Advances in High Energy Physics

Volume 2012 (2012), Article ID 804652, 13 pages

http://dx.doi.org/10.1155/2012/804652

## Quasi-Analytical Solutions of DKP Equation under the Deng-Fan Interaction

Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran

Received 4 April 2012; Revised 6 June 2012; Accepted 8 June 2012

Academic Editor: S. H. Dong

Copyright © 2012 S. 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

Quasianalytical solutions of Duffin-Kemmer-Petiau equation under scalar and vector Deng-Fan potentials are reported via a novel ansatz.

#### 1. Introduction

A yet open challenge in the annals of wave mechanics is the equivalence or nonequivalence of Duffin-Kemmer-Petiau (DKP) and Klein-Gordon (KG) equations. While the latter solely describes relativistic spin-zero bosons, the former can investigate spin-one particles as well. This equation was introduced 1930s in search for a linear relativistic wave equation similar to Dirac equation for relativistic bosons [1–4]. In its present format, the DKP equation is normally represented in two five and ten-dimensional versions that respectively work for spin-zero and spin-one bosons. At the present, to our best knowledge, the equivalence of KG and spin-zero DKP equations is doubted [5–14]. This is also true for the other counterpart of DKP equation, that is the Proca equation which is a relativistic framework to study spin-one bosons. There are papers that investigate related problems via these equations and report motivating data [15–20]. Here, our focus is on the spin-zero version of the equation, which has many applications [21–25].

In solving wave equations of mechanics, various analytical techniques including supersymmetry quantum mechanics (SUSYQM), Nikiforov-Uvarov (NU), quantization rule, Lie algebras, WKB approximation, point canonical transformation (PCT), and series expansion have been applied [26–40]. Nevertheless, we face situations in which these methodologies do not work very well. A very successful approach in such cases is proposing a physical ansatz solution that is on the one hand consistent with the requirements of quantum mechanics and on the other hand satisfies the equation. This is in many cases a definitely cumbersome, and in some cases an impossible task. Here we follow this rich quasianalytical technique to solve the DKP equation under the Deng-Fan potential [41]. There are, however, a few papers which discuss this attractive potential. This is partially due to the complex structure of this interaction. As far as we know, the published works include references [42–46] that consider the potential under either relativistic or nonrelativistic equations. The structure of the present paper is as follows. In Section 2, we briefly introduce the DKP equation and a Pekeris-type approximation. In the next part, we introduce some transformations as well as an attractive ansatz by which the problem is solved in a quasianalytical manner.

#### 2. The DKP Equation

The DKP Hamiltonian for scalar and vector interactions is where and the upper and lower components, respectively, are

The engaged matrices are with , and , respectively, being zero matrices

The equation, in (3+0)-dimensions, is written as [1–4] where . In (2.4a) and (2.4b), is a simultaneous eigenfunction of and , that is, and the general solution is considered as where spherical harmonics are of order , are the normalized vector spherical harmonics, and , and represent the radial wavefunctions. The above equations yield the following coupled differential equations [30–38]: which give [39–41] Where and . When , we recover the well-known formula [30–32]

We consider the Deng-Fan vector and scalar potentials [42–46] By a change of variable of the form

(2.9) is written as where

From (2.11), (2.13) can be written as

Here, we use the following approximations for the centrifugal term [47]:

Equation (2.16a) is a quite logical alternative for (see Figure 1), and (2.16a) and (2.16b) brings (2.15) into the form

By introducing and making the transformation , we obtain which after decomposition of fractions gives with

Let us now consider an ansatz of the form with Substitution of the latter in (2.20) yields

For the sake of simplicity, here we consider only the nodless solutions. Substitution of the proposed ansatz solution and equating the corresponding powers on both sides give For the fixed values of , in particular, the system of ten equations (2.24) determines the sets of variables , and . Therefore, the spectrum and eigenfunctions of the system are easily obtained for a particular system. For the higher states, the mathematical process is more cumbersome and complicated but can be followed by the same token we did here, that is, by choosing for the first node, for the second node, and so forth.

