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Advances in High Energy Physics

VolumeΒ 2011Β (2011), Article IDΒ 458087, 6 pages

http://dx.doi.org/10.1155/2011/458087

## Approximate Solutions of Klein-Gordon Equation with Kratzer Potential

^{1}Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran^{2}Computer Engineering Department, Shahrood University of Technology, Shahrood, Iran^{3}Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Received 17 March 2011; Revised 26 June 2011; Accepted 1 August 2011

Academic Editor: A.Β Petrov

Copyright Β© 2011 H. Hassanabadi 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

Approximate solutions of the *D*-dimensional Klein-Gordon equation are obtained for the scalar and vector general Kratzer potential for any by using the ansatz method. The energy behavior is numerically discussed.

#### 1. Introduction

The Kratzer potential is amongst the most attractive physical potentials as it contains a degeneracy-removing inverse square term besides the common Coulomb term. It appears in a wide class of physical and chemical sciences including the atomic and molecular physics providing quite motivating results [1β7]. When we deal with this potential within the framework of SchrΓΆdinger equation, the problem is simply solved via the analogy with familiar example of 3-dimesnional Coulomb Hamiltonian or many other techniques including series expansions, supersymmetry quantum mechanics (SUSY) [8β10], the Nikiforov-Uvarov (NU) [11], point canonical transformation (PCT) [12β14], and so forth. Such investigations have been done by many authors in the annals of wave equations [15β24]. The problem just arises when we intend to study the problem via the Klein-Gordon (KG) equation. This is because we have to deal with an equivalent potential which includes Coulomb, inverse square, inverse cubic and inverse quadric terms. Until now, no exact analytical solution has been reported for the problem. Within the present study, we study the problem via an Ansatz approach proposed by Dong [25] and numerically report the results.

#### 2. *D*-Dimensions Klein-Gordon Equation

The radial Klein-Gordon equation for a spherically symmetric potential in *D*-dimensions is
For the scalar and vector potentials we choose
where denotes the hyperradius and , , , and are constant coefficients. For the mass, instead of constant one, we consider a position-dependent mass of the form
The transformation , after inserting (2.2) brings (2.1) into the form
Choosing
Equation (2.4) is more neatly written as
The SchrΓΆdinger analogue of this problem has been analyzed by Dong [25]. We choose [25]
where
Substitution of (2.9), (2.8), and (2.7) in (2.4), after equating the corresponding coefficients on both sides, gives
From (2.5) and (2.10), the energy of the nodeless state is obtained as
with its corresponding eigenfunction being obtained by substitution of (2.8), (2.9), and (2.10) in (2.7) as
In Table 1, we have reported the eigenvalues for s and s. Repeating the same procedure for the first node, the eigenvalues are found as
where
And the corresponding eigenfunction is
Also in Table 2, as well as Figures 1 and 2, we have reported the energy behavior for various conditions. The figures well illustrate the symmetries of energy relation.

#### 3. Conclusion

Approximate analytical solutions of Klein-Gordon equation are reported for the Kratzer potential using the Ansatz method. The behavior of energy eigenvalues on dimension and quantum numbers is numerically calculated. The results are applicable to some branches of physics, particularly atomic, molecular, and chemical physics, where a spin-0 system is being investigated.

#### Acknowledgment

The authors would like to thank Professor Shi-Hai Dong for several useful suggestions.

