About this Journal Submit a Manuscript Table of Contents 
Advances in High Energy Physics
VolumeΒ 2011Β (2011), Article IDΒ 458087, 6 pages
doi:10.1155/2011/458087
Research Article

Approximate Solutions of Klein-Gordon Equation with Kratzer Potential

H. Hassanabadi,1Β H. Rahimov,2Β and S. Zarrinkamar3

1Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran
2Computer Engineering Department, Shahrood University of Technology, Shahrood, Iran
3Department 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 𝑙 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 [1–7]. When we deal with this potential within the framework of Schrödinger 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) [8–10], the Nikiforov-Uvarov (NU) [11], point canonical transformation (PCT) [12–14], and so forth. Such investigations have been done by many authors in the annals of wave equations [15–24]. 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 βˆ’ 𝑑 2 𝑅 𝑛 , 𝑙 ( π‘Ÿ ) 𝑑 π‘Ÿ 2 βˆ’ 𝑑 βˆ’ 1 π‘Ÿ 𝑅 𝑛 , 𝑙 ( π‘Ÿ ) + ξ‚Έ 𝑑 π‘Ÿ 𝑙 ( 𝑙 + 𝐷 βˆ’ 2 ) π‘Ÿ 2 + ( π‘š ( π‘Ÿ ) + 𝑆 ( π‘Ÿ ) ) 2 βˆ’ ξ€· 𝐸 𝑛 , 𝑙 ξ€Έ βˆ’ 𝑉 ( π‘Ÿ ) 2 ξ‚Ή 𝑅 𝑛 , 𝑙 ( π‘Ÿ ) = 0 . ( 2 . 1 ) For the scalar and vector potentials we choose 𝑉 𝑉 ( π‘Ÿ ) = 0 π‘Ÿ + 𝑉 1 π‘Ÿ 2 𝑆 , 𝑆 ( π‘Ÿ ) = 0 π‘Ÿ + 𝑆 1 π‘Ÿ 2 , ( 2 . 2 ) where π‘Ÿ denotes the hyperradius and 𝑉 0 , 𝑆 0 , 𝑉 1 , and 𝑆 1 are constant coefficients. For the mass, instead of constant one, we consider a position-dependent mass of the form π‘š ( π‘Ÿ ) = π‘š 0 + π‘š 1 π‘Ÿ . ( 2 . 3 ) The transformation 𝑅 𝑛 , 𝑙 ( π‘Ÿ ) = π‘Ÿ βˆ’ ( 𝐷 βˆ’ 1 ) / 2 π‘ˆ 𝑛 , 𝑙 ( π‘Ÿ ) , after inserting (2.2) brings (2.1) into the form ξƒ― 𝑑 2 𝑑 π‘Ÿ 2 + βˆ’ 2 𝐸 𝑛 , 𝑙 𝑉 0 + 2 π‘š 0 π‘š 1 βˆ’ 2 π‘š 0 𝑆 0 π‘Ÿ + 𝑉 2 0 βˆ’ 2 𝐸 𝑛 , 𝑙 𝑉 1 βˆ’ π‘š 2 1 βˆ’ 𝑆 2 0 βˆ’ 2 π‘š 0 𝑆 1 βˆ’ 2 π‘š 1 𝑆 0 βˆ’ ( 𝐷 + 2 𝑙 βˆ’ 1 ) ( 𝐷 + 2 𝑙 βˆ’ 3 ) / 4 π‘Ÿ 2 + 2 𝑉 0 𝑉 1 βˆ’ 2 𝑆 0 𝑆 1 βˆ’ 2 π‘š 1 𝑆 1 π‘Ÿ 3 + 𝑉 2 1 βˆ’ 𝑆 2 1 π‘Ÿ 4 + ξ€· 𝐸 2 𝑛 , 𝑙 βˆ’ π‘š 2 0 ξ€Έ ξƒ° π‘ˆ 𝑛 , 𝑙 ( π‘Ÿ ) = 0 . ( 2 . 