#### 3. Conclusion

Motivation behind our study was the high number of spin-zero relativistic systems that we frequently face as well as the attractive structure of the Deng-Fan potential. The corresponding ordinary differential equation was too complicated to be solved by common analytical techniques. We, therefore, performed some novel transformations and applied an acceptable approximation to the centrifugal term. We next proposed an interesting physical ansatz solution by which we were able to find a quasi-analytical solution. Our results are particularly useful in particle and nuclear physics and can be directly used after prerequisite fits performed.

#### Acknowledgment

It is a great pleasure for the authors to thank the referee for his careful comments on the paper.

#### References

- N. Kemmer, “Quantum theory of Einstein-Bose particles and nuclear interaction,”
*Proceedings of the Royal Society A*, vol. 166, no. 924, pp. 127–153, 1938. View at Publisher · View at Google Scholar - R. J. Duffin, “On the characteristic matrices of covariant systems,”
*Physical Review*, vol. 54, no. 12, article 1114, 1938. View at Publisher · View at Google Scholar · View at Scopus - N. Kemmer, “The particle aspect of Meson theory,”
*Proceedings of the Royal Society A*, vol. 173, no. 952, pp. 91–116, 1939. View at Publisher · View at Google Scholar - G. Petiau,
*[Ph.D. thesis]*, University of Paris, Académie Royale de Belgique. Classe des. Sciences. Mémoires. Collection in-8o 16, no. 2, 1936. - T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Effects Due to a scalar coupling on the particle-antiparticle production in the Duffin-Kemmer-Petiau theory,”
*The International Journal of Theoretical Physics*, vol. 49, no. 1, pp. 10–17, 2010. View at Publisher · View at Google Scholar - H. Hassanabadi, B. 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, 4 pages, 2011. View at Publisher · View at Google Scholar - T. R. Cardoso, L. B. Castro, and A. S. de Castro, “Inconsistencies of a purported probability current in the Duffin-Kemmer-Petiau theory,”
*Physics Letters A*, vol. 372, no. 38, pp. 5964–5967, 2008. View at Publisher · View at Google Scholar · View at Scopus - L. Chetouani, M. Merad, T. Boudjedaa, and A. Lecheheb, “Solution of Duffin-Kemmer-Petiau equation for the step potential,”
*The International Journal of Theoretical Physics*, vol. 43, no. 4, pp. 1147–1159, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. Oudi S. Hassanabadi, A.A. Rajabi, and H. Hasanabadi, “Approximate bound state solutions of DKP equation for any J state in the presence of Woods-Saxon potential,”
*Communications in Theoretical Physics*, vol. 57, no. 1, pp. 15–18, 2012. View at Publisher · View at Google Scholar - A. S. de Castro, “Bound states of the Duffin-Kemmer-Petiau equation with a mixed minimal-nonminimal vector cusp potential,”
*Journal of Physics A*, vol. 44, no. 3, Article ID 035201, 2011. View at Publisher · View at Google Scholar - M. Nowakowski, “The electromagnetic coupling in Kemmer-Duffin-Petiau theory,”
*Physics Letters A*, vol. 244, no. 5, pp. 329–337, 1998. View at Publisher · View at Google Scholar - J. T. Lunardi, B. M. Pimentel, R. G. Teixeira, and J. S. Valverde, “Remarks on Duffin-Kemmer-Petiau theory and gauge invariance,”
*Physics Letters A*, vol. 268, no. 3, pp. 165–173, 2000. View at Publisher · View at Google Scholar · View at Scopus - M. Riedel,
*Relativistische Gleichungen Fuer Spin-1-Teilchen*, Diplomarbeit, Institute for Theoretical Physics, Johann Wolfgang Goethe-University, Frankfurt, Germany, 1979. - E. Fischbach, M. M. Nieto, and C. K. Scott, “Duffin-Kemmer-Petiau subalgebras: representations and applications,”
