Advances in High Energy Physics

Volume 2013 (2013), Article ID 910419, 10 pages

http://dx.doi.org/10.1155/2013/910419

Research Article

Relativistic Symmetries of Hulthén Potential Incorporated with Generalized Tensor Interactions

A. N. Ikot,¹ H. Hassanabadi,² E. Maghsoodi,² and S. Zarrinkamar³

¹Theoretical Physics Group, Department of Physics, University of Uyo, Nigeria

²Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran

³Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Received 5 May 2013; Accepted 2 August 2013

Academic Editor: Shi-Hai Dong

Copyright © 2013 A. N. Ikot et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Spin and pseudospin symmetries of the Dirac equation for a Hulthén potential with a novel tensor interaction, that is, a combination of the Coulomb and Yukawa potentials, are investigated using the Nikiforov-Uvarov method. The bound-state energy spectra and the radial wave functions are approximately obtained in the case of spin and pseudospin symmetries. The tensor interactions and the degeneracy-removing role are presented in details.

1. Introduction

The exact solutions of wave equations are still an interesting problem in fundamental quantum mechanics. Unfortunately, there are only a few potentials for which the Schrödinger, Dirac, Klein-Gordon, and Duffin-Kemmer-Petiau (DKP) equations can be exactly solved. Several potential models have been introduced to explore the relativistic and nonrelativistic energy spectra and the corresponding wave functions [1–5]. Jia et al. [6] have derived the bound-state solution of the Klein-Gordon equation under unequal scalar and vector kink-like potentials. By using the series expansion method, the authors in [7] have obtained the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential. Pseudospin symmetry solution of Dirac equation for the modified Rosen-Morse potential is investigated in [8]. The DKP equation under a scalar Coulomb interaction is solved in [9] where the authors have used the ansatz approach. Ginocchio et al. [10–14] established the relationship between the pseudospin and the Dirac equation by recognizing that pseudo-orbital angular momentum is nothing but the usual orbital angular momentum of the lower component of the Dirac spinor. They went further to show that, within the framework of the Dirac theory, the spin symmetry occurs when the difference of the potential between the repulsive Lorentz vector potential and the attractive Lorentz scalar potential is a constant; that is, . On the other hand, the pseudospin symmetry arises when the sum of the potentials is a constant; that is, . Many researchers in the field have investigated the symmetries in the presence of various interactions. The list includes Manning-Rosen [15], Eckart [16], Hyllerass [17], Deng-Fan [18], Mobious square [19], Tietz [20], and Hyperbolical potentials [21]. The analysis of the symmetries has been done by using various methods such as Nikiforov-Uvarov (NU) technique [22], supersymmetric quantum mechanics (SUSYQM) [23], and many others [24, and references therein]. Due to the mathematical structure of the problem, the tensor interaction is often considered as a Coulomb or Cornell interaction [25]. Hassanabadi et al. were the first who introduced the Yukawa tensor interaction [26] besides the Coulomb-like term.

Here, we intend to report the solution of Dirac equation for the Hulthén potential under the generalized Coulomb-like and Yukawa-like potentials as tensor interactions. The Hulthén potential is defined as [27, 28]

where , are two positive real parameters and represent the strength and the screening range of the potential, respectively.

2. Dirac Equation with a Tensor Interaction

The Dirac equation under an attractive scalar potential , a repulsive vector potential , and a tensor potential in the relativistic unit is [13, 14]

where is the relativistic energy of the system, denotes the three-dimensional momentum operator, and stands for the mass of the fermionic particle. , are the Dirac matrices given as

where is unitary matrix and are the Pauli three-vector matrices as follows:

The eigenvalues of the spin-orbit coupling operator are , for unaligned and the aligned spin, respectively. The set forms a complete set of conserved quantities. Thus, we can write the spinors as

where , represent the upper and lower components of the Dirac spinors. , are the spin and pseudospin spherical harmonics and is the projection on the -axis. With other known identities,

as well as

we find the following two coupled first-order Dirac equations:

where

Decoupling the components, we obtain the second-order Schrödinger-like equation as

where , .

3. Pseudospin Symmetry Limit

The pseudospin symmetry limit occurs in Dirac equation when or . In this limit, we take as the Hulthén potential:

In addition, we propose a novel generalized tensor interaction of the form

where and are the Coulomb-like and Yukawa-like potentials defined as

with

where is the Coulomb radius and and denote the charges of the projectile a and the target nuclei , respectively. Also, is the depth of the potential. Substituting (14) into (13), we can write the tensor potential as

Substitution of (16) and (29) into (9) gives

It is well known that the above equation cannot be exactly solved due to the centrifugal term . In order to get rid of the centrifugal term, we make use of the following approximation [27, 29] (see Figure 1):

where and for and , respectively. Using a new variable of the form and introducing

where

(17) is transformed to

Comparing (23) with , we find

From one can determine the rest of coefficients as

Substitution of the values of the parameters given by (25) into and gives us the following relations for the eigenfunctions and energy eigenvalues:

where

On the other hand, the upper component can be found by using the following relation:

Figure 1: and its approximations for  fm.

