- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in High Energy Physics

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

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

## Relativistic Symmetries of Hulthén Potential Incorporated with Generalized Tensor Interactions

^{1}Theoretical Physics Group, Department of Physics, University of Uyo, Nigeria^{2}Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran^{3}Department 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:

#### 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.

#### 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.

#### References

- Y. Xu, S. He, and C. S. Jia, “Approximate analytical solutions of the Klein-Gordon equation with the Pöschl-Teller potential including the centrifugal term,”
*Physica Scripta*, vol. 81, Article ID 045001, 2010. View at Publisher · View at Google Scholar - X. Y. Gu and S. H. Dong, “Energy spectrum of the Manning-Rosen potential including centrifugal term solved by exact and proper quantization rules,”
*Journal of Mathematical Chemistry*, vol. 49, pp. 2053–2062, 2011. View at Publisher · View at Google Scholar - M. R. Pahlavani and S. A. Alavi, “Solutions of Woods-Saxon potential with Spin-Orbit and centrifugal terms through Nikiforov-Uvarov method,”
*Communications in Theoretical Physics*, vol. 58, p. 739, 2012. View at Publisher · View at Google Scholar - G. F. Wei and S. H. Dong, “A novel algebraic approach to spin symmetry for Dirac equation with scalar and vector second Pöschl-Teller potentials,”
*European Physical Journal A*, vol. 43, pp. 185–190, 2010. View at Publisher · View at Google Scholar - C.-Y. Chen, D.-S. Sun, and F.-L. Lu, “The relativistic bound states of the Hartmann potentials,”
*Physica Scripta*, vol. 74, no. 4, article 001, pp. 405–409, 2006. View at Publisher · View at Google Scholar · View at Scopus - C. S. Jia, X. P. Li, and L. H. Zhang, “Exact solutions of the Klein-Gordon equation with position-dependent mass for mixed vector and scalar kink-like potentials,”
*Few-Body Systems*, vol. 52, pp. 11–18, 2012. View at Publisher · View at Google Scholar - S. H. Dong and G. H. Sun, “The series solutions of the non-relativistic equation with the Morse potential,”
*Physics Letters A*, vol. 314, pp. 261–266, 2003. View at Publisher · View at Google Scholar - G. F. Wei and S. H. Dong, “Pseudospin symmetry for modified Rosen-Morse potential including a Pekeris-type approximation to the pseudo-centrifugal term,”
*European Physical Journal A*, vol. 46, pp. 207–212, 2010. View at Publisher · View at Google Scholar - H. Hassanabadi, B. H. Yazarloo, S. Zarrinkamar, and A. A. Rajabi, “Duffin-Kemmer-Petiau equation under a scalar Coulomb interaction,”
*Physical Review C*, vol. 84, no. 6, Article ID 064003, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. N. Ginocchio, “Relativistic harmonic oscillator with spin symmetry,”
*Physical Review C*, vol. 69, no. 3, Article ID 034318, 2004. View at Publisher · View at Google Scholar · View at Scopus - J. N. Ginocchio, “Pseudospin as a relativistic symmetry,”
*Physical Review Letters*, vol. 78, pp. 436–439, 1997. - P. R. Page, T. Goldman, and J. N. Ginocchio, “Relativistic symmetry suppresses quark spin-orbit splitting,”
*Physical Review Letters*, vol. 66, pp. 204–207, 2001. - J. N. Ginocchio, “Relativistic symmetries in nuclei and hadrons,”
*Physics Reports*, vol. 414, no. 4-5, pp. 165–261, 2005. - J. N. Ginocchio, A. Leviatan, J. Meng, and S.-G. Zhou, “Test of pseudospin symmetry in deformed nuclei,”
*Physical Review C*, vol. 69, no. 3, Article ID 034303, 2004. View at Publisher · View at Google Scholar · View at Scopus - F. Taskin, “Approximate solutions of the Dirac equation for the Manning-Rosen potential including the spin-orbit coupling term,”
