Exact solution of Dirac equation for Scarf potential with new tensor coupling potential for spin and pseudospin symmetries using Romanovski polynomials

The bound state solutions of Dirac equations for a trigonometric Scarf potential with a new tensor potential under spin and pseudospin symmetry limits are investigated using Romanovski polynomials. The proposed new tensor potential is inspired by superpotential form in supersymmetric(SUSY) quantum mechanics. The Dirac equations with trigonometric Scarf potential coupled by a new tensor potential for the pseudospin and spin symmetries reduce to Schrdinger-type equations with a shape invariant potential since the proposed new tensor potential is similar to the superpotential of trigonometric Scarf potential. The relativistic wave functions are exactly obtained in terms of Romanovski polynomials and the relativistic energy equations are also exactly obtained in the approximation scheme of centrifugal term. The new tensor potential removes the degeneracies both for pseudospin and spin symmetries.     

Relativistic symmetry of position-dependent mass particles in a Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials,including a tensor interaction under the spin and pseudospin symmetric limits.Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ.Some numerical results are also given,and the effect of tensor interaction on the bound states is presented.It is shown that tensor interaction removes the degeneracy between two states in the spin doublets.We also investigate the effects of the spatially-dependent mass on the bound states under spin symmetric limit conditions in the absence of tensor interaction.     

Relativistic symmetry of position-dependent mass particle in Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction under the spin and pseudospin symmetric limit. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit in the absence of tensor interaction.     

Spin and pseudospin symmetric Dirac particles in the field of Tietz-Hua potential including Coulomb tensor interaction＊

Approximate analytical solutions of the Dirac equation for Tietz-Hua (TH) potential including Coulomb-like tensor (CLT) potential with arbitrary spin-orbit quantum number κ are obtained within the Pekeris approximation scheme to deal with the spin-orbit coupling terms κ(κ± 1)r-2 . Under the exact spin and pseudospin symmetric limitation, bound state energy eigenvalues and associated unnormalized two-component wave functions of the Dirac particle in the field of both attractive and repulsive TH potential with tensor potential are found using the parametric Nikiforov-Uvarov (NU) method. The cases of the Morse oscillator with tensor potential, the generalized Morse oscillator with tensor potential, and the non-relativistic limits have been investigated.     

Relativistic symmetries in the Rosen-Morse potential and tensor interaction using the Nikiforov-Uvarov method

Approximate analytical bound-state solutions of the Dirac particle in the fields of attractive and repulsive Rosen- Morse (RM) potentials including the Coulomb-like tensor (CLT) potential are obtained for arbitrary spin-orbit quantum number κ. The Pekeris approximation is used to deal with the spin-orbit coupling terms κ(κ± 1)r 2 . In the presence of exact spin and pseudospin (p-spin) symmetries, the energy eigenvalues and the corresponding normalized two-component wave functions are found by using the parametric generalization of the Nikiforov-Uvarov (NU) method. The numerical results show that the CLT interaction removes degeneracies between the spin and p-spin state doublets.     

Relativistic symmetries in the Hulthn scalar-vector-tensor interactions

In the presence of spin and pseudospin (p-spin) symmetries, the approximate analytical bound states of the Dirac equation for scalar-vector-tensor Hulthn potentials are obtained with any arbitrary spin-orbit coupling number κ using the Pekeris approximation. The Hulthn tensor interaction is studied instead of the commonly used Coulomb or linear terms. The generalized parametric Nikiforov-Uvarov (NU) method is used to obtain energy eigenvalues and corresponding wave functions in their closed forms. It is shown that tensor interaction removes degeneracy between spin and p-spin doublets. Some numerical results are also given.     

Spin and pseudospin symmetries of the Dirac equation with shifted Hulthe′n potential using supersymmetric quantum mechanics

In this paper, we obtain approximate analytical solutions of the Dirac equation for the shifted Hulthe′n potential within the framework of spin and pseudospin symmetry limits for arbitrary spin–orbit quantum number κ using the supersymmetry quantum mechanics. The energy eigenvalues and the corresponding Dirac wave functions are obtained in closed forms.     

Spin and pseudospin symmetries of the Dirac equation with shifted Hulthén potential using supersymmetric quantum mechanics

In this paper, we obtain approximate analytical solutions of the Dirac equation for the shifted Hulthén potential within the framework of spin and pseudospin symmetry limits for arbitrary spin–orbit quantum number κ using the supersymmetry quantum mechanics. The energy eigenvalues and the corresponding Dirac wave functions are obtained in closed forms.     

Spin and Pseudospin Symmetries of Hellmann Potential with Three Tensor Interactions Using Nikiforov–Uvarov Method

The Dirac equation with Hellmann potential is presented in the presence of Coulomb-like tensor(CLT),Yukawa-like tensor(YLT),and Hulthen-type tensor(HLT) interactions by using Nikiforov–Uvarov method. The bound state energy spectra and the radial wave functions are obtained approximately within the framework of spin and pseudospin symmetries limit. We have also reported some numerical results and figures to show the effects of the tensor interactions. Special cases of the potential are also discussed.     

Bound states of the Schrdinger equation for the Pschl-Teller double-ring-shaped Coulomb potential

Põschl--Teller double-ring-shaped Coulomb (PTDRSC) potential, the Coulomb potential surrounded by Põschl--Teller and double-ring-shaped inversed square potential, is put forward. In spherical polar coordinates, PTDRSC potential has supersymmetry and shape invariance in φ, θ and r coordinates. By using the method of supersymmetry and shape invariance, exact bound state solutions of Schrõdinger equation with PTDRSC potential are presented. The normalized φ, θ angular wave function expressed in terms of Jacobi polynomials and the normalized radial wave function expressed in terms of Laguerre polynomials are presented. Energy spectrum equations are obtained. Wave function and energy spectrum equations of the system are related to three quantum numbers and parameters of PTDRSC potential. The solutions of wave functions and corresponding eigenvalues are only suitable for the PTDRSC potential.     

