potential including the spin-orbit centrifugal term. In the framework of the spin and pseudospin symmetry concept, we obtain the analytic bound state energy spectra and corresponding two-component upper-and lower-spinors of the two Dirac particles, in closed form, by means of the Nikiforov-Uvarov method. The special cases of the s-wave κ = ±1 (l = l = 0) Rosen-Morse potential, the Eckart-type potential, the PT-symmetric Rosen-Morse potential and non-relativistic limits are briefly studied. Within the framework of the Dirac equation the spin symmetry arises if the magnitude of the attractive scalar potential S(r) and repulsive vector potential are nearly equal, S(r) ∼ V (r) in the nuclei (i.e., when the difference potential ∆(r) = V (r) − S(r) = C s = constant). However, the pseudospin symmetry occurs if 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 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] and 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 introduced in nuclear theory refers to a quasi-degeneracy of the single-nucleon doublets and can be characterized with 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 pseudoangular 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]. Alhaidari et al. [12] investigated in detail the physical interpretation on the three-dimensional Dirac equation in the context of spin symmetry limitation ∆(r) = 0 and pseudospin symmetry limitation Σ(r) = 0. Some authors have applied the spin and pseudospin symmetry on several physical potentials, such as the harmonic oscillator [12][13][14][15], the Woods-Saxon potential [16], the Morse potential [17,18], the Hulthén potential [19], the Eckart potential [20][21][22], the molecular diatomic three-parameter potential [23], the Pöschl-Teller potential [24], the Rosen-Morse potential [25] and the generalized Morse potential [26].The exact solutions of the Dirac equation for the exponential-type potentials are possible only for the s-wave (l = 0 case). However, for l-states an approximation scheme has to be used to deal with the centrifugal and pseudo-centrifugal terms. Many authors have used different methods to study the partially exactly solvable and exactly solvable Schrödinger, Klein-Gordon (KG) and Dirac equations in 1D, 3D and/or any D-dimen...

Bound States of the Klein-Gordon for Exponential-Type Potentials in D-Dimensions

“…Also, the KG equation with equally mixed scalar and vector Rosen-Morse-type potentials can be solved exactly for s-wave bound states ( case). The calculated energy equations of these potentials are seen to be complicated transcendental equations in the relativistic model [39]. The non-relativistic limit can be easily reached by making a mapping on the parameters and/or solving the original Schrödinger equation.…”

“…The present solutions include energy equation and un-normalized wave functions which have been expressed in terms of the Jacobi polynomials (or hypergeometric functions). Additionally, in making appropriate changes in the Eckarttype potential parameters, one can easily generate new energy spectrum formulas for various types of the wellknown molecular potentials such as the Rosen-Morse well [39], the Eckart potential, the Hulthén potential [13], the Woods-Saxon potential [7] and the Manning-Rosen potential [31] and others. It is also noted that under the PT-symmetry property, the exponential potentials can be reduced to the trigonometric potentials with real bound state solutions.…”

“…An example of its application to the vibrational states of NH₃ was given by Rosen and Morse in [39,71]. Making the corresponding parameter replacements in Equation (69), we obtain the energy equation for the Rosen-Morse well in the s-wave KG theory with equally mixed potentials, where n N   is a normalization constant.…”

“…Quite recently, we have also proposed a new approximation scheme for the centrifugal term [13,14]. The Nikiforov-Uvarov (NU) method [60] and other methods have also been used to solve the D-dimensional Schrödinger equation [61] and relativistic D-dimensional KG equation [62], Dirac equation [6,15,39,40,63] and spinless Salpeter equation [64].…”

“…Recently, the study of exponential-type potentials has attracted much attention from many authors (for example, cf, ). These physical potentials include the Woods-Saxon [7,8], Hulthén [9][10][11][12][13][14][15][16][17][18][19][20][21][22], modified hyperbolic-type [23], ManningRosen [24][25][26][27][28][29][30][31], the Eckart [32][33][34][35][36][37], the Pöschl-Teller [38] and the Rosen-Morse [39,40] potentials.…”