Approximate Solutions of the Dirac Equation for the Rosen-Morse Potential in the Presence of the Spin-Orbit and Pseudo-Orbit Centrifugal Terms

Abstract

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 (Arima et al., 1969;Hecht & Adler, 1969;Ginocchio, 2004 ). 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 pseudo-spin angular momentum. Meng et al., (1998) deduced that in real nuclei, the PSS is only an approximation and the quality of approximation depends on the pseudo-centrifugal potential and pseudo-spin orbital potential. The orbital and pseudo-orbital angular momentum quantum numbers for SS ℓ and PSS ℓ refer to the upper-and lower-spinor components (for instance, F n,κ (r) and G n,κ (r), respectively. Ginocchio (1997); (1999); (2004); (2005a); (2005b) and Meng et al., (1998) showed that SS occurs when the difference between the vector potential V(r) and scalar potential S(r) in the Dirac Hamiltonian is a constant (that is, ∆(r)=V(r) − S(r)) and PSS occurs when the sum of two potential is a constant (that is, Σ(r)=V(r)+S(r)). A large number of investigations have been carried out on the SS and PSS by solving the Dirac equation with various methods (Alberto et al

Eigensolution techniques, their applications and Fisherʼs information entropy of the Tietz–Wei diatomic molecular model

“…In other to find energy eigenvalues and the wave function (eigensolution) of the secondorder homogeneous linear differential equations of the above form, there have been several eigensolution techniques developed to exactly solve the quantum systems. Some of these techniques include the AIM [29,30,[33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50], the Feynman integral formalism [51][52][53], FAA [54][55][56][57][58][59], the exact quantization rule method [60][61][62][63][64][65], the proper quantization rule [66,67], the Nikiforov-Uvarov method [68][69][70][71][72][73][74][75][76][77][78], SU...…”

“…= − a n). This method has been employed by many researchers to obtain an eigensolution of quantum mechanical problems in both the relativistic and the non-relativistic case [54][55][56][57][58][59].…”

“…where W(r) is named as a superpotential in supersymmetric quantum mechanics [92][93][94]. Now, by substituting equation (55) into equation (53), we obtain the following equation for W(r): where the ∼ E s 0 denotes the ground-state energy. Since it is required that the superpotential should be made compatible with the right-hand side of the nonlinear Riccati equation (56), therefore, we take the superpotential W(r) as…”

Approximate analytical solutions of the relativistic equations with the Deng–Fan molecular potential including a Pekeris-type approximation to the (pseudo or) centrifugal term

“…which does not have analytic solutions for l = 0. In order to solve for any l-state solutions, we use the following Pekeris-type approximation (or a new improved approximation) [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64]:…”

“…Since analytical solutions of the wave equations (both relativistic and non-relativistic equations) with some exponential-type potentials are impossible for any arbitrary l-wave (or κ-wave) states, approximation schemes such as the Pekeris-type (or a new improved approximation scheme) need to be employed to deal with the (pseudo or) centrifugal term [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64]. These exponential-type potentials include the Pöschl-Teller potential [56,57,66,78], Manning-Rosen potential [40,48,58,59,71], Wood-Saxon potential [21,38,[60][61][62], Rosen-Morse-type potentials [63][64][65], Hulthén potential [13,18,67,68] and Morse potential [52,69,70].…”