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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
Research Article

Approximate Solutions of Klein-Gordon Equation with Kratzer Potential

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 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) [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 isβˆ’π‘‘2𝑅𝑛,𝑙(π‘Ÿ)π‘‘π‘Ÿ2βˆ’π‘‘βˆ’1π‘Ÿπ‘…π‘›,𝑙(π‘Ÿ)+ξ‚Έπ‘‘π‘Ÿπ‘™(𝑙+π·βˆ’2)π‘Ÿ2+(π‘š(π‘Ÿ)+𝑆(π‘Ÿ))2βˆ’ξ€·πΈπ‘›,π‘™ξ€Έβˆ’π‘‰(π‘Ÿ)2𝑅𝑛,𝑙(π‘Ÿ)=0.(2.1) For the scalar and vector potentials we choose 𝑉𝑉(π‘Ÿ)=0π‘Ÿ+𝑉1π‘Ÿ2𝑆,𝑆(π‘Ÿ)=0π‘Ÿ+𝑆1π‘Ÿ2,(2.2) where π‘Ÿ denotes the hyperradius and 𝑉0, 𝑆0, 𝑉1, and 𝑆1 are constant coefficients. For the mass, instead of constant one, we consider a position-dependent mass of the form π‘š(π‘Ÿ)=π‘š0+π‘š1π‘Ÿ.(2.3) The transformation 𝑅𝑛,𝑙(π‘Ÿ)=π‘Ÿβˆ’(π·βˆ’1)/2π‘ˆπ‘›,𝑙(π‘Ÿ), after inserting (2.2) brings (2.1) into the form𝑑2π‘‘π‘Ÿ2+βˆ’2𝐸𝑛,𝑙𝑉0+2π‘š0π‘š1βˆ’2π‘š0𝑆0π‘Ÿ+𝑉20βˆ’2𝐸𝑛,𝑙𝑉1βˆ’π‘š21βˆ’π‘†20βˆ’2π‘š0𝑆1βˆ’2π‘š1𝑆0βˆ’(𝐷+2π‘™βˆ’1)(𝐷+2π‘™βˆ’3)/4π‘Ÿ2+2𝑉0𝑉1βˆ’2𝑆0𝑆1βˆ’2π‘š1𝑆1π‘Ÿ3+𝑉21βˆ’π‘†21π‘Ÿ4+𝐸2𝑛,π‘™βˆ’π‘š20ξ€Έξƒ°π‘ˆπ‘›,𝑙(π‘Ÿ)=0.(2.4) Choosing 𝐹=2𝐸𝑛,𝑙𝑉0βˆ’2π‘š0π‘š1+2π‘š0𝑆0,𝐢=βˆ’π‘‰20+2𝐸𝑛,𝑙𝑉1+π‘š21+𝑆20+2π‘š0𝑆1+2π‘š1𝑆0+14(𝐷+2π‘™βˆ’1)(𝐷+2π‘™βˆ’3),𝐡=βˆ’2𝑉0𝑉1+2𝑆0𝑆1+2π‘š1𝑆1,𝐴=βˆ’π‘‰21+𝑆21,πœ€π‘›,𝑙=𝐸2𝑛,π‘™βˆ’π‘š20ξ€Έ.(2.5) Equation (2.4) is more neatly written as𝑑2π‘‘π‘Ÿ2βˆ’πΉπ‘Ÿβˆ’πΆπ‘Ÿ2βˆ’π΅π‘Ÿ3βˆ’π΄π‘Ÿ4+πœ€π‘›,π‘™ξ‚Όπ‘ˆπ‘›,𝑙(π‘Ÿ)=0.(2.6) The SchrΓΆdinger analogue of this problem has been analyzed by Dong [25]. We choose [25]π‘ˆπ‘›,𝑙(π‘Ÿ)=β„Žπ‘›ξ€Ίπ‘˜(π‘Ÿ)exp𝑙(π‘Ÿ),(2.7) whereβ„Žπ‘›βŽ§βŽͺ⎨βŽͺ⎩(π‘Ÿ)=1𝑛=0𝑛𝑖=1ξ€·π‘Ÿβˆ’π›Ώπ‘›π‘–ξ€Έπ‘˜π‘›β‰₯1,(2.8)π‘™π‘Ž(π‘Ÿ)=π‘Ÿ+π‘π‘Ÿ+𝑐lnπ‘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, givesβˆšπ‘Ž=βˆ’βˆšπ΄,𝑏=βˆ’βˆ’πœ€0,𝑙,2π‘Ž(1βˆ’π‘)=𝐡,βˆ’π‘+𝑐2βˆ’2π‘Žπ‘=𝐢,2𝑏𝑐=𝐹.(2.10) From (2.5) and (2.10), the energy of the nodeless state is obtained as𝐸20,π‘™βˆ’π‘š20ξ€Έ1=βˆ’ξ‚ƒβˆš16𝐴𝐢±𝐢2ξ‚„βˆ’2𝐡𝐹,(2.11) with its corresponding eigenfunction being obtained by substitution of (2.8), (2.9), and (2.10) in (2.7) asπ‘ˆ0,𝑙(π‘Ÿ)=𝑁0,π‘™π‘Ÿβˆšβˆ’πΉ/2βˆ’πœ€0,π‘™ξƒ¬βˆ’βˆšexpπ΄π‘Ÿβˆ’βˆšβˆ’πœ€0,π‘™π‘Ÿξƒ­.(2.12) In Table 1, we have reported the eigenvalues for 𝐷s and 𝑙s. Repeating the same procedure for the first node, the eigenvalues are found as𝐸21,π‘™βˆ’π‘š20ξ€ΈΒ±βŽ›βŽœβŽœβŽœβŽξ‚ƒπΆ=βˆ’πΆΒ±2ξ‚€π›Ώβˆ’4𝐹1(1)+βˆšπ΄ξ‚€βˆšξ‚ξ‚€1+𝐡/4𝐴1/24𝛿1(1)+βˆšπ΄ξ‚βŽžβŽŸβŽŸβŽŸβŽ 2,(2.13) where𝛿11=βˆšξ‚€ξ‚€π΅/4𝐴+2π·ξƒ―ξƒ©π΅πΆβˆ’1+√𝐡4𝐴ξƒͺ2+βˆšβˆ’βˆš4𝐴ξƒͺ𝐴.(2.14) And the corresponding eigenfunction isπ‘ˆ1,𝑙(π‘Ÿ)=𝑁1,π‘™ξ‚€π‘Ÿβˆ’π›Ώ1(1)ξ‚π‘Ÿπ‘ξ‚ƒπ‘Žexpπ‘Ÿξ‚„+π‘π‘Ÿ.(2.15) 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‘š0s.
tab2
Table 2: Energy for 𝑙=1 and various 𝐷s.
458087.fig.001
Figure 1: Energy versus dimension for 𝑙=1.
458087.fig.002
Figure 2: Energy versus dimension for various π‘š1s.

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.

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