Research Article  Open Access
M. Eshghi, M. Hamzavi, S. M. Ikhdair, "Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method", Advances in High Energy Physics, vol. 2012, Article ID 873619, 15 pages, 2012. https://doi.org/10.1155/2012/873619
Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method
Abstract
The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (pspin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spinorbit quantum number . Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatiallydependent mass on the bound states under the conditions of the spin symmetric limit and in the absence of tensor interaction .
1. Introduction
Dirac equation has become one of the most appealing relativistic wave equations for spin1/2 particles. However, solving such a wave equation is still a very challenging problem even if it has been derived more than 80 years ago and has been utilized profusely. It is always useful to investigate the relativistic effects [1–4]. For example, in the relativistic treatment of nuclear phenomena the Dirac equation is used to describe the behavior of the nuclei in nucleus and also in solving many problems of highenergy physics and chemistry. For this reason, it has been used extensively to study the relativistic heavy ion collisions, heavy ion spectroscopy, and more recently in lasermatter interaction (for a review, see [5] and references therein) and condensed matter physics [6].
On the other hand, systems with positiondependent mass (PDM) have been found to be very useful in studying the physical properties of various microstructures [7–14]. Recently, there has been increased interest in searching for analytical solutions of the Dirac equation with PDM and with constant mass under the spin and pspin symmetries [15–34].
Here, we shall attempt to solve the Dirac equation by using the Laplace transform method (LTM). The LTM is an integral transform and is comprehensively useful in physics and engineering [35] and recently used by many authors to solve the Schrödinger equation for different potential forms [36–40]. This method could be a nearly new formalism in the literature and serve as a powerful algebraic treatment for solving the secondorder differential equations. As a result, the LTM describes a simple way for solving of radial and onedimensional differential equations. The other advantage of this method is that a secondorder equation can be converted into more simpler form whose solutions may be obtained easily [36]. In this paper, we obtain solution of the Dirac equation both PDM and tensor interaction for attractive scalar and repulsive vector Coulomb potential under the spin and pspin symmetry limits. We give some numerical results.
2. Review to Dirac Equation including Tensor Coupling
The Dirac equation which describes a nucleon in repulsive vector and attractive scalar and a tensor potentials is written as where is the effective mass of the fermionic particle, is the relativistic energy of the system, is the threedimensional momentum operator. and are the Dirac matrices give as where is unitary matrix and are threevector spin matrices The total angular momentum operator and spinorbit , where is orbital angular momentum, of the spherical nucleons commute with the Dirac Hamiltonian. The eigenvalues of spinorbit coupling operator are and for unaligned spin and the aligned spin , respectively. () can be taken as the complete set of the conservative quantities. Thus, the spinor wave functions can be classified according to their angular momentum , spinorbit quantum number and the radial quantum number can be written as follows: where is the upper (large) component and is the lower (small) component of the Dirac spinors. and are spin and pspin spherical harmonics, respectively, and is the projection of the angular momentum on the axis. Substituting (2.4) into (2.1) and using the following relations [41] as and with the following properties: one obtains two coupled differential equations for upper and lower radial wave functions and as where Eliminating and from (2.5e) and (2.5f), we finally obtain the following two Schrödingerlike differential equations for the upper and lower radial spinor components, respectively: where and . These radial wave functions are required to satisfy the necessary boundary conditions. The spinorbit quantum number is related to the quantum numbers for spin symmetry and pspin symmetry as and the quasidegenerate doublet structure can be expressed in terms of a pspin angular momentum and pseudoorbital angular momentum , which is defined as where . For example, and can be considered as pspin doublets. In the next section, we will consider the spin and pspin symmetry cases.
3. Relativistic Bound State Solutions
3.1. Spin Symmetry Case
Equation (2.5i) cannot be solved analytically because of the last term . Therefore, in solving the mathematical relation [42, 43], we can then exactly solve (2.5i). In this stage, we take the vector potential in the form of an attractive Coulomblike field [18] as where is being a vector dimensionless real parameter coupling constant and is being a constant with dimension. Also, it is convenient to take the mass function [18] as where and stand for the rest mass of the fermionic particle and the perturbed mass, respectively. Further, is the dimensionless real constant to be set to zero for the constant mass case and is the Comptonlike wavelength in fm units. Further, the tensor interaction takes the simple form: where is the coulomb radius, and stand for the charges of the projectile particle and the target nucleus , respectively.