#### References

- W. C. Qiang, βBound states of the Klein-Gordon and Dirac equations for potential $V(r)=A{r}^{-\text{2}}-B{r}^{-\text{1}}$,β
*Chinese Physics*, vol. 12, no. 10, pp. 1054β1057, 2003. View at Publisher Β· View at Google Scholar Β· View at Scopus - Y. F. Cheng and T. Q. Dai, βExact solution of the Schrödinger equation for the modified Kratzer potential plus a ring-shaped potential by the Nikiforov-Uvarov method,β
*Physica Scripta*, vol. 75, no. 3, pp. 274β277, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - C. Berkdemir, A. Berkdemir, and J. Han, βBound state solutions of the Schrödinger equation for modified Kratzer's molecular potential,β
*Chemical Physics Letters*, vol. 417, no. 4–6, pp. 326β329, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus - S. M. Ikhdair and R. Sever, βExact solutions of the modified kratzer potential plus ring-shaped potential in the
*D*-dimensional Schrödinger equation by the nikiforovuvarov method,β*International Journal of Modern Physics C*, vol. 19, no. 2, pp. 221β235, 2008. View at Publisher Β· View at Google Scholar - R. J. Leroy and R. B. Bernstein, βDissociation energy and long-range potential of diatomic molecules from vibrational spacings of higher levels,β
*The Journal of Chemical Physics*, vol. 52, no. 8, pp. 3869β3879, 1970. View at Scopus - C. L. Pekeris, βThe rotation-vibration coupling in diatomic molecules,β
*Physical Review*, vol. 45, no. 2, pp. 98β103, 1934. View at Publisher Β· View at Google Scholar Β· View at Scopus - C. Y. Chen and S. H. Dong, βExactly complete solutions of the Coulomb potential plus a new ring-shaped potential,β
*Physics Letters. A*, vol. 335, no. 5-6, pp. 374β382, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - B. K. Bagchi,
*Supersymmetry in Quantum and Classical Mechanics*, vol. 116, Chapman and Hall, Boca Raton, Fla, USA, 2001. - F. Cooper, A. Khare, and U. Sukhatme, βSupersymmetry and quantum mechanics,β
*Physics Reports*, vol. 251, no. 5-6, pp. 267β385, 1995. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - G. Junker,
*Supersymmetric Methods in Quantum and Statistical Physics*, Springer, Berlin, Germany, 1996. - A. F. Nikiforov and V. B. Uvarov,
*Special Functions of Mathematical Physics*, Birkhäuser, Basel, Germany, 1988. - R. De, R. Dutt, and U. Sukhatme, βMapping of shape invariant potentials under point canonical transformations,β
*Journal of Physics A: Mathematical and General*, vol. 25, no. 13, pp. L843βL850, 1992. View at Publisher Β· View at Google Scholar - A. Gangopadhyaya, P. K. Panigrahi, and U. P. Sukhatme, βInter-relations of solvable potentials,β
*Helvetica Physica Acta*, vol. 67, no. 4, pp. 363β368, 1994. View at Zentralblatt MATH - A. Gangopadhyaya, J. V. Mallow, C. Rasinariu, and U. P. Sukhatme,
*Supersymmetric Quantum Mechanics: An Introduction*. - A. Lahiri, P. K. Roy, and B. Bagchi, βSupersymmetry in atomic physics and the radial problem,β
*Journal of Physics. A. Mathematical and General*, vol. 20, no. 12, pp. 3825β3832, 1987. View at Publisher Β· View at Google Scholar - A. D. Alhaidari, βSolutions of the nonrelativistic wave equation with position-dependent effective mass,β
*Physical Review A*, vol. 66, no. 4, pp. 421161β421167, 2002. - O. Mustafa and M. Znojil, βPT-symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes,β
*Journal of Physics A: Mathematical and General*, vol. 35, no. 42, pp. 8929β8942, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet Β· View at Scopus - M. Znojil, β
*PT*-symmetric harmonic oscillators,β*Physics Letters. A*, vol. 259, no. 3-4, pp. 220β223, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - O. Mustafa, βEnergy-levels crossing and radial Dirac equation: supersymmetry and quasi-parity spectral signatures,β
*International Journal of Theoretical Physics*, vol. 47, no. 5, pp. 1300β1311, 2008. View at Publisher Β· View at Google Scholar - O. Mustafa and S. H. Mazharimousavi, β
*d*-dimensional generalization of the point canonical transformation for a quantum particle with position-dependent mass,β*Journal of Physics. A. Mathematical and General*, vol. 39, no. 33, pp. 10537β10547, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. D. Alhaidari, H. Bahlouli, and A. Al-Hasan, βDirac and Klein-Gordon equations with equal scalar and vector potentials,β
*Physics Letters. A*, vol. 349, no. 1–4, pp. 87β97, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. Akcay, βDirac equation with scalar and vector quadratic potentials and Coulomb-like tensor potential,β
*Physics Letters. A*, vol. 373, no. 6, pp. 616β620, 2009. View at Publisher Β· View at Google Scholar - S. Zarrinkamar, A. A. Rajabi, and H. Hassanabadi, βDirac equation for the harmonic scalar and vector potentials and linear plus coulomb-like tensor potential; The SUSY approach,β
*Annals of Physics*, vol. 325, no. 11, pp. 2522β2528, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus - A. Lahiri, P. K. Roy, and B. Bagchi, βSupersymmetry and the three-dimensional isotropic oscillator problem,β
*Journal of Physics A: Mathematical and General*, vol. 20, no. 15, article 052, pp. 5403β5404, 1987. View at Publisher Β· View at Google Scholar Β· View at Scopus - S.-H. Dong, βSchrödinger equation with the potential $V(r)=A{r}^{-4}+B{r}^{-3}+C{r}^{-2}+D{r}^{-1}$,β
*Physica Scripta*, vol. 64, no. 4, pp. 273β276, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet