Research Article | Open Access

# The Nonrelativistic Scattering States of the Deng-Fan Potential

**Academic Editor:**Kadayam S. Viswanathan

#### Abstract

The approximately analytical scattering state solution of the Schrodinger equation is obtained for the Deng-Fan potential by using an approximation scheme to the centrifugal term. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated. We consider and verify two special cases: the and the -wave Hulthén potential.

#### 1. Introduction

One of the interesting problems in quantum mechanics is to investigate the energy spectra and the wavefunctions of a quantum system under different potentials because one can obtain all the necessary information regarding the quantum system under consideration. In order to understand the studied quantum system completely, we should study the bound states and the scattering states for a given quantum system. Among various potential models, we consider the Deng-Fan potential. In 1957, Deng and Fan [1] introduced Deng-Fan potential in an attempt to find a more suitable diatomic potential to describe the vibrational spectrum. The Deng-Fan potential is a molecular potential, and it is qualitatively similar to the Morse potential. In the description of the motion of nucleons this potential is applicable. As the forthcoming (2) reveals, this potential, in some ranges of the potential perimeter, is very similar to the Kratzer potential. The Deng-Fan potential is consistent with quantum requirements and can be a suitable choice to study physical systems besides the Coulomb or linear terms. This potential has been investigated by some authors under different wave equations of quantum mechanics [2–5]. In [3], arbitrary state solutions of the Schrödinger equation with the Deng-Fan molecular potential is reported. Dong in [2] obtained the energy spectra of the Klein-Gordon equation under the equal scalar and vector potentials by using a proper approximation to the centrifugal term. To deal with the wave equations in quantum mechanics such as Klein-Gordon [6], Dirac [7], Duffin-Kemmer-Petiau [8], and Schrodinger [3] equations different methods have been used; these methods include the supersymmetry (SUSY) method [9], Nikiforov-Uvarov method [10], the quantization rules [11], series expansion [12], and ansatz method [13].

Here, we report solutions of the scattering states of Schrodinger equation for the Deng-Fan potential for any states. The reasons for which we write this paper are as follows. Firstly, we have not yet found the scattering states related to this potential. Secondly, theoretical prediction of many properties of diatomic molecules requires the knowledge of the radial wavefunctions of scattering states and the phase shifts. In [14] the authors have obtained the properties of scattering state solutions of the Klein-Gordon equation for a Coulomb like scalar plus vector potentials. The analytical scattering state solutions of the -wave Schrodinger equation for the Eckart potential is given in [15]. The scattering states of Schrodinger and the Klein-Gordon equations under different potentials are reported in [16–19].

The rest of this paper is organized as follow. In Section 2, we obtain the solutions of scattering states. The normalized radial wave functions of scattering states and the calculation formula of phase shifts are presented. In Section 3, we consider and verify two special cases: the and the -wave Hulthén potential. And finally, our conclusion is given in Section 4.

#### 2. Scattering States of the Arbitrary -Wave Schrodinger Equation

The radial Schrodinger equation has the form where and are the reduced mass and the Planck’s constant, respectively. Here, we consider the Deng-Fan potential where denotes the hyper radius and , and are constant coefficients. Substitution of (2) into (1) gives To get rid of the centrifugal term, we make use of the elegant approximation [2]

Equation (3) changes into where Introducing brings (5) as To proceed on, we choose Substitution of (8) into (7) leads to Equation (9) can be written as Equation (10) is the hypergeometric equation, and its solution is the hypergeometric function, so we have where Therefore the total wavefunction of the system is Or equivalently () To obtain a finite solution, or must be a negative integer. This gives the following equality: where the energy eigenvalue equation can be found from the previous equation.

Here, to obtain the normalized constant and phase shifts we recall the following properties of hypergeometric function:By using (16a) and (16b) for can be written as By inserting the following relations in the previous equation We obtain By taking and inserting in (19) we arrive at Therefore, we have the asymptotic form of the formula (14) for By comparing (21) with the boundary condition [20] phase shifts and the normalized constant can be given by

#### 3. Discussion

In this section we study two special cases. First, we discuss the special case . In this case, we have Therefore, the results which is obtained in the previous section are reduced to the those of the exact solutions of -wave scattering state for the Deng-Fan potential as follows: Second, we consider the case of in which the Deng-Fang potential reduces to the Hulthen potential. In the case of , we have The results given in (22) are reduced to which are same as the -wave scattering state for the Hulthén potential [20] if we choose , .

#### 4. Conclusion

Due to the application of the Deng-Fan potential for theoretical physicists especially for molecular system, we have discussed the approximate bound and scattering state solutions of the Schrodinger equation for the Deng-Fan potential. We have obtained the energy eigenvalues, normalized wave functions, and scattering phase shifts by using an approximation for the centrifugal term. We should mention that from the relation of the scattering phase shifts and the general theory of the partial-wave method one can obtain the scattering amplitude. Also, we have studied two special cases for and the -wave of the Hulthén potential which is a special case of the Deng-Fan potential. Results are useful in quantum mechanics and particle physics.

