Advances in High Energy Physics

Volume 2014 (2014), Article ID 840907, 9 pages

http://dx.doi.org/10.1155/2014/840907

## Scattering and Bound States of Duffin-Kemmer-Petiau Particles for -Parameter Hyperbolic Pöschl-Teller Potential

Department of Physics, Mersin University, 33143 Mersin, Turkey

Received 1 March 2014; Revised 13 June 2014; Accepted 29 June 2014; Published 14 July 2014

Academic Editor: Frank Filthaut

Copyright © 2014 Hilmi Yanar 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. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The Duffin-Kemmer-Petiau (DKP) equation in the presence of a scalar potential is solved in one spatial dimension for the vector -parameter Hyperbolic Pöschl-Teller (HPT) potential. In obtaining complete solutions we used the weak interaction approach and took the scalar and vector potentials in a correlated form. By looking at the asymptotic behaviors of the solutions, we identify the bound and scattering states. We calculate transmission () and reflection () probability densities and analyze their dependence on the potential shape parameters. Also we investigate the dependence of energy eigenvalues of the bound states on the potential shape parameters.

#### 1. Introduction

In the early 20th century, many scientific studies have been done to explain the structure of nuclei, atoms, and molecules. In the explanation of these physical systems, particle equations describing fermions or bosons were solved for the potentials that vary depending on the physical systems. Also scattering and bound states were examined. Within this context, some potentials used in the previous studies are Morse potential [1], Rosen-Morse potential [2], Yukawa potential [3], Coulomb potential [4], Hylleraas potential [5], Manning-Rosen potential [6], Woods-Saxon potential [7], Hulthen potential [8], Eckart potential [9], Trigonometric Pöschl-Teller potential [10–12], Generalized Pöschl-Teller potential [13–15], and Hyperbolic Pöschl-Teller potential [10, 16–19]. Recently some of these potentials have been introduced in terms of hyperbolic functions whose hyperbolic parts have been extended by a parameter due to the suggestion of Arai [20]. Of these potentials the one called -parameter Hyperbolic Pöschl-Teller potential (HPT) is given by the following form: where is the deformation parameter and and is the height of the potential barrier and not equal to and . Also, is related to the range of the potential barrier. The graph of the potential against the position is displayed in Figure 1 for different values of the potential parameters. This potential has a feature of shape invariant as obtained from the supersymmetric quantum mechanics. By defining the deformation parameter energy levels and wavefunctions are altered and, depending on the values that parameter takes whether or , the number of energy levels varies. Although the usual PT-type potential provides limited knowledge in describing of atomic interactions, namely, at most two parameters, addition of the deformation parameter to the potential extends the applications of such kind of potential to the other fields of the physics. PT-type potentials are used for analyzing the bound energies of the -particle in hypernuclei in nuclear physics. Therefore, the HPT type potential can play an important role in describing the interactions not only in molecular and atomic physics but also in nuclear physics. Moreover, the -deformed hyperbolic potential is supposed to be used to characterize the curvature in spaces of negative constant curvature [16, 21, 22].

The Schrödinger equation has been solved in one dimension for this potential by using Nikiforov-Uvarov and Path integral methods and bound energy states have been obtained [22, 23]. Also the relativistic Klein-Gordon (spin-0) and Dirac (spin-) equations were examined in the existence of the position-dependent mass or scalar potential [24–27]. Usually, in one spatial dimension, position-dependent mass and scalar potential are selected to be equal to the vector potential (external potential). Therefore, the scalar potential can be considered as counterpart of the position-dependent mass which is useful in the cases of quantum dots [28] and electronic properties of semiconductors [29].

Another relativistic equation that defines massive spin-0 (scalar bosons) and spin-1 (vector bosons) particles is called DKP equation. Unlike the Klein-Gordon and Dirac equations, the DKP equation has much more components for wavefunction, namely, it has sixteen components, which makes it difficult to solve exactly. This difficulty makes it harder to study the DKP equation compared to the other relativistic equations.

In contrast to the Klein-Gordon and Dirac equations, the DKP equation has not been studied for the HPT potential. So, it is the goal of this study to obtain the bound and scattering states of the DKP equation in the presence of a scalar potential for the HPT potential in one dimension and analyze the transmission () and reflection () probability densities numerically.

The scheme of this paper is as follows. In Section 2, we solve the DKP equation for the HPT potential. In Section 3, the transmission () and reflection () probability densities are calculated by using the left and right solutions. In Section 4, the bound energy states are found for the HPT potential well. Finally, in Section 5 we discuss the obtained results.

