Research Article | Open Access
The Effect of Tensor Interaction in Splitting the Energy Levels of Relativistic Systems
We solve approximately Dirac equation for Eckart plus Hulthen potentials with Coulomb-like and Yukawa-like tensor interaction in the presence of spin and pseudospin symmetry for . The formula method is used to obtain the energy eigenvalues and wave functions. We also discuss the energy eigenvalues and the Dirac spinors for Eckart plus Hulthen potentials with formula method. To show the accuracy of the present model, some numerical results are shown in both pseudospin and spin symmetry limits.
One of the interesting problems in nuclear and high energy physics is to obtain analytical solution of the Klein-Gordon, Duffin-Kemmer-Petiau, and Dirac equations for mixed vector and scalar potentials . The study of relativistic effect is always useful in some quantum mechanical systems [2, 3]. Therefore, the Dirac equation has become the most appealing relativistic wave equation for spin-1/2 particles. For example, in the relativistic treatment of nuclear phenomena, the Dirac equation is used to describe the behavior of the nuclei in nucleus and also to solve many problems of high energy physics and chemistry. For this reason, it has been used extensively to study the relativistic heavy ion collisions and heavy ion spectroscopy and more recently in laser-matter interaction (for a review, see  and references therein) and condensed matter physics [5, 6]. The idea about spin symmetry and pseudospin symmetry with the nuclear shell model has been introduced in 1969 by Arima et al. (1969) and Hecht and Adler (1969) [7, 8]. Spin and pseudospin symmetries are symmetries of a Dirac Hamiltonian with vector and scalar potentials. They are realized when the difference, , or the sum, , is constant. The near realization of these symmetries may explain degeneracy in some heavy meson spectra (spin symmetry) or in single-particle energy levels in nuclei (pseudospin symmetry), when these physical systems are described by relativistic mean-field theories (RMF) with scalar and vector potentials . Recently, some authors have studied various types of potential with a tensor potential, under the conditions of pseudospin and spin symmetry. They have found out that the tensor interaction removes the degeneracy between two states in the pseudospin and spin doublet [10–13]. The pseudospin and spin symmetry appearing in nuclear physics refers to a quasidegeneracy of the single-nucleon doublets and can be characterized with the nonrelativistic quantum numbers () and (), where , , and are the single-nucleon radial, orbital, and total angular momentum quantum numbers for a single particle, respectively [7, 8]. These kinds of various methods have been used for the analytical solutions of the Klein-Gordon equation and Dirac equation such as the super symmetric quantum mechanics [14–16], asymptotic iteration method (AIM) [17, 18], factorization method [19, 20], Laplace transform approach , GPS method [22, 23] and the path integral method [24–26], and Nikiforov-Uvarov method [27–29]. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics with some typical kinds of potential by using different methods . For example, Kratzer potential [31, 32], Woods-Saxon potential [33, 34], Scarf potential [35, 36], Hartmann potential [37, 38], Rosen Morse potential, [39, 40], Hulthen potential , and Eckart potential [42, 43].
In this paper, we attempt to solve approximately Dirac wave equation for for Eckart plus Hulthen potentials for the spin and pseudospin symmetry with a tensor potential by using the formula method. The organization of this paper is as follows: in Section 2, the formula method is reviewed . In Section 3 we review basic Dirac equations briefly. In Sections 3.1 and 3.2, solutions of Dirac wave equation for the spin and pseudospin symmetry of these potentials in the presence of Coulomb-like tensor interaction are presented, respectively. In Sections 3.3 and 3.4, solutions of Dirac wave equation for the spin and pseudospin symmetry of these potentials in the presence of Coulomb-like plus Yukawa-like tensor interaction are presented, respectively. In Section 4, we provide results and discussion. The conclusion is given in Section 5.
2. Review of Formula Method
The formula method has been used to solve the Schrodinger, Dirac, and Klein-Gordon wave equations for a certain kind of potential. In this method, the differential equations can be written as follows :For a given Schrödinger-like equation in the presence of any potential model which can be written in the form of (1), the energy eigenvalues and the corresponding wave function can be obtained by using the following formulas, respectively :where And is the normalization constant. In special case where , the energy eigenvalues and the corresponding wave function can be obtained as The solution provides a valuable means for checking and improving models and numerical methods introduced for solving complicated quantum systems.
3. Basic Dirac Equations
In the relativistic description, the Dirac equation of a single nucleon with the mass moving in an attractive scalar potential and a repulsive vector potential can be written as where is the relativistic energy, is the mass of a single particle, and and are the 4 × 4 Dirac matrices. For a particle in a central field, the total angular momentum and commute with the Dirac Hamiltonian, where is the orbital angular momentum. For a given total angular momentum , the eigenvalues of the are , where negative sign is for aligned spin and positive sign is for unaligned spin. The wave functions can be classified according to their angular momentum and spin-orbit quantum number as follows:where and are upper and lower components and and are the spherical harmonic functions. is the radial quantum number and is the projection of the angular momentum on the -axis. The orbital angular momentum quantum numbers and represent the spin and pseudospin quantum numbers. Substituting (7) into (6), we obtain couple equations for the radial part of the Dirac equation as follows by :where and are the difference and the sum of the potentials and , respectively, and is a tensor potential. We obtain the second-order Schrodinger-like equation asWe consider bound state solutions that demand the radial components satisfying , and .
3.1. Solution Spin Symmetric with Coulomb-Like Tensor Interaction
Under the condition of the spin symmetry—that is, or —the upper component Dirac equation can be written asThe potential is taken as the Eckart [42, 43] plus Hulthen potentials : where the parameters , , , and are real parameters; these parameters describe the depth of the potential well, and the parameter is related to the range of the potential.
For the tensor term, we consider the Coulomb-like potential ,where is the Coulomb radius and and stand for the charges of the projectile particle and the target nucleus , respectively.
Equation (13) is exactly solvable only for the case of . In order to obtain the analytical solutions of (13), we employ the improved approximation scheme suggested by Greene and Aldrich [47, 48] and replace the spin-orbit coupling term with the expression that is valid for :The behavior of the improved approximation is plotted in Figure 1. We can see the good agreement for small values.
By applying the transformation , (13) is brought into the formwhere the parameters , , and are considered as follows:Now by comparing (15) with (1), we can easily obtain the coefficients () as follows:The values of the coefficients () are also found from (4) as below:Thus, by the use of energy equation (2) for energy eigenvalues, we findIn Tables 1–3, we give the numerical results for the spin symmetric energy eigenvalues (in units of fm−1).
Let us find the corresponding wave functions. In reference to (3) and (18), we can obtain the upper wave function aswhere is the normalization constant; on the other hand, the lower component of the Dirac spinor can be calculated fromWe have obtained the energy eigenvalues and the wave function of the radial Dirac equation for Eckart plus Hulthen potentials with Coulomb-like tensor interaction in the presence of the spin symmetry for .
3.2. Solution Pseudospin Symmetry with Coulomb-Like Tensor Interaction
For pseudospin symmetry—that is, or —the lower component Dirac equation can be written asWe consider the scalar, vector, and tensor potentials as the following : Substituting (23) into (22), we obtain the lower radial equation of Dirac equation aswhere , and .
By using the transformation and employing the improved approximation, (24) is brought into the formwhere the parameters , , and are considered as follows:We can easily obtain the coefficients () by comparing (25) with (1) as follows:The values of the coefficients () are also found from (4) as below:We have (29) for energy eigenvalues by the use of (2):In Tables 4–6, we give the numerical results for the pseudospin symmetric energy eigenvalues (in units of fm−1).