Substituting (3.1)–(3.3) into (2.5i) considering the spin symmetry case where ., that is, [44, 45]. Thus, the equation obtained for the upper component of the Dirac spinor becomes Further, defining the new parameters and introducing , then (3.4) turns into the formSetting with is a constant and then inserting into (3.6a), we have Now, to obtain a finite wave function when , if we take in (3.7) then it becomes The LTM [46, 47] leads to an equation Equation (3.10) is a firstorder differential equation and therefore we may directly make use of the integral to get the expression where is a constant. Noting that is a multivalued function and the wave functions are required to be singlevalued, we must take which gives singlevalues wave functions. Using this requirement and further expanding (3.11) into series, we obtain where is a constant. In terms of a simple extension of the inverse Laplace transformation [46, 47], we can immediately obtain and from , we then obtain where is a constant. On the other hand, the confluent hypergeometric functions is defined as a series expansion [48] So, we find the upperspinor component of wave function as where is normalization constant. By using the normalization condition given as , and the relation between the Laguerre polynomials and confluent hypergeometric functions as [47], the normalization constant in (3.17) is written as where we have used [47] Inserting the parameters in (3.5) and (3.6b) into (3.12), one obtain the transcendental energy eigenvalue equation as follows which is identical to [18] when and the tensor interaction is removed, that is, . For the constant mass case, we have In case when , , , , and , the above equation reduces to the nonrelativistic energy limit (in units of ): The wave function (3.17) is finite in the entire range and at the boundaries, that is, and .
The lower spinor wave function can be obtained via where we have used In Tables 1, 2, and 3 with , we give some numerical results for the energy eigenvalues from energy formula (3.20).



3.2. pSpin Symmetry Case
To avoid repetition in the solution of (2.5j), the negative energy solution for pspin symmetry can be obtained directly from those of the above positive energy solution for spin symmetry by using the parameter mapping [18]: Following the previous results with the above transformations, we finally arrive at the transcendental energy equation as which is identical to [18] when and the tensor interaction is removed, that is, . For the constant mass case, we have the lowerspinor component of wave function as where , and is normalization constant The upperspinor wave function can be obtained via
4. Numerical Results
In Tables 1 to 3, we see that energies of bound states such as: , , , (where each pair is considered as a spin doublet) in the absence of the tensor potential are degenerate but in the presence of the tensor potential, the degeneracies are removed. Also, we investigate the effects of the and parameters on the bound states under the conditions of the spin symmetry limits for . The results are given in Tables 1 to 3. It is readily seen that if and parameters increases, the value of the bound state energy eigenvalues of this potential increases for several states. It is shown that the energy eigenvalues decrease with decreasing and when increase with increasing the tensor strength for and . In Table 3, for constant values of and , the energy increases when increasing. The decrease in the energy values is large without tensor interaction while small in presence of tensor interaction and then being large. Further, when increases, the energy increasing.
Finally, we plot the relativistic energy eigenvalues under spin and pspin symmetry limitations in Figures 1 to 4. In Figure 2, we plot the energy eigenvalues of spin symmetry limit versus the perturbated mass . It is seen that when increases, the energy increases too. In Figure 2, we have shown the variation of the energy as a function of . We can see the degeneracy removes between spin doublets and also they become far from each other, when the parameter increases. In Figures 3 to 4, we plot the energy states of the pseudospin symmetry limit for different levels as functions of parameters and , respectively. The variation of energy can also be seen from these figures.
5. Conclusion
In this paper, the relativistic equation for particles with spin 1/2 was solved exactly with both spatiallydependent mass and tensor interaction for attractive scalar and repulsive vector Coulomb potentials under the spin symmetry limit via the Laplace transformation method. Some numerical results are given for specific values of the model parameters. Effects of the tensor interaction on the bound states were presented that tensor interaction removes degeneracy between two states in spin doublets. We also investigated the effects of the spatiallydependent mass on the bound states under the conditions of the spin symmetry limits for .