#### 2. Solution of DKP Equation For HPT Potential

The DKP equation with a scalar potential [30] describing massive spin-0 and spin-1 particles is given by (in natural units ) where is the four-vector potential, is the scalar potential, and is the sixteen components wavefunction of the DKP equation. are the Kemmer matrices which obey the following commutation relation [31, 32]: where are the Dirac matrices. For -dimensional case, the Pauli spin matrices are used instead of the Dirac gamma matrices in (3). In this case the beta matrices become as follows [33]: With the insertion of matrices into (2), we obtain where the wavefunction is given by Equation (5) reduces to four coupled differential equations with the choice of , : After some simple algebra, we find a second order differential equation in the form The scalar potential we are dealing with has the form , where represents the strength of the weak interaction. It is very small compared with the mass of the particle. Considering this effect the term including the first order derivative can be ignored since Then (8) reduces to where , , , , and .

There are two reasons for choosing the scalar and vector potentials in a correlated form; they are as follows.(i)Mathematically, with this choice, (8) reduces to a solvable form. In the original form the second order differential equation has a fourth order singular point caused by the external vector HPT potential. Taking the vector and scalar potential in similar forms removes this difficulty and the resulting equation given in (10) is a solvable form.(ii)Another reason is that we have no information about the form of potential describing the weak interactions. Our approach to the problem is to take the scalar and vector potentials in a correlated form and to investigate whether we are able to get reasonable quantum mechanical parameters like and .

We solve (10) by defining different variables for and cases.

For case, by substituting the HPT potential given by (1) into (10) and defining the variable we obtain the following equation: where and .

Equation (12) has singularities at and . Therefore we may suggest the wavefunction in the following form to find the exact solution: By using this definition we obtain the following differential equation: where and . This is the hypergeometric differential equation and its solution is given by [34] In this case the left () side solutions are obtained in the following form: For case, we define the variable as follows: In this case (10) becomes By setting we obtain in the following differential equation: where and . In this case . The solution of this differential equation is and the right () side solutions are obtained in the following form: The other components of the wavefunction can be obtained by using (7) and the recurrence formula of the hypergeometric functions [34]:

#### 3. Calculation of Transmission and Reflection Probability Densities

Transmission () and Reflection () probability densities are defined, respectively, in the following form: where , , and are transmitted, reflected, and incident probability current densities, respectively. Probability of the current density for the DKP equation is given by where . Its explicit form is In order to calculate the transmission () and reflection () probability densities, we use asymptotic expressions of the wavefunctions.

The incoming wavefunction and its asymptotic behavior as , , are defined as follows, respectively: where we used The reflected wavefunction and its asymptotic behavior as , , are defined as follows, respectively: where we used The transmitted wavefunction and its asymptotic behavior as , , are defined as follows, respectively: where we take to obtain (31) because there is no the reflected wavefunction in the region of the potential and Transmission () and reflection () probability densities are found by substituting the obtained asymptotic expressions of wavefunctions into (26): In order to correlate the coefficients in (34), we use the continuity condition for the DKP equation that is given by By using (35), we obtain where the constants are given in Table 1.

#### 4. Bound Energy States

To discuss the bound states, the potential must be in the well form. Also, energy values must be smaller than the mass (). Therefore, the HPT potential given in (1) is rewritten in the following form: Accordingly, becomes where . In that case the solutions will be in the same form of the once obtained in Section 1, except the above definitions. We will apply the boundary conditions to the solution given in (16) as follows.(i)First boundary condition: as (), the wavefunction must be equal to zero: In order to satisfy the wavefunction to be continuous, should be zero. In this case (16) becomes where .(ii) Second boundary condition: as (), the wavefunction must vanish.

As (), the hypergeometric functions are defined as follows [34]: By using (40) and (41), we obtain the following one: where the factor depending on the variable of the second term of the right side goes to the infinity. Therefore, in order to satisfy this boundary condition, the argument of gamma functions in the denominator of second term should be equal to negative integers: where is finite and positive integer. By using one of these equations, the energy relation for bound states is obtained in the following form: We give Table 2 in order to evaluate numbers of the energy eigenvalues of the bound states for the HPT potential against several values of , , and . From Table 2 we see that numbers of energy eigenvalues depend on the HPT potential parameters which are , , and .