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pschl-Teller(tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ± 1)r-2.In view of spin and pseudo-spin(p-spin) symmetries,the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method(AIM).We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ.The non-relativistic limit is also obtained.     

Eigen-spectra in the Dirac-attractive radial problem plus a tensor interaction under pseudospin and spin symmetry with the SUSY approach

We approximately solve the Dirac equation for attractive radial potential including a Coulomb-like tensor interaction under pseudospin and the spin symmetry limit for any arbitrary spin-orbit quantum number, by employing the supersymmetric （SUSY） quantum mechanics and supersymmetric shape invariance technique. We obtain the energy eigenvalue equation under the pseudospin and spin conditions. Some numerical results are compared with those obtained by the Nikiforove-Uvarov （NU） method.     

Relativistic symmetries in the Hulth6n scalar-vector-tensor interactions

In the presence of spin and pseudospin （p-spin） symmetries, the approximate analytical bound states of the Dirac equation for scalar-vector-tensor Hulth6n potentials are obtained with any arbitrary spin-orbit coupling number using the Pekeris approximation. The Hulth6n tensor interaction is studied instead of the commonly used Coulomb or linear terms. The generalized parametric Nikiforov-Uvarov （NU） method is used to obtain energy eigenvalues and corresponding wave functions in their closed forms. It is shown that tensor interaction removes degeneracy between spin and p-spin doublets. Some numerical results are also given.     

Relativistic particle scattering states with tensor potential and spatially-dependent mass

We investigate the relativistic equation for particles with spin 1/2 in the q-parameter modified Pöschl-Teller potential, including Coulomb-like tensor interaction with spatially-dependent mass for the D-dimension. We present approximate solutions of the Dirac equation with these potentials for any spin-orbit quantum number κ under spin symmetry. The normalized wave functions are expressed in terms of the hyper-geometric series of the scattering states on the k/2π scale. We also give the formula for the phase shifts, and use the Nikiforov-Uvarov method to obtain the energy eigen-values equation.     

Effective Mass Dirac--Morse Problem with any κ-value

The Dirac-Morse problem is investigated within the framework of an approximation to the term proportional to 1/r^2 in the view of the position-dependent mass formalism. The energy eigenvalues and corresponding wave functions are obtained by using the parametric generalization of the Nikiforov-Uvarov method for any R-value. We also study the approximate energy eigenvalues, and the corresponding wave functions in the case of the constant-mass for pseudospin, and spin cases, respectively.     

Relativistic symmetries with the trigonometric Poeschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Poeschl-Teller （tPT） potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ（κ± i 1）r^-2. In view of spin and pseudo-spin （p-spin） symmetries, the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method （AIM）. We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ. The non-relativistic limit is also obtained.     

Relativistic effect of pseudospin symmetry and tensor coupling on the Mie-type potential via Laplace transformation method

A relativistic Mie-type potential for spin-1/2 particles is studied. The Dirac Hamiltonian contains a scalar S（r） and a vector V（r） Mie-type potential in the radial coordinates, as well as a tensor potential U（r） in the form of Coulomb potential. In the pseudospin（p-spin） symmetry setting Σ = Cps and Δ = V（r）, an analytical solution for exact bound states of the corresponding Dirac equation is found. The eigenenergies and normalized wave functions are presented and particular cases are discussed with any arbitrary spin–orbit coupling number κ. Special attention is devoted to the caseΣ = 0 for which p-spin symmetry is exact. The Laplace transform approach（LTA） is used in our calculations. Some numerical results are obtained and compared with those of other methods.     

Relativistic symmetries of Deng Fan and Eckart potentials with Coulomb-like and Yukawa-like tensor interactions

Relativistic symmetries of the Dirac equation under spin and pseudo-spin symmetries are investigated and a combina- tion of Deng-Fan and Eckart potentials with Coulomb-like and Yukawa-like tensor interaction terms are considered. The energy equation is obtained by using the Nikiforov-Uvarov method and the corresponding wave functions are expressed in terms of the hypergeometric functions. The effects of the Coulomb and Yukawa tensor interactions are numerically discussed as well.     

Relativistic interpretation of the nature of the nuclear tensor force

The spin-dependent nature of the nuclear tensor force is studied in detail within the relativistic HartreeFock approach. The relativistic formalism for the tensor force is supplemented with an additional Lorentz-invariant tensor formalism in the a-scalar channel, so as to take into account almost fully the nature of the tensor force brought about by the Fock diagrams in realistic nuclei. Specifically, the tensor sum rules are tested for the spin and pseudospin partners with and without nodes, to further understand the nature of the tensor force within the relativistic model. It is shown that the interference between the two components of nucleon spinors causes distinct violations of the tensor sum rules in realistic nuclei, mainly due to the opposite signs on the κ quantities of the upper and lower components, as well as the nodal difference. However, the sum rules can be precisely reproduced if the same radial wave functions are taken for the spin/pseudo-spin partners in addition to neglecting the lower/upper components,revealing clearly the nature of the tensor force.     

Approximate any 1-state solutions of the Dirac equation for modified deformed Hylleraas potential by using the Nikiforov-Uvarov method

<正>We investigate the spin and pseudospin symmetries of the Dirac equation under modified deformed Hylleraas potential via a Pekeris approximation and the Nikiforov-Uvarov technique.A tensor interaction of Coulomb form is considered and its degeneracy-removing role is discussed in detail.The solutions are reported for an arbitrary quantum number in a compact form and useful numerical data are included.