Abstract
Spin and pseudospin symmetries of Dirac equation are solved under scalar and vector generalized isotonic oscillators and Cornell potential as a tensor interaction for arbitrary quantum number via the analytical ansatz approach. The spectrum of the system is numerically reported for typical values of the potential parameters.
1. Introduction
The solution of the fundamental dynamical equations is an interesting phenomenon in many fields of physics. The relativistic Dirac equation which describes the motion of spin particle [1]. Within the framework of Dirac equation, pseudospin and spin symmetries are used to study features of deformed nuclei, superdeformation, and effective shell model [2]. The concept of SUSYQM provides theoretical physicists with a powerful tool to deal with nonrelativistic Schrödinger equation. The pseudospin symmetry is referred to as a quasidegeneracy of single nucleon doublets with nonrelativistic quantum number and (, , ), where , , and are single nucleon radial, orbital, and total angular quantum numbers [3] and it was shown that the exact pseudospin symmetry occurs in the Dirac equation when ; that is, ., where and are repulsive vector potential and attractive scalar potential, respectively. Details of recent review of spin and pseudospin symmetries are given in [4]. These symmetries, under various phenomenological potentials, have been investigated using various methods such as Nikiforov-Uvarov (NU), supersymmetric quantum mechanics (SUSSYQM) and shape invariance (SI), ansatz approach, and asymptotic iteration (AIM) [5–15]. In recent years, the Dirac equation with different potentials in relativistic quantum mechanics with spin and pseudospin symmetry has been investigated [16–25]. The main aim of the present paper is to obtain approximate solutions of the Dirac equation with scalar and vector generalized isotonic oscillators and Cornell tensor interaction under the above mentioned symmetry limits. The isotonic oscillator potential consists of the harmonic oscillator plus centrifugal barrier which is of great interest in the theory of coherent states [26] and quantum optics [27] and in the analysis of the isochronous oscillator [28]. This potential is important because of its relation with supersymmetric quantum mechanics [29]. The two-dimensional version of the isotonic potential is superintegrable and usually is referred to as the Smorodinsky-Winternitz potential [30]. Here, we make use of the ansatz approach to deal with this complicated equation. A survey on the application of this technique to other wave equations including Dirac, Schrödinger, Klein-Gordon, spinless-Salpeter, and DKP equations can be found in [31–39]. The paper is organized as follows. In Section 2, we give a brief introduction of the supersymmetry quantum mechanics (SUSYQM). In Section 3, the Dirac equation is written for spin and pseudospin including the Cornell tensor interaction term. We next propose a physical ansatz solution to the equation and we solve the Dirac equation under these symmetries in Section 4. Finally, conclusion is presented in Section 5.
2. Dirac Equation including Tensor Coupling
In spherical coordinates, the Dirac equation with both scalar potential and vector potential can be expressed as [1, 2] where is the relativistic energy of the system; and are the 4 × 4 Dirac matrices; p is the momentum operator, . For a particle in a spherical field, the total angular momentum operator and spin-orbit matrix operator commute with the Dirac Hamiltonian, where and are the Pauli matrix and orbital angular momentum, respectively. The eigenvalues of are for aligned spin (, , etc.) and for unaligned spin (, , etc.). The complete set of the conservative quantities can be taken as , , , . As shown in [15], by taking the spherically symmetric Dirac spinor wave functions as where and are the radial wave functions of the upper and lower components, respectively, and , respectively, stand for spin and pseudospin spherical harmonics that are coupled to the angular momentum and is the projection of the angular momentum on the -axis. The orbital angular momentum quantum numbers and refer to the upper and lower components, respectively. The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum and pseudoorbital angular momentum , which is defined as for aligned spin and for unaligned spin . As shown in [1, 2], substituting (2) into (1) yields two coupled differential equations as follows:
We consider the difference potential and sum potential as and , respectively.
The spin-orbit quantum number is related to the orbital angular momentum quantum number . By eliminating in (3) and in (4), we obtain the following two second-order differential equations for the upper and lower components:
We have and .
3. Pseudospin and Spin Symmetric Solutions
3.1. Pseudospin Symmetry Limit
The pseudospin symmetry occurs when or equivalently . [1, 2]. Here, we work on
For the tensor term, we consider the Cornell potential
Substitution of these two terms into (6) gives where and for and , respectively.
3.2. Spin Symmetry Limit
In the spin symmetry limit or . [15], we consider
Substitution of the later into (5) gives where and for and , respectively.
4. The Ansatz Solution
4.1. Solution of the Pseudospin Symmetry Limit
In the previous section, we obtained a Schrodinger-like equation of the form where
The latter fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square where
By substitution of and into (14) and then taking the second-order derivative of the obtained equation, we can get By considering the case , from (14)–(16), we find By comparing the corresponding powers of (12) and (18), we have
Equation (19) gives
Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take from (19) as
Form (13), (19), and (20), the ground-state energy satisfies
From (14), (16), and (20), we simply have the upper and lower components of the wave function as
4.2. Solution of the Spin Symmetry Limit
In this case, our ordinary differential equation is where
In this case, we introduce the ansatz where substitution of the proposed ansatz gives
For , we have
By comparing the corresponding powers of (24) and (30), we have
To have physically acceptable solutions, we pick up the value by considering the above equation; the first node eigenvalue satisfies From (26), (28), and (31), the upper component of the wave function isand for the lower component of the wave function, we have
We have given some numerical values of the energy eigenvalues in Tables 1–6 for various states. We have investigated the dependence of the bound-state energy levels on potential parameter . The results in Tables 1 and 2 have found that case is contrary to case under the condition of the pseudospin and spin symmetries, respectively. Tables 3 and 4 present the dependence of the bound-state energy levels on parameters and in view of the pseudospin and spin symmetry limits. It is seen in Tables 3 and 4 that although bound states obtained in view of spin symmetry become more bounded with increasing , they become less bounded in the pseudospin symmetry limit with increasing . We show the effects of the -parameter on the bound states under the conditions of the pseudospin and spin symmetry limits. The results are given in Tables 5 and 6. It is seen that if the -parameter increases, the bound states become less bounded for both the pseudospin and the spin symmetry limits. In Figure 1, the wave functions are plotted for spin and pseudospin symmetry limits.
(a)
(b)
5. Conclusion
In this paper, we have obtained the approximate solutions of the Dirac equation for the isotonic oscillator potential including the Cornell tensor interaction within the framework of pseudospin and spin symmetry limits using the ansatz approach which stands as a strong tool of mathematical physics. We have obtained the energy eigenvalues and corresponding lower and upper wave functions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors wish to give their sincere gratitude to the referees for a technical comment on the paper.