#### 5. Conclusion

The main purpose of this study is to determine the solutions of the DKP equation and to obtain the bound and scattering states of the DKP particles for the HPT potential which have not been done in previous studies. The DKP equation written for the HPT potential has fourth order singular points. Therefore the equation can not be solved. In order to solve it we make the weak interaction approach and take the scalar and vector potentials in a correlated form. Then, by using the condition that the wavefunction obtained as a solution must be finite as and by analyzing the gamma functions, we find the relation which gives energy eigenvalues for the bound states. As it is seen in Table 2, the number of the bound states () depends on the HPT potential parameters. For example, when evaluating variation of the energy against for , , it can be seen that the number of the bound states equals for . Similarly, when one evaluates variation of the energy according to for , , one can see that the number of the bound states equals for . If one analyzes variation of the energy versus which determines depth of the HPT potential for , , , one can find that the number of the bound states equals for . By adjusting the values of , , and the number of bound states can be increased or decreased.

The transmission and reflection probability densities are calculated by using asymptotic behavior of the wavefunction and their dependence on the potential shape parameters is analyzed numerically. The plot of energy against the transmission () and reflection () probability densities is given in Figure 2. We see from this plot that the unitarity condition () is satisfied. The effect of , , and parameters on the transmission probability density is represented in Figures 3, 4, and 5, respectively. From these figures, it is seen that the transmission probability density decreases as increases and it increases with raising and parameters. These findings are the expected results, since the increase in the also supports the height of the potential, which is shown in Figure 1. Therefore, the transmission probability density decreases. As it can be seen from Figure 1, unlike , the height of the HPT potential decreases due to the increase of and parameters. In this case, the transmission probability density increases.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors wish to thank the referee for suggested improvements to the paper. This work is partially supported by the Scientific Research Unit of Mersin University under Project no. BAP-FBE F (HY) 2011-1 YL.