*Journal of Mathematical Physics*, vol. 14, no. 12, pp. 1760–1774, 1973. View at Google Scholar · View at Scopus - B. C. Clark, S. Hama, and G. R. Kalbermann, “Relativistic impulse approximation for Meson-nucleus scattering in the Kemmer-Duffin-Petiau formalism,”
*Physics Review Letters*, vol. 55, no. 6, pp. 592–595, 1985. View at Publisher · View at Google Scholar - G. Kalbermann, “Kemmer-Duffin-Petiau equation approach to pionic atoms,”
*Physical Review C*, vol. 34, no. 6, pp. 2240–2243, 1986. View at Publisher · View at Google Scholar - R. E. Kozack, B. C. Clark, S. Hama, and V. K. Mishra, “Relativistic deuteron-nucleus scattering in the Kemmer-Duffin-Petiau formalism,”
*Physical Review C*, vol. 37, no. 6, pp. 2898–2901, 1988. View at Publisher · View at Google Scholar - R. E. Kozack, “Spin-one Kemmer-Duffin-Petiau equations and intermediate-energy deuteron-nucleus scattering,”
*Physical Review C*, vol. 40, no. 5, pp. 2181–2194, 1989. View at Publisher · View at Google Scholar - V. K. Mishra, S. Hama, and B. C. Clark, “Implications of various spin-one relativistic wave equations for intermediate-energy deuteron-nucleus scattering,”
*Physical Review C*, vol. 43, no. 2, pp. 801–811, 1991. View at Publisher · View at Google Scholar - B. C. Clark, R.J. Furnstahl, L. K. Kerr, J. Rusnak, and S. Hama, “Pion-nucleus scattering at medium energies with densities from chiral effective field theories,”
*Physics Letters B*, vol. 427, no. 3-4, pp. 231–234, 1998. View at Publisher · View at Google Scholar - V. Gribov, “QCD at large and short distances (annotated version),”
*The European Physical Journal C*, vol. 10, no. 1, pp. 71–90, 1999. View at Publisher · View at Google Scholar - I. V. Kanatchikov, “On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory,”
*Reports on Mathematical Physics*, vol. 46, no. 1-2, pp. 107–112, 2000. View at Publisher · View at Google Scholar - H. Hassanabadi, S. F. Forouhandeh, H. Rahimov, S. Zarrinkamar, and B. H. Yazarloo, “Duffin-Kemmer-Petiau equation under a scalar and vector Hulthen potential; an ansatz solution to the corresponding Heun equation,”
*Canadian Journal of Physics*, vol. 90, no. 3, pp. 299–304, 2012. View at Publisher · View at Google Scholar - J. T. Lunardi, L.A. Manzoni, B.M. Pimentel, and J.S. Valverde, “Duffin-Kemmer-Petiau theory in the causal approach,”
*International Journal of Modern Physics A*, vol. 17, no. 2, pp. 205–227, 2000. View at Publisher · View at Google Scholar - M. de Montigny, F. C. Khanna, A. E. Santana, E. S. Santos, and J. D. M. Vianna, “Galilean covariance and the Duffin-Kemmer-Petiau equation,”
*Journal of Physics A*, vol. 33, no. 31, pp. L273–L278, 2000. View at Publisher · View at Google Scholar · View at Scopus - S. Zarrinkamar, A. A. Rajabi, and H. Hassanabadi, “Dirac 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 · View at Google Scholar · View at Scopus - H. Hassanabadi, H. Rahimov, and S. Zarrinkamar, “Cornell and Coulomb interactions for the D-dimensional Klein-Gordon equation,”
*Annalen der Physik*, vol. 523, no. 7, pp. 566–575, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. D. Alhaidari, H. Bahlouli, and A. Al-Hasan, “Dirac and Klein-Gordon equations with equal scalar and vector potentials,”
*Physics Letters A*, vol. 349, no. 1–4, pp. 87–97, 2008. View at Publisher · View at Google Scholar - R. L. Hall, N. Saad, and K. D. Sen, “Soft-core Coulomb potentials and Heun's differential equation,”
*Journal of Mathematical Physics*, vol. 51, no. 2, Article ID 022107, 2010. View at Google Scholar - Y. Nedjadi and R. C. Barrett, “On the properties of the Duffin-Kemmer-Petiau equation,”
*Journal of Physics G*, vol. 19, no. 1, pp. 87–98, 1993. View at Publisher · View at Google Scholar - Y. Nedjadi, S. Ait-Tahar, and R. C. Barrett, “An extended relativistic quantum oscillator for $\text{S}=1$ particles,”
