Amplitude-phase methods for analyzing the radial Dirac equation: calculation of scattering phase shifts

Applied Mathematics and Computation

Sever

2010

105

122

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 ...

Section: Introductionmentioning

confidence: 99%

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

Applied Mathematics and Computation

Sever

2010

105

122

“…However, for l-states an approximation scheme has to be used to deal with the spinorbit centrifugal κ(κ + 1)/r 2 (pseudo-centrifugal, κ(κ − 1)/r 2 ) term. In this direction, many works have been done to solve the Dirac equation with large number of potentials to obtain the energy equation and the two-component spinor wave functions [35][36][37][38][39][40][41][42]. It has been concluded that the values of energy spectra may not depend on the spinor structure of the particle [43], i.e., whether one has a spin-1/2 or a spin-0 particle.…”

Section: Introductionmentioning

confidence: 99%

Approximate bound states of the Dirac equation with some physical quantum potentials

Applied Mathematics and Computation

Sever

2012

The approximate analytical solutions of the Dirac equations with the reflectionless-type and Rosen-Morse potentials including the spin-orbit centrifugal (pseudo-centrifugal) term are obtained.Under the conditions of spin and pseudospin (pspin) symmetry concept, we obtain the bound state energy spectra and the corresponding two-component upper-and lower-spinors of the two Dirac particles by means of the Nikiforov-Uvarov (NU) method in closed form. The special cases of the s-wave κ = ±1 (l = l = 0) Dirac equation and the non-relativistic limit of Dirac equation are briefly studied.

“…Zou et al [50] solved the Dirac equation with equal Eckart scalar and vector potentials in terms of SUSYQM method, shape invariance approach and function analysis method. Wei and Dong [51] obtained approximately the analytical bound state solutions of the Dirac equation with the Manning-Rosen for arbitrary spin-orbit coupling quantum number κ. Thylwe [52] presented the approach inspired by amplitude-phase method for analyzing the radial Dirac equation to calculate phase shifts by including the spin-and pseudo-spin symmetries of relativistic spectra. Alhaidari [53] solved Dirac equation by separation of variables in spherical coordinates for a large class of noncentral electromagnetic potentials.…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Mathematical Physics

2010

114

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...