Research Article | Open Access
F. Pakdel, A. A. Rajabi, M. Hamzavi, "Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation", Advances in High Energy Physics, vol. 2014, Article ID 867483, 7 pages, 2014. https://doi.org/10.1155/2014/867483
Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation
In the presence of spin symmetry case, we obtain bound and scattering states solutions of the Dirac equation for the equal scalar and vector Yukawa potentials for any spin-orbit quantum number . The approximate analytical solutions are presented for the bound and scattering states and scattering phase shifts.
For studying the quantum mechanical systems, it is necessary to pay attention to two points. These two points are to study bound states to take the necessary information about the system under consideration and also solving scattering states for a system under the effect of a potential. Solving both of these problems gives us complete information about a quantum mechanical system under consideration.
The solutions of scattering and/or bound state problem have been investigated for the well-known potentials by applying different methods [1–10]. The analytical scattering state solution of the-wave Schrödinger Equation for the Eckart potential has been obtained in . The solution of the Schrödinger equation for the modified Morse potential has been studied by Wei and Chen . Rojas and Villalba have found the solutions of the Klein-Gordon equation for one-dimensional Wood-Saxon potential by hypergeometric functions . The exact solutions of scattering state have been studied for the-wave Schrödinger equation with the Manning-Rosen potential by using standard method . Low momentum scattering states of the Dirac equation have been studied in . Properties of scattering state solutions of the Klein-Gordon equation Coulomb scalar plus vector potential have been studied in .
In this work, we have studied bound state and scattering state of the Dirac equation with the Yukawa potential. The Dirac equation describes the particle dynamics in the relativistic quantum mechanics [17, 18]. Thus, solving the Dirac equation is very significant in describing the nuclear shell structure [19, 20]. Also the Yukawa potential has many applications in different areas of physics like high-energy physics  and atomic, molecular, and plasma physics .
This paper is organized as follows. In Section 2, we briefly introduce the Dirac equation with scalar and vector potential with any spin-orbit quantum number . The bound and scattering states of the Yukawa potential within the Dirac equation are presented in Section 3. Finally, our concluding remarks are given in Section 4.
2. Dirac Equation with Scalar and Vector Potential
The Dirac equation with scalar and vector potential ( and ) is  where is the relativistic energy of the system and is the three-dimensional momentum operator. and are the usual Dirac matrices given as where is theunitary matrix and the three Pauli matrices are given as where is the orbital angular momentum of the spherical nucleons and the total angular momentum operator and spin-orbit commute with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are and for unaligned spin and the aligned spin , respectively. Thus, in the Pauli-Dirac representation, where is the upper component and is the lower component of the Dirac spinors. and are spin and pseudospin spherical harmonics and is the projection of the angular momentum on the-axis. Substituting (4) into (1), one obtains two coupled differential equations for the upper and the lower radial wave functions as follows: where Solving (5) leads to a second-order Schrödinger-like differential equation for the upper and the lower components of the Dirac wavefunctions as follows: where and.
3. Dirac Equation with the Yukawa Potential
3.1. Bound State Solutions
For the bound state, by taking the following variable (forandfor) one can obtain (13) in the following form: Taking the form of the wave function and substituting this equation into (15), one gets a hypergeometric-type equation as follows : where Comparing (17) with the hypergeometric equation of the form  we can obtain the wavefunction as the hypergeometric function: where Then, with (14) and (16), we have the upper-spinor component of wavefunction for the Dirac equation with the Yukawa potential as follows: where is the normalization constant. With (9), the lower-spinor component can be obtained as Therefore, Finally, the spinor wave function under the condition of equal scalar and vector potentials with (4), (7b), (22), and (24) becomes By considering the finiteness of the solution, the quantum condition is given by It is the energy eigenvalue equation of bound state for the upper component of the Dirac equation with the Yukawa potential.
3.2. Scattering State
Now, we turn to solve (10) for scattering state. For this purpose, we use a new variable as follows: and obtain Considering the boundary condition of the scattering state, we take the following trial wavefunction: And inserting this equation into (28), we obtain the following equation: where This equation is a hypergeometric type. Thus, (29) is as follows: ( is the normalization constant) that is, the upper component of wavefunction. The lower component is And therefore, total wavefunction of scattering state with (4), (7b), (32), and (33) is where According to (26), we obtain the following form of energy eigenvalue equation for scattering states: Now, by finding the asymptotic form of (32) for large , we try to obtain the scattering phase shifts. For this purpose, we use the following property of the hypergeometric function: Therefore, we obtain By using this definition the upper component of wavefunction is as follows: Now, with (38) and (39) in the limitwe have Then The general boundary condition of the scattering state wave function on the “ scale” is Therefore, we have Thus, we obtain the scattering phase shifts as follows: Then we have the scattering phase shifts of the upper component of wavefunction as For the lower component, with (34), we have Then we have where Thus with (43), (44), (45), and (49), we have the following: And finally we have Therefore, the scattering phase shifts of two components are equal but in a constant coefficient.