*Journal of Physics A*, vol. 31, no. 16, pp. 3867–3874, 1998. View at Publisher · View at Google Scholar · View at Scopus - A. Boumali, “On the eigensolutions of the one-dimensional Duffin-Kemmer-Petiau oscillator,”
*Journal of Mathematical Physics*, vol. 49, no. 2, Article ID 022302, 10 pages, 2008. View at Publisher · View at Google Scholar - I. Boztosun, M. Karakoc, and A. Durmus, “Asymptotic iteration method solutions to the relativistic Duffin-Kemmer-Petiau equation,”
*Journal of Mathematical Physics*, vol. 47, no. 6, Article ID 062301, 11 pages, 2006. View at Publisher · View at Google Scholar - B. Boutabia-Chéraitia and T. Boudjedaa, “Solution of DKP equation in Woods-Saxon potential,”
*Physics Letters A*, vol. 338, no. 2, pp. 97–107, 2005. View at Publisher · View at Google Scholar · View at Scopus - M. Merad, “DKP equation with smooth potential and position-dependent mass,”
*The International Journal of Theoretical Physics*, vol. 46, no. 8, pp. 2105–2118, 2007. View at Publisher · View at Google Scholar - Y. Chargui, A. Trabelsi, and L. Chetouani, “Bound-states of the ($1+1$)-dimensional DKP equation with a pseudoscalar linear plus Coulomb-like potential,”
*Physics Letters A*, vol. 374, no. 29, pp. 2907–2913, 2010. View at Publisher · View at Google Scholar · View at Scopus - K. Sogut and A. Havare, “Scattering of vector bosons by an asymmetric Hulthen potential,”
*Journal of Physics A*, vol. 43, no. 22, Article ID 225204, 14 pages, 2010. View at Publisher · View at Google Scholar - F. Yaşuk, C. Berkdemir, A. Berkdemir, and C. Önem, “Exact solutions of the Duffin-Kemmer-Petiau equation for the deformed hulthen potential,”
*Physica Scripta*, vol. 71, no. 4, pp. 340–343, 2005. View at Google Scholar · View at Scopus - A. Okninski, “Supersymmetric content of the Dirac and Duffin-Kemmer-Petiau equations,”
*The International Journal of Theoretical Physics*, vol. 50, no. 3, pp. 729–736, 2011. View at Publisher · View at Google Scholar - S. Zarrinkamar, A. A. Rajabi, H. Rahimov, and H. Hassanabadi, “DKP equation under a vector Hulthén-type potential: an approximate solution,”
*Modern Physics Letters A*, vol. 26, no. 22, pp. 1621–1629, 2011. View at Publisher · View at Google Scholar · View at Scopus - H. Deng and Y. P. Fan,
*Shandong University Journal*, vol. 7, article 162, 1957. - S. H. Dong and X. Y. Gu, “Arbitrary l state solutions of the Schrödinger equation with the Deng-Fan molecular potential,”
*Journal of Physics: Conference Series*, vol. 96, no. 1, Article ID 012109, 2008. View at Google Scholar - S. H. Dong, “Relativistic treatment of spinless particles subject to a rotating Deng-Fan oscillator,”
*Communications in Theoretical Physics*, vol. 55, no. 6, article 969, 2011. View at Publisher · View at Google Scholar - A. D. S. Mesa, C. Quesne, and Yu. F. Smirnov, “Generalized Morse potential: symmetry and satellite potentials,”
*Journal of Physics A*, vol. 31, no. 1, pp. 321–335, 1998. View at Publisher · View at Google Scholar - Z. M. Rong, H. G. Kjaergaard, and M. L. Sage, “Comparison of the Morse and Deng-Fan potentials for X-H bonds in small molecules,”
*Molecular Physics*, vol. 101, no. 14, pp. 2285–2294, 2003. View at Publisher · View at Google Scholar - H. Hassanabadi, B. Yazarloo, S. Zarrinkamar, and H. Rahimov, “Deng-Fan potential for relativistic spinless particles—an Ansatz solution,”
*Communications in Theoretical Physics*, vol. 57, no. 3, pp. 339–342, 2012. View at Publisher · View at Google Scholar - H. Hassanabadi, S. Zarrinkamar, and H. Rahimov, “Approximate solution of D-dimensional Klein-Gordon equation with Hulthén-type potential via SUSYQM,”
*Communications in Theoretical Physics*, vol. 56, no. 3, pp. 423–428, 2011. View at Publisher · View at Google Scholar