4 ) Choosing 𝐹 = 2 𝐸 𝑛 , 𝑙 𝑉 0 βˆ’ 2 π‘š 0 π‘š 1 + 2 π‘š 0 𝑆 0 , 𝐢 = βˆ’ 𝑉 2 0 + 2 𝐸 𝑛 , 𝑙 𝑉 1 + π‘š 2 1 + 𝑆 2 0 + 2 π‘š 0 𝑆 1 + 2 π‘š 1 𝑆 0 + 1 4 ( 𝐷 + 2 𝑙 βˆ’ 1 ) ( 𝐷 + 2 𝑙 βˆ’ 3 ) , 𝐡 = βˆ’ 2 𝑉 0 𝑉 1 + 2 𝑆 0 𝑆 1 + 2 π‘š 1 𝑆 1 , 𝐴 = βˆ’ 𝑉 2 1 + 𝑆 2 1 , πœ€ 𝑛 , 𝑙 = ξ€· 𝐸 2 𝑛 , 𝑙 βˆ’ π‘š 2 0 ξ€Έ . ( 2 . 5 ) Equation (2.4) is more neatly written as ξ‚» 𝑑 2 𝑑 π‘Ÿ 2 βˆ’ 𝐹 π‘Ÿ βˆ’ 𝐢 π‘Ÿ 2 βˆ’ 𝐡 π‘Ÿ 3 βˆ’ 𝐴 π‘Ÿ 4 + πœ€ 𝑛 , 𝑙 ξ‚Ό π‘ˆ 𝑛 , 𝑙 ( π‘Ÿ ) = 0 . ( 2 . 6 ) The Schrödinger analogue of this problem has been analyzed by Dong [25]. We choose [25] π‘ˆ 𝑛 , 𝑙 ( π‘Ÿ ) = β„Ž 𝑛 ξ€Ί π‘˜ ( π‘Ÿ ) e x p 𝑙 ξ€» ( π‘Ÿ ) , ( 2 . 7 ) where β„Ž 𝑛 ⎧ βŽͺ ⎨ βŽͺ ⎩ ( π‘Ÿ ) = 1 𝑛 = 0 𝑛  𝑖 = 1 ξ€· π‘Ÿ βˆ’ 𝛿 𝑛 𝑖 ξ€Έ π‘˜ 𝑛 β‰₯ 1 , ( 2 . 8 ) 𝑙 π‘Ž ( π‘Ÿ ) = π‘Ÿ + 𝑏 π‘Ÿ + 𝑐 l n π‘Ÿ , π‘Ž < 0 , 𝑏 < 0 . ( 2 . 9 ) Substitution of (2.9), (2.8), and (2.7) in (2.4), after equating the corresponding coefficients on both sides, gives √ π‘Ž = βˆ’ √ 𝐴 , 𝑏 = βˆ’ βˆ’ πœ€ 0 , 𝑙 , 2 π‘Ž ( 1 βˆ’ 𝑐 ) = 𝐡 , βˆ’ 𝑐 + 𝑐 2 βˆ’ 2 π‘Ž 𝑏 = 𝐢 , 2 𝑏 𝑐 = 𝐹 . ( 2 . 1 0 ) From (2.5) and (2.10), the energy of the nodeless state is obtained as ξ€· 𝐸 2 0 , 𝑙 βˆ’ π‘š 2 0 ξ€Έ 1 = βˆ’  √ 1 6 𝐴 𝐢 Β± 𝐢 2 ξ‚„ βˆ’ 2 𝐡 𝐹 , ( 2 . 1 1 ) with its corresponding eigenfunction being obtained by substitution of (2.8), (2.9), and (2.10) in (2.7) as π‘ˆ 0 , 𝑙 ( π‘Ÿ ) = 𝑁 0 , 𝑙 π‘Ÿ √ βˆ’ 𝐹 / 2 βˆ’ πœ€ 0 , 𝑙  βˆ’ √ e x p 𝐴 π‘Ÿ βˆ’ √ βˆ’ πœ€ 0 , 𝑙 π‘Ÿ ξƒ­ . ( 2 . 1 2 ) In Table 1, we have reported the eigenvalues for 𝐷 s and 𝑙 s. Repeating the same procedure for the first node, the eigenvalues are found as ξ€· 𝐸 2 1 , 𝑙 βˆ’ π‘š 2 0 ξ€Έ Β± βŽ› ⎜ ⎜ ⎜ ⎝  𝐢 = βˆ’ 𝐢 Β± 2 ξ‚€ 𝛿 βˆ’ 4 𝐹 1 ( 1 ) + √ 𝐴 ξ‚€ √  ξ‚€ 1 + 𝐡 / 4 𝐴   ξ‚„ 1 / 2 4 ξ‚€ 𝛿 1 ( 1 ) + √ 𝐴  ⎞ ⎟ ⎟ ⎟ ⎠ 2 , ( 2 . 1 3 ) where 𝛿 1 1 = √ ξ‚€ ξ‚€ 𝐡 /   4 𝐴 + 2 𝐷 ξƒ―  𝐡 𝐢 βˆ’ 1 + √ 𝐡 4 𝐴 ξƒͺ  2 + √ βˆ’ √ 4 𝐴 ξƒͺ ξƒ° 𝐴 . ( 2 . 1 4 ) And the corresponding eigenfunction is π‘ˆ 1 , 𝑙 ( π‘Ÿ ) = 𝑁 1 , 𝑙 ξ‚€ π‘Ÿ βˆ’ 𝛿 1 ( 1 )  π‘Ÿ 𝑐  π‘Ž e x p π‘Ÿ ξ‚„ + 𝑏 π‘Ÿ . ( 2 . 1 5 ) 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.

tab1
Table 1: Energy for various 𝐷 and π‘š 0 s.
tab2