*International Journal of Theoretical Physics*, vol. 48, pp. 1142–1149, 2000. - C. S. Jia, P. Guo, and X. L. Peng, “Exact solution of the Dirac-Eckart problem with spin and pseudospin symmetry,”
*International Journal of Theoretical Physics*, vol. 39, p. 7737, 2006. - H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, and H. Rahimov, “Approximate any l-state solutions of Dirac equation for modified deformed Hylleraas potential by the Nikiforov-Uvarov method,”
*Journal of Mathematical Physics*, vol. 21, no. 2, Article ID 120302, 2012. View at Publisher · View at Google Scholar - E. Maghsoodi, H. Hassanabadi, and S. Zarrinkamar, “Spectrum of Dirac Equation under Deng-Fan scalar and vector potentials and a Coulomb tensor interaction by SUSYQM,”
*Few-Body Systems*, vol. 53, pp. 528–538, 2012. View at Publisher · View at Google Scholar · View at Scopus - E. Maghsoodi, H. Hassanabadi, H. Rahimov, and S. Zarrinkamar, “Arbitrary-state solutions of the Dirac equation for a Möbius square potential using the Nikiforov-Uvarov method,”
*Chinese Physics C*, vol. 37, Article ID 04105, 2013. View at Publisher · View at Google Scholar - H. Hassanabadi, E. Maghsoodi, and S. Zarrinkamar, “Relativistic symmetries of Dirac equation and the Tietz potential,”
*European Physical Journal Plus*, vol. 127, no. 3, pp. 1–14, 2012. View at Publisher · View at Google Scholar · View at Scopus - A. N. Ikot, “Solutions of Dirac Equation for generalized hyperbolical potential including Coulomb-Like tensor potential with spin symmetry,”
*Few-Body Systems*, vol. 53, pp. 549–555, 2012. View at Publisher · View at Google Scholar - A. F. Nikiforov and V. B. Uvarov,
*Special Functions of Mathematical Physics*, Birkhauser, Basel, Switzerland, 1988. - F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,”
*Physics Report*, vol. 251, no. 5-6, pp. 267–385, 1995. View at Scopus - A. N. Ikot, E. Maghsoodi, S. Zarrinkamar, and H. Hassanabadi, “Relativistic spin and pseudospin symmetries of inversely quadratic Yukawa-like plus Mobius Square potentials including a Coulomb-like tensor interaction,”
*Few-Body Systems*, 2013. - M. Hamzavi, S. M. Ikhdair, and B. I. Ita, “Approximate spin and pseudospin solutions to the Dirac equation for the inversely quadratic Yukawa potential and tensor interaction,”
*Physica Scripta*, vol. 85, no. 4, Article ID 045009, 2012. View at Publisher · View at Google Scholar · View at Scopus - H. Hassanabadi, E. Maghsoodi, and S. Zarrinkamar, “Spin and pseudospin symmetries of Dirac Equation and the Yukawa potential as the tensor interaction,”
*Communications in Theoretical Physics*, vol. 58, p. 807, 2012. View at Publisher · View at Google Scholar - O. Aydoğdua, E. Maghsoodi, and H. Hassanabadi, “Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction,”
*Chinese Physics B*, vol. 22, Article ID 010302, 2013. - H. Hassanabadi, S. Zarrinkamar, and H. Rahimov, “Approximate solution of D-dimensional Klein-Gordon equation with Hulthen-type potential via SUSYQM,”
*Communications in Theoretical Physics*, vol. 56, pp. 423–428, 2011. - C. L. Pekeris, “The rotation-vibration coupling in diatomic molecules,”
*Physical Review*, vol. 45, no. 2, pp. 98–103, 1934. View at Publisher · View at Google Scholar · View at Scopus - H. Hassanabadi, E. Maghsoodi, A. N. Ikot, and S. Zarrinkamar, “Dirac equation under Manning-Rosen potential and Hulthén tensor interaction,”
*European Physical Journal Plus*, vol. 128, article 79, 2013. View at Publisher · View at Google Scholar - Y. Sun, S. He, and C. S. Jia, “Equivalence of the deformed modified Rosen-Morse potential energy model and the Tietz potential energy model,”
*Physica Scripta*, vol. 87, Article ID 025301, 2013. View at Publisher · View at Google Scholar - C. Tezcan and R. Sever, “A general approach for the exact solution of the Schrödinger equation,”
*International Journal of Theoretical Physics*, vol. 48, pp. 337–350, 2009. View at Publisher · View at Google Scholar