4. Spin Symmetry Limit

In this section, we consider the spin symmetry limit where or . As in the previous section, we consider

Substitution of (16) and (22) into (10) gives

with , , and and for and , respectively. Substituting (18) into (30), we have

where

Introducing a new transformation of the form , we arrive at the Schrödinger-like equation:

where

Using the same approach we used to solve (23), the energy equation is obtained as

where

and the corresponding upper and lower radial wave functions are

5. Discussion and Numerical Results

We obtained the energy eigenvalues in the absence and the presence of the Coulomb-like plus Yukawa tensor interaction for various values of the quantum numbers and . The results are reported in Tables 1 and 2 under the condition of the pseudospin and spin symmetries, respectively, where we can see the way the tensor interaction affects the degeneracy of the system. If we set , the potential reduces into the Hulthén potential and our result is consistent with that of Aydoğdua et al. [27]. In Tables 1 and 2, we have compared our results with [27]. We represent the effects of the -parameter on the bound states for in Figure 2. It is seen that if the -parameter increases, the bound states become more bounded in both for symmetry limits. Figures 3 and 4 present the effect of the tensor interaction on the bound states. We have plotted the energy versus in Figure 5. Figure 6 shows the effects of and on the bound-states. In Figures 7 and 8, the wave functions are plotted for pseudospin and spin symmetry limits with and without a tensor interaction, respectively. It is seen in Figures 7 and 8 that the tensor interaction affects only the shape of the wave functions and does not change the node structure of the radial upper and lower components of the Dirac spinors.

Table 1: Bound state for the pseudospin symmetry , , , and .

Ref. [27]

Ref. [27]

1
1, −1

−0.00862755, −0.05738712
−0.00255933, −0.02961889
0, 2

−0.00862755, −0.05738712
−0.01389734, −0.10389112

2
1, −2

−0.01538462, −0.10000000
−0.00721029, −0.06124936
0, 3

−0.01538462, −0.10000000
−0.02260173, −0.15622180

3
1, −3

−0.02413384, −0.15233675
−0.01389734, −0.10389112
0, 4

−0.02413384, −0.15233675
−0.03335506, −0.21664494

4
1, −4

−0.03492388, −0.21278254
−0.02260173, −0.15622180
0, 5

−0.03492388, −0.21278254
−0.04621304, −0.28340834

1
2, −1

−0.01538462, −0.10000000
−0.00721029, −0.06124936
1, 2

−0.01538462, −0.10000000
−0.02260173, −0.15622180

2
2, −2

−0.02413384, −0.15233675
−0.01389734, −0.10389112
1, 3

−0.02413384, −0.15233675
−0.03335506, −0.21664494

3
2, −3

−0.03492388, −0.21278254
−0.02260173, −0.15622180
1, 4

−0.03492388, −0.21278254
−0.04621304, −0.28340834

4
2, −4

−0.04781723, −0.27957698
−0.03335506, −0.21664494
1, 5

−0.04781723, −0.27957698
−0.06125073, −0.35469754

Table 2: Bound state for the spin symmetry , , , and .

Ref. [27]

Ref. [27]

1
0, −2

0.06784763, 0.00145930
0.01709159, 0.00036476
0, 1

0.06784763, 0.00145930
0.15074992, 0.00328431

2
0, −3

0.15074992, 0.00328431
0.06784763, 0.00145929
0, 2

0.15074992, 0.00328431
0.26338976, 0.00584101

3
0, −4

0.26338976, 0.00584101
0.15074992, 0.00328431
0, 3

0.26338976, 0.00584101
0.40263365, 0.00913106

4
0,−5

0.40263365, 0.00913106
0.26338976, 0.00584101
0, 4

0.40263365, 0.00913106
0.56482502, 0.01315663

1
1, −2

0.15074992, 0.00328431
0.06784763, 0.00145929
1, 1

0.15074992, 0.00328431
0.26338976, 0.00584101

2
1, −3

0.26338976, 0.00584101
0.15074992, 0.00328431
1, 2

0.26338976, 0.00584101
0.40263365, 0.00913106

3
1, −4

0.40263365, 0.00913106
0.26338976, 0.00584101
1, 3

0.40263365, 0.00913106
0.56482502, 0.01315663

4
1, −5

0.56482502, 0.01315663
0.40263365, 0.00913106
1, 4

0.56482502, 0.01315663
0.74599944, 0.01792038

Figure 2: PSS: energy versus for pseudospin symmetry limit for , , , , and . SS: energy versus for spin symmetry limit for , , , , and .

Figure 3: PSS: energy versus for pseudospin symmetry limit for , , , , and . SS: energy versus for spin symmetry limit for , , , , and .

Figure 4: PSS: energy versus for pseudospin symmetry limit for , , , , and . SS: energy versus for spin symmetry limit for , , , , and .

Figure 5: PSS: energy versus for pseudospin symmetry limit for , , , , and . SS: energy versus for spin symmetry limit for , , , , and .

Figure 6: PSS: energy versus for pseudospin symmetry limit for , , , , and . SS: energy versus for spin symmetry limit for , , , , and .

Figure 7: Wavefunction for pseudospin symmetry limit for , , , and .

Figure 8: Wavefunction for spin symmetry limit for , , , and .

6. Conclusion

In this paper, we obtained the approximate analytical solutions of the Dirac equation for the Hulthén potential with a novel generalized tensor interaction consisting of the Coulomb and Yukawa interactions within the framework of pseudospin and spin symmetry limits using the NU technique. We have obtained the energy levels in a closed form and the corresponding wave functions in terms of the Jacobi polynomials. We also included some numerical results to investigate the way the combination of Coulomb and Yukawa potentials. Finally, the results of our work find many applications in both nuclear and Hadron physics and therefore provide more general solutions compared to other previous works performed in [30, 31].

Appendix

The NU method solves many linear second-order differential equations by reducing them to a generalized equation of hypergeometric type. Here, instead of the original formulation, we use the parametric version which enables us to solve a second-order differential equation of the form [22, 32]

According to the NU method, the eigenfunction is

and the energy of the system satisfies

where

And is Jacobi polynomial.

Acknowledgment

The authors wish to give their sincere gratitude to the referees for their careful and technical critiques.

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