References
 I. C. Wang and C. Y. Wong, “Finitesize effect in the Schwinger particleproduction mechanism,” Physical Review D, vol. 38, pp. 348–359, 1988. View at: Publisher Site  Google Scholar
 G. Mao, “Effect of tensor couplings in a relativistic Hartree approach for finite nuclei,” Physical Review C, vol. 67, Article ID 044318, 12 pages, 2003. View at: Publisher Site  Google Scholar
 R. J. Furnstahl, J. J. Rusnak, and B. D. Serot, “The nuclear spinorbit force in chiral effective field theories,” Nuclear Physics A, vol. 632, no. 4, pp. 607–623, 1998. View at: Publisher Site  Google Scholar
 P. Alberto, R. Lisboa, M. Malheiro, and A. S. de Castro, “Tensor coupling and pseudospin symmetry in nuclei,” Physical Review C, vol. 71, Article ID 034313, 7 pages, 2005. View at: Publisher Site  Google Scholar
 Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic highpower lasermatter interactions,” Physics Reports, vol. 427, no. 23, pp. 41–155, 2006. View at: Publisher Site  Google Scholar
 M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nature Physics, vol. 2, no. 9, pp. 620–625, 2006. View at: Publisher Site  Google Scholar
 M. R. Galler and W. Kohn, “Quantum mechanics of electrons in crystals with graded composition,” Physical Review Letters, vol. 70, pp. 3103–3106, 1993. View at: Publisher Site  Google Scholar
 A. Puente and M. Gasas, “Nonlocal energy density functional for atoms and metal clusters,” Computational Materials Science, vol. 2, pp. 441–449, 1994. View at: Publisher Site  Google Scholar
 F. Arias De Saavedra, J. Boronat, A. Polls, and A. Fabrocini, “Effective mass of one ^{4}He atom in liquid ^{3}He,” Physical Review B, vol. 50, no. 6, pp. 4248–4251, 1994. View at: Publisher Site  Google Scholar
 M. Barranco, M. Pi, S. M. Gatica, E. S. Hernandez, and J. Navarro, “Structure and energetics of mixed ^{4}He^{3}He drops,” Physical Review B, vol. 56, pp. 8997–9003, 1997. View at: Publisher Site  Google Scholar
 A. Puente, L. I. Serra, and M. Casas, “Dipole excitation of Na clusters with a nonlocal energy density functional,” Zeitschrift Für Physik D, vol. 31, pp. 283–286, 1994. View at: Publisher Site  Google Scholar
 L. Serra and E. Lipparini, “Spin response of unpolarized quantum dots,” Europhysics Letters, vol. 40, no. 6, pp. 667–672, 1997. View at: Publisher Site  Google Scholar
 G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de physique, Les Ulis, France, 1988.
 A. G. M. Schmidt, “Wavepacket revival for the Schrödinger equation with positiondependent mass,” Physics Letters A, vol. 353, pp. 459–462, 2006. View at: Publisher Site  Google Scholar
 M. Eshghi and H. Mehraban, “Solution of the dirac equation with positiondependent mass for qparameter modified pöschlteller and coulomblike tensor potential,” FewBody Systems, vol. 52, pp. 41–47, 2012. View at: Publisher Site  Google Scholar
 E. Maghsoodi, H. Hassanabadi, and S. Zarrinkamar, “Spectrum of dirac equation under dengfan scalar and vector potentials and a coulomb tensor interaction by SUSYQM,” FewBody Systems, vol. 53, no. 34, pp. 525–538, 2012. View at: Publisher Site  Google Scholar
 C. S. Jia and A. de Souza Dutra, “Extension of PTsymmetric quantum mechanics to the Dirac theory with positiondependent mass,” Annals of Physics, vol. 323, no. 3, pp. 566–579, 2008. View at: Publisher Site  Google Scholar
 S. M. Ikhdair and R. Sever, “Solutions of the spatiallydependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomblike potential,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 545–555, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. Xu, S. He, and C.S. Jia, “Approximate analytical solutions of the Dirac equation with the PöschlTeller potential including the spinorbit coupling term,” Journal of Physics A, vol. 41, no. 25, p. 255302, 8, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Akcay, “Dirac equation with scalar and vector quadratic potentials and Coulomblike tensor potential,” Physics Letters A, vol. 373, no. 6, pp. 616–620, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Eshghi and M. Hamzavi, “Spin symmetry in diracattractive radial problem and tensor potential,” Communications in Theoretical Physics, vol. 57, article 355, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Eshghi, “Pseudoharmonic oscillatory ringshaped potential in a relativistic equation,” Chinese Physics Letters, vol. 29, no. 11, Article ID 110304, 2012. View at: Google Scholar