#### References

- P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,”
*Physical Review*, vol. 34, no. 1, pp. 57–64, 1929. View at Publisher · View at Google Scholar · View at Scopus - N. Rosen and P. M. Morse, “On the vibrations of polyatomic molecules,”
*Physical Review*, vol. 42, no. 2, pp. 210–217, 1932. View at Publisher · View at Google Scholar · View at Scopus - 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 Zentralblatt MATH · View at Scopus - S. Flügge,
*Practical Quantum Mechanics*, Springer, Berlin, Germany, 2nd edition, 1994. View at MathSciNet - E. A. Hylleraas, “Energy formula and potential distribution of diatomic molecules,”
*The Journal of Chemical Physics*, vol. 3, no. 9, article 595, 1935. View at Publisher · View at Google Scholar · View at Scopus - M. F. Manning and N. Rosen, “Potential function of Diatomic Molecules,”
*Physical Review*, vol. 44, p. 953, 1933. View at Google Scholar - R. D. Woods and D. S. Saxon, “Diffuse surface optical model for nucleon-nuclei scattering,”
*Physical Review*, vol. 95, no. 2, pp. 577–578, 1954. View at Publisher · View at Google Scholar · View at Scopus - L. Hulthen, “Uber die Eigenlösungen der Schrödinger chung des Deutrons,”
*Arkiv för Matematik, Astronomi och Fysik*, vol. 28A, no. 5, 1942. View at Google Scholar - C. Eckart, “The penetration of a potential barrier by electrons,”
*Physical Review*, vol. 35, no. 11, article 1303, 1930. View at Publisher · View at Google Scholar · View at Scopus - G. Pöschl and E. Teller, “Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,”
*Zeitschrift für Physik*, vol. 83, no. 3-4, pp. 143–151, 1933. View at Google Scholar - X. Liu, G. Wei, X. Cao, and H. Bai, “Spin symmetry for Dirac equation with the trigonometric Pöschl-Teller potential,”
*International Journal of Theoretical Physics*, vol. 49, no. 2, pp. 343–348, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - B. J. Falaye and S. M. Ikhdair, “Relativistic symmetries with the trigonometric Pöschl—teller potential plus Coulomb-like tensor interaction,”
*Chinese Physics B*, vol. 22, no. 6, Article ID 060305, 2013. View at Publisher · View at Google Scholar · View at Scopus - T. Chen, Y. Diao, and C. Jia, “Bound state solutions of the Klein-Gordon equation with the generalized Pöschl-Teller potential,”
*Physica Scripta*, vol. 79, no. 6, Article ID 065014, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - C. Jia, T. Chen, and L. Cui, “Approximate analytical solutions of the Dirac equation with the generalized Pöschl-Teller potential including the pseudo-centrifugal term,”
*Physics Letters A*, vol. 373, no. 18-19, pp. 1621–1626, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G.-F. Wei and S.-H. Dong, “The spin symmetry for deformed generalized Pöschl-Teller potential,”
*Physics Letters A*, vol. 373, no. 29, pp. 2428–2431, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Chang-Yuan, L. Fa-Lin, and Y. Yuan, “Scattering states of modified Pöschl-Teller potential in
*D*-dimension,”*Chinese Physics B*, vol. 21, Article ID 030302, 2012. View at Google Scholar - D. Agboola, “Solutions to the modified Pöschl-teller potential in D-dimensions,”
*Chinese Physics Letters*, vol. 27, Article ID 040301, 2010. View at Publisher · View at Google Scholar - S. Dong and S. Dong, “An alternative approach to study the dynamical group for the modified Pöschl-Teller potential,”
*Czechoslovak Journal of Physics*, vol. 52, no. 6, pp. 753–764, 2002. View at Publisher · View at Google Scholar · View at Scopus - S. Cruz y Cruz, Ş. Kuru, and J. Negro, “Classical motion and coherent states for Pöschl-Teller potentials,”
*Physics Letters A*, vol. 372, no. 9, pp. 1391–1405, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Arai, “Exactly solvable supersymmetric quantum mechanics,”
*Journal of Mathematical Analysis and Applications*, vol. 158, no. 1, pp. 63–79, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Suparmi, C. Cari, and H. Yuliani, “Energy spectra and wave function analysis of q-deformed modified poschl-teller and hyperbolic scarf II potentials using NU method and a mapping method,”
*Advances in Physics Theories and Applications*, vol. 16, pp. 64–74, 2013. View at Google Scholar - C. Grosche, “Path integral solutions for deformed Pöschl–Teller-like and conditionally solvable potentials,”
*Journal of Physics A: Mathematical and General*, vol. 38, no. 13, pp. 2947–2958, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - S. Meyur and S. Debnath, “Schroedinger equation with Woods-Saxon plus Poeschl-Teller potential,”
*Bulgarian Journal of Physics*, vol. 36, no. 1, pp. 17–34, 2009. View at Google Scholar - K. J. Oyewumi, T. T. Ibrahim, S. Ajibola, and D. Ajadi, “Relativistic treatment of the spin-zero particles subject to the q-deformed hyperbolic modified Pöschl-Teller potential,”
*Journal of Vectorial Relativity*, vol. 5, pp. 19–26, 2010. View at Google Scholar - A. Arda, R. Sever, and C. Tezcan, “Analytical solutions to the Klein-Gordon equation with position-dependent mass for q-parameter poschl-teller potential,”
*Chinese Physics Letters*, vol. 27, Article ID 010306, 2010. View at Google Scholar - M. Eshghi and H. Mehraban, “Solution of the Dirac equation with position-dependent mass for q-parameter modified pöschl-teller and coulomb-like tensor potential,”
*Few-Body Systems*, vol. 52, no. 1-2, pp. 41–47, 2012. View at Publisher · View at Google Scholar · View at Scopus - X. Zhao, C. Jia, and Q. Yang, “Bound states of relativistic particles in the generalized symmetrical double-well potential,”
*Physics Letters A*, vol. 337, no. 3, pp. 189–196, 2005. View at Publisher · View at Google Scholar · View at Scopus - L. Serra and E. Lipparini, “Spin response of unpolarized quantum dots,”
*Europhysics Letters*, vol. 40, no. 6, pp. 667–672, 1997. View at Publisher · View at Google Scholar · View at Scopus - G. Bastard,
*Wave Mechanics Applied to Semiconductor Hetrostructures*, Les Editions de Physique, Les Ulis, France, 1992. - 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 · View at Scopus - N. Unal, “Kernel of the classical zitterbewegung,”
*Foundations of Physics*, vol. 27, no. 5, pp. 747–758, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - N. Ünal, “Path integral quantization of a spinning particle,”
*Foundations of Physics*, vol. 28, no. 5, pp. 755–762, 1998, 10.1023/A:1018897719975. View at Google Scholar - N. Unal, “Duffin-Kemmer-Petiau equation, proca equation and Maxwells equation in 1+1 D,”
*Concepts of Physics*, vol. 2, pp. 273–282, 2005. View at Google Scholar - R. P. Boas,
*Special Functions of Mathematical Physics*, Birkhauser, Basel, Switzerland, 1988.