#### Acknowledgment

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

#### References

- Z. H. Deng and Y. P. Fan, “A potential function of diatomic molecules,”
*Shandong University Journal*, vol. 7, article 162, 1957. 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: Google Scholar - S. H. Dong and X. Y. Gu, “Arbitrary l state solutions of the Schrِdinger equation with the Deng-Fan molecular potential,”
*Journal of Physics Conference Series*, vol. 96, no. 1, Article ID 012109, 2008. View at: Publisher Site | Google Scholar - H. Hassanabadi, B. H. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. J. Oluwadare, K. J. Oyewumi, and O. A. Babalola, “Exact S-Wave Solution of the Klein-Gordon Equation with the Deng-Fan Molecular Potential using the Nikiforov-Uvarov (NU) Method,”
*The African Review of Physics*, vol. 7, article 0016, 2012. View at: Google Scholar - L. L. Lu, B. H. Yazarloo, S. Zarrinkamar, G. Liu, and H. Hassanabadi, “Calculation of the oscillator strength for the Klein-Gordon equation with Tietz potential,”
*Few-Body Systems*, vol. 53, pp. 573–581, 2012. View at: Google Scholar - G. F. Wei and S. H. Dong, “A novel algebraic approach to spin symmetry for Dirac equation with scalar and vector second Pöschl-Teller potentials,”
*European Physical Journal A*, vol. 43, pp. 185–190, 2010. View at: Google Scholar - S. Hassanabadi, A. A. Rajabi, B. H. Yazarloo, S. Zarrinkamar, and H. Hassanabadi, “Quasi-analytical solutions of DKP equation under the Deng-Fan interaction,”
*Advances in High Energy Physics*, vol. 2012, Article ID 804652, 13 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - H. Hassanabadi S. Zarrinkamar and B. H. Yazarloo, “Spectrum of a Hyperbolic Potential via SUSYQM within the Semi-Relativistic Formalism,”
*Chinese Journal of Physics*, vol. 50, no. 5, pp. 788–794, 2012. View at: Google Scholar - B. H. Yazarloo, H. Hassanabadi, and S. Zarrinkamar, “Oscillator strengths based on the Mِbius square potential under Schrِdinger equation,”
*The European Physical Journal Plus*, vol. 127, article 51, 2012. View at: Google Scholar - X.-Y. Gu and S.-H. Dong, “Energy spectrum of the Manning-Rosen potential including centrifugal term solved by exact and proper quantization rules,”
*Journal of Mathematical Chemistry*, vol. 49, no. 9, pp. 2053–2062, 2011. View at: Publisher Site | Google Scholar | MathSciNet - S.-H. Dong and G.-H. Sun, “The series solutions of the non-relativistic equation with the Morse potential,”
*Physics Letters A*, vol. 314, no. 4, pp. 261–266, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S.-H. Dong, “A new approach to the relativistic Schrödinger equation with central potential: ansatz method,”
*International Journal of Theoretical Physics*, vol. 40, no. 2, pp. 559–567, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-Y. Chen, D.-S. Sun, and F.-L. Lu, “Scattering states of the Klein-Gordon equation with Coulomb-like scalar plus vector potentials in arbitrary dimension,”
*Physics Letters A*, vol. 330, no. 6, pp. 424–428, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. F. Wei, W. C. Qiang, and W. L. Chen, “Approximate analytical solution of continuous states for the l-wave Schrِdinger equation with a diatomic molecule potential,”
*Central European Journal of Physics*, vol. 8, no. 4, pp. 574–579, 2010. View at: Google Scholar - A. Arda, O. Aydogdu, and R. Sever, “Scattering and bound state solutions of the asymmetric Hulthén potential,”
*Physica Scripta*, vol. 84, no. 2, pp. 25004–25009, 2011. View at: Google Scholar - C. Rojas and V. M. Villalba, “Scattering of a Klein-Gordon particle by a Woods-Saxon potential,”
*Physical Review A*, vol. 71, no. 5, Article ID 052101. View at: Google Scholar - A. Arda and R. Sever, “Effective-mass Klein-Gordon-Yukawa problem for bound and scattering states,”
*Journal of Mathematical Physics*, vol. 52, no. 9, 2011. View at: Publisher Site | Google Scholar | MathSciNet - G.-F. Wei, X.-Y. Liu, and W.-L. Chen, “The relativistic scattering states of the Hulthén potential with an improved new approximate scheme to the centrifugal term,”
*International Journal of Theoretical Physics*, vol. 48, no. 6, pp. 1649–1658, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G.-F. Wei, C.-Y. Long, and S.-H. Dong, “The scattering of the Manning-Rosen potential with centrifugal term,”
*Physics Letters A*, vol. 372, no. 15, pp. 2592–2596, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Bentol Hoda Yazarloo 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.