We have studied the Dirac equation with the Yukawa potential and have obtained bound and scattering states of this problem. The energy eigenvalues, eigenstates, and scattering phase shifts have been presented. The numerical results of the energy eigenvalues of this work have been compared with the ones obtained in the literature in Table 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the anonymous referees for the valuable comments and suggestions.
- J. Y. Guo and X. Z. Fang, “Scattering of a Klein-Gordon particle by a Hulthén potential,” Canadian Journal of Physics, vol. 87, no. 9, pp. 1021–1024, 2009.
- W. C. Qiang and S. H. Dong, “Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term,” Physics Letters A, vol. 368, no. 1-2, pp. 13–17, 2007.
- G. F. Wei and S. H. Dong, “Spin symmetry in the relativistic symmetrical well potential including a proper approximation to the spin-orbit coupling term,” Physica Scripta, vol. 81, no. 3, Article ID 035009, 2010.
- G. F. Wei and S. H. Dong, “Algebraic approach to pseudospin symmetry for the Dirac equation with scalar and vector modified Pöschl-Teller potentials,” Europhysics Letters, vol. 87, no. 4, Article ID 40004, 2009.
- 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.
- G. F. Wei and S. H. Dong, “Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry,” Physics Letters A, vol. 373, no. 1, pp. 49–53, 2008.
- 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,” The European Physical Journal A, vol. 43, no. 2, pp. 185–190, 2010.
- G. F. Wei and S. H. Dong, “Pseudospin symmetry in the relativistic Manning-Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal term,” Physics Letters B, vol. 686, no. 4-5, pp. 288–292, 2010.
- G. F. Wei and S. H. Dong, “Pseudospin symmetry for modified Rosen-Morse potential including a Pekeris-type approximation to the pseudo-centrifugal term,” The European Physical Journal A, vol. 46, no. 2, pp. 207–212, 2010.
- V. M. Villalba and L. A. Gonzalez-Arraga, “Tunneling and transmission resonances of a Dirac particle by a double barrier,” Physica Scripta, vol. 81, no. 2, Article ID 025010, 2010.
- G. F. Wei, W. C. Qiang, and W. L. Chen, “Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential,” Central European Journal of Physics, vol. 8, no. 4, pp. 574–579, 2010.
- G. F. Wei and W. L. Chen, “Continuum states of modified Morse potential,” Chinese Physics B, vol. 19, no. 9, Article ID 090308, 2010.
- 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, p. 4, 2005.
- C. Y. Chen, F. L. Lu, and D. S. Sun, “Exact solutions of scattering states for the s-wave Schrödinger equation with the Manning-Rosen potential,” Physica Scripta, vol. 76, no. 5, pp. 428–430, 2007.
- N. Dombey and P. Kennedy, “Low momentum scattering in the Dirac equation,” Journal of Physics A, vol. 35, no. 31, pp. 6645–6657, 2002.
- 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.
- W. Greiner, Relativistic Quantum Mechanics, Wave Equations, Springer, Berlin, Germany, 3rd edition, 2000.
- L. H. Zhang, X. P. Li, and C. S. Jia, “Analytical approximation to the solution of the Dirac equation with the Eckart potential including the spin-orbit coupling term,” Physics Letters A, vol. 372, no. 13, pp. 2201–2207, 2008.
- C. S. Jia, P. Guo, and X. L. Peng, “Exact solution of the Dirac-Eckart problem with spin and pseudospin symmetry,” Journal of Physics A, vol. 39, no. 24, pp. 7737–7744, 2006.
- S. M. Ikhdair and R. Sever, “Exact solution of the Klein-Gordon equation for the PT-symmetric generalized Woods-Saxon potential by the Nikiforov-Uvarov method,” Annalen der Physik, vol. 16, no. 3, pp. 218–232, 2007.
- H. Yukawa, “The prediction and discovery of pions and muons,” Proceedings of the Physico-Mathematical Society of Japan, vol. 17, no. 3, p. 48, 1935.
- A. D. Alhaidari, H. Bahlouli, and M. S. Abdelmonem, “Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies,” Journal of Physics A, vol. 41, no. 3, Article ID 032001, p. 9, 2008.
- R. Sever and C. Tezcan, “Hypervirial solution for the generalized exponential cosine-screened Coulomb potential,” Physical Review A, vol. 41, no. 9, pp. 5250–5208, 1990.
- R. L. Green and C. Aldrich, “Variational wave functions for a screened Coulomb potential,” Physical Review A, vol. 14, no. 6, pp. 2363–2366, 1976.
- M. Hamzavi, M. Movahedi, K. E. Thylwe, and A. A. Rajabi, “Approximate analytical solution of the Yukawa potential with arbitrary angular momenta,” Chinese Physics Letters, vol. 29, no. 8, Article ID 080302, 2012.
- M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Department of Commerce, National Bureau of Standards, New York, NY, USA, 1965.
Copyright © 2014 F. Pakdel 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 SCOAP3.