Table 2: Energy for 𝑙 = 1 and various 𝐷 s.
458087.fig.001
Figure 1: Energy versus dimension for 𝑙 = 1 .
458087.fig.002
Figure 2: Energy versus dimension for various π‘š 1 s.

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

  1. W. C. Qiang, β€œBound states of the Klein-Gordon and Dirac equations for potential V(r)=Ar−2−Br−1,” Chinese Physics, vol. 12, no. 10, pp. 1054–1057, 2003. View at Publisher Β· View at Google Scholar Β· View at Scopus
  2. Y. F. Cheng and T. Q. Dai, β€œExact solution of the Schrödinger equation for the modified Kratzer potential plus a ring-shaped potential by the Nikiforov-Uvarov method,” Physica Scripta, vol. 75, no. 3, pp. 274–277, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  3. C. Berkdemir, A. Berkdemir, and J. Han, β€œBound state solutions of the Schrödinger equation for modified Kratzer's molecular potential,” Chemical Physics Letters, vol. 417, no. 4–6, pp. 326–329, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  4. S. M. Ikhdair and R. Sever, β€œExact solutions of the modified kratzer potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the nikiforovuvarov method,” International Journal of Modern Physics C, vol. 19, no. 2, pp. 221–235, 2008. View at Publisher Β· View at Google Scholar
  5. R. J. Leroy and R. B. Bernstein, β€œDissociation energy and long-range potential of diatomic molecules from vibrational spacings of higher levels,” The Journal of Chemical Physics, vol. 52, no. 8, pp. 3869–3879, 1970. View at Scopus
  6. C. L. Pekeris, β€œThe rotation-vibration coupling in diatomic molecules,” Physical Review, vol. 45, no. 2, pp. 98–103, 1934. View at Publisher Β· View at Google Scholar Β· View at Scopus
  7. C. Y. Chen and S. H. Dong, β€œExactly complete solutions of the Coulomb potential plus a new ring-shaped potential,” Physics Letters. A, vol. 335, no. 5-6, pp. 374–382, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. B. K. Bagchi, Supersymmetry in Quantum and Classical Mechanics, vol. 116, Chapman and Hall, Boca Raton, Fla, USA, 2001.