 O. Panella, S. Biondini, and A. Arda, “New exact solution of the onedimensional Dirac equation for the WoodsSaxon potential within the effective mass case,” Journal of Physics A, vol. 43, Article ID 325302, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. D. Alhaidari, “Solution of the Dirac equation with positiondependent mass in the Coulomb field,” Physics Letters A, vol. 322, no. 12, pp. 72–77, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 X.L. Peng, J.Y. Liu, and C.S. Jia, “Approximation solution of the Dirac equation with positiondependent mass for the generalized Hulthén potential,” Physics Letters A, vol. 352, no. 6, pp. 478–483, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Eshghi and H. Mehraban, “Eigen spectra for ManningRosen potential including a Coulomblike tensor interaction,” International Journal of Physical Sciences, vol. 6, no. 29, pp. 6643–6652, 2011. View at: Google Scholar
 M. Hamzavi, M. Eshghi, and S. M. Ikhdair, “Effect of tensor interaction in the Diracattractive radial problem under pseudospin symmetry limit,” Journal of Mathematical Physics, vol. 53, Article ID 082101, 10 pages, 2012. View at: Publisher Site  Google Scholar
 M.C. Zhang and G.Q. HuangFu, “Solution of the Dirac equation in the tridiagonal representation with pseudospin symmetry for an anharmonic oscillator and electric dipole ringshaped potential,” Annals of Physics, vol. 327, no. 3, pp. 841–850, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 O. Aydogdu and R. Sever, “Exact solution of the Dirac equation with the Mietype potential under the pseudospin and spin symmetry limit,” Annals of Physics, vol. 325, pp. 373–383, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Eshghi and H. Mehraban, “Eigen spectra in the Dirachyperbolic problem with tensor coupling,” Chinese Journal of Physics, vol. 50, no. 4, p. 533, 2012. View at: Google Scholar
 M. Eshghi, “Dirachyperbolic scarf problem including a coulomblike tensor potential,” Acta Scientiarum, vol. 34, p. 207, 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 PöschlTeller potentials,” The European Physical Journal A, vol. 43, pp. 185–190, 2010. View at: Publisher Site  Google Scholar
 V. G. C. S. dos Santos, A. de Souza Dutra, and M. B. Hott, “Real spectra for the nonHermitian Dirac equation in $1+1$ dimensions with the most general coupling,” Physics Letters A, vol. 373, no. 38, pp. 3401–3406, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Hamzavi, A. A. Rajabi, and H. Hassanabadi, “Exact spin and pseudospin symmetry solutions of the Dirac equation for mietype potential including a coulomblike tensor potential,” FewBody Systems, vol. 48, no. 2, pp. 171–182, 2010. View at: Publisher Site  Google Scholar
 G. Doetsch, Guide to the Applications of Laplace Transforms, Princeton University press, Princeton, NJ, USA, 1961.
 G. Chen, “The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms,” Physics Letters A, vol. 326, no. 12, pp. 55–57, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Arda and R. Sever, “Exact solutions of the Schrödinger equation via Laplace transform approach: pseudoharmonic potential and Mietype potentials,” Journal of Mathematical Chemistry, vol. 50, no. 4, pp. 971–980, 2012. View at: Publisher Site  Google Scholar
 E. Schrodinger, “Quantisierung als Eigenwertproblem. I and II,” Annalen Der Physik, vol. 79, pp. 361–376, 1926. View at: Google Scholar
 R. A. Swainson and G. W. F. Drake, “A unified treatment of the nonrelativistic and relativistic hydrogen atom. I. The wavefunctions,” Journal of Physics A, vol. 24, no. 1, pp. 79–94, 1991. View at: Publisher Site  Google Scholar
 G. Chen, “Exact solutions of Ndimensional harmonic oscillator via Laplace transformation,” Chinese Physics, vol. 14, no. 6, p. 1075, 2005. View at: Publisher Site  Google Scholar
 J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGrawHill, New York, NY, USA, 1964.
 D. Agboola, “DiracHulthen Problem with Positiondependent Mass in Ddimensions,” http://arxiv.org/abs/1011.2368. View at: Google Scholar
 A. Arda, R. Sever, and C. Tezcan, “Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any κvalue,” Central European Journal of Physics, vol. 8, no. 5, pp. 843–849, 2010. View at: Publisher Site  Google Scholar
 J. Meng, K. SugawaraTanabe, S. Yamaji, P. Ring, and A. Arima, “Pseudospin symmetry in relativistic mean field theory,” Physical Review C, vol. 58, no. 2, pp. R628–R631, 1998. View at: Publisher Site  Google Scholar
 J. Meng, K. SugawaraTanaha, S. Yamaji, and A. Arima, “Pseudospin symmetry in Zr and Sn isotopes from the proton drip line to the neutron drip line,” Physical Review C, vol. 59, pp. 154–163, 1999. View at: Publisher Site  Google Scholar
 M. R. Spiegel, Schaum's Outline of Theory and Problems of Laplace Transforms, Schaum Publishing, New York, NY, USA, 1965.
 F. Doetsch, Guides to the Application of Laplace Transforms, Princeton University press, Princeton, NJ, USA, 1961.
 M. Abramowitz and I. A. Stegun, Handbook Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1970.
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