  9. F. Cooper, A. Khare, and U. Sukhatme, β€œSupersymmetry and quantum mechanics,” Physics Reports, vol. 251, no. 5-6, pp. 267–385, 1995. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  10. G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer, Berlin, Germany, 1996.
  11. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Basel, Germany, 1988.
  12. R. De, R. Dutt, and U. Sukhatme, β€œMapping of shape invariant potentials under point canonical transformations,” Journal of Physics A: Mathematical and General, vol. 25, no. 13, pp. L843–L850, 1992. View at Publisher Β· View at Google Scholar
  13. A. Gangopadhyaya, P. K. Panigrahi, and U. P. Sukhatme, β€œInter-relations of solvable potentials,” Helvetica Physica Acta, vol. 67, no. 4, pp. 363–368, 1994. View at Zentralblatt MATH
  14. A. Gangopadhyaya, J. V. Mallow, C. Rasinariu, and U. P. Sukhatme, Supersymmetric Quantum Mechanics: An Introduction.
  15. A. Lahiri, P. K. Roy, and B. Bagchi, β€œSupersymmetry in atomic physics and the radial problem,” Journal of Physics. A. Mathematical and General, vol. 20, no. 12, pp. 3825–3832, 1987. View at Publisher Β· View at Google Scholar
  16. A. D. Alhaidari, β€œSolutions of the nonrelativistic wave equation with position-dependent effective mass,” Physical Review A, vol. 66, no. 4, pp. 421161–421167, 2002.
  17. O. Mustafa and M. Znojil, β€œPT-symmetric pseudo-perturbation recipe: an imaginary cubic oscillator with spikes,” Journal of Physics A: Mathematical and General, vol. 35, no. 42, pp. 8929–8942, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet Β· View at Scopus
  18. M. Znojil, β€œPT-symmetric harmonic oscillators,” Physics Letters. A, vol. 259, no. 3-4, pp. 220–223, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  19. O. Mustafa, β€œEnergy-levels crossing and radial Dirac equation: supersymmetry and quasi-parity spectral signatures,” International Journal of Theoretical Physics, vol. 47, no. 5, pp. 1300–1311, 2008. View at Publisher Β· View at Google Scholar
  20. O. Mustafa and S. H. Mazharimousavi, β€œd-dimensional generalization of the point canonical transformation for a quantum particle with position-dependent mass,” Journal of Physics. A. Mathematical and General, vol. 39, no. 33, pp. 10537–10547, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  21. A. D. Alhaidari, H. Bahlouli, and A. Al-Hasan, β€œDirac and Klein-Gordon equations with equal scalar and vector potentials,” Physics Letters. A, vol. 349, no. 1–4, pp. 87–97, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  22. H. Akcay, β€œDirac equation with scalar and vector quadratic potentials and Coulomb-like tensor potential,” Physics Letters. A, vol. 373, no. 6, pp. 616–620, 2009. View at Publisher Β· View at Google Scholar
  23. S. Zarrinkamar, A. A. Rajabi, and H. Hassanabadi, β€œDirac equation for the harmonic scalar and vector potentials and linear plus coulomb-like tensor potential; The SUSY approach,” Annals of Physics, vol. 325, no. 11, pp. 2522–2528, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  24. A. Lahiri, P. K. Roy, and B. Bagchi, β€œSupersymmetry and the three-dimensional isotropic oscillator problem,” Journal of Physics A: Mathematical and General, vol. 20, no. 15, article 052, pp. 5403–5404, 1987. View at Publisher Β· View at Google Scholar Β· View at Scopus
  25. S.-H. Dong, β€œSchrödinger equation with the potential V(r)=Ar−4+Br−3+Cr−2+Dr−1,” Physica Scripta, vol. 64, no. 4, pp. 273–276, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet