Solutions of the Dirac Equation with Gravitational plus Exponential Potential

ABSTRACT

The solutions of the Alhaidari formalism of the Dirac equation for the gravitational plus exponential potential have been presented using the parametric Nikiforov-Uvarov method. The energy eigenvalues and the corresponding unnormalized eigenfunctions are obtained in terms of Laguerre polynomials.

Cite this paper

B. Ita and A. Ikeuba, "Solutions of the Dirac Equation with Gravitational plus Exponential Potential,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 1-6. doi: 10.4236/am.2013.410A3001.

B. Ita and A. Ikeuba, "Solutions of the Dirac Equation with Gravitational plus Exponential Potential,"

References

[1] M. K. Bahar and F. Yasuk, “Fermionic Particles with Position-Dependent Mass in the Presence of Inversely Quadratic Yukawa Potential and Tensor Interaction,” Pramana Journal of Physics, Vol. 80, No. 2, 2013, pp. 187-197. http://dx.doi.org/10.1007/s12043-012-0483-2

[2] A. N. Ikot and E. Maghsoodi, “Relativistic Spin and Pseudospin Symmetries of Inversely Quadratic Yukawa Like plus Mobius Square Potentials Including a Cou lomb-Like Tensor Interaction,” Few-Body Systems, 2013. http://dx.doi.org/10.1007/s00601-013-0701-6

[3] K. V. Koshelev, “Another Method to Solve Dirac’s One Electron Equation Numerically,” arXiv:

0811.3912v1 [Physics.gen-ph], 2008.

[4] H. Eleuch and H. Bahlouli, “Analytical Solution to the Dirac Equation in 3 + 1 Space-Time Dimensions,” Ap plied Mathematics and Information Science, Vol. 6, 2012, pp. 153-156.

[5] H. Nakashima and H. Nakatsuji, “Relativistic Free Com plement Method for Correctly Solving the Dirac Equation with the Applications to Hydrogen Isoelectronic Atoms,” Theoretical Chemistry Account, Vol. 129, No. 3-5, 2011, pp. 567-574. http://dx.doi.org/10.1007/s00214-011-0899-7

[6] M. Eshghi, “Relativistic Problem by Choosing Spa tially-Dependent Mass Coupled With Tensor Potential,” World Applied Sciences Journal, Vol. 6, No. 3, 2012, pp. 415-420.

[7] M. Hamzavi, A. A. Rajabi and H. Hassanabi, “Exact Pseudospin Symmetry Solution of the Dirac Equation for Spatially-Dependent Mass Coulomb Potential Including Coulomb-Like Tensor Interaction via Asymptotic Itera tion Method,” Physics Letters A, Vol. 374, No. 42, pp. 4303-4307.

http://dx.doi.org/10.1016/j.physleta.2010.08.065

[8] M. Eshghi and H. Mehraban, “Eigen Spectra for q-De formed Hyperbolic Scarf Potential Including a Coulomb Like Tensor Interaction,” Journal of Scientific Research, Vol. 3, No. 2, 2011, pp. 239-247.

[9] H. Akcay and C. Tezcan, “Exact Solutions of the Dirac Equation with Harmonic Oscillator Potential Including a Coulomb-Like Tensor Interaction,” International Journal of Modern Physics C, Vol. 20, No. 6, 2010, pp. 931-940. http://dx.doi.org/10.1142/S0129183109014084

[10] H. Akcay, “Dirac Equation with Scalar and Vector Quad ratic Potentials and Coulomb-Like Tensor Potential,” Physics Letters A, Vol. 373, No. 6, 2009, pp. 616-620.

http://dx.doi.org/10.1016/j.physleta.2008.12.029

[11] M. Hamzavi, H. Hassanabadi and A. A. Rajabi, “Ap proximate Pseudospin Solutions of the Dirac Equation with the Eckart Potential Including a Coulomb-Like Ten sor Potential,” International Journal of Theoretical Phys ics, Vol. 50, No. 2, 2011, pp. 454-464. http://dx.doi.org/10.1007/s10773-010-0552-6

[12] O. Aydogdu and R. Sever, “Exact Pseduspin Symmetric Solution of the Dirac Equation for Pseudoharmonic Po tential in the Presence of Tensor Potential,” Few-Body Systems, vol. 47, No. 3, 2010, pp. 193-200. http://dx.doi.org/10.1007/s00601-010-0085-9

[13] S. M. Ikhdair and R. Sever, “Approximate Bound State Solutions of Dirac Equation with Hulthen Potential In cluding Coulomb-Like Tensor Potential,” Applied Mathe matics and Communication, Vol. 216, No. 3, 2010, pp. 911-923. http://dx.doi.org/10.1016/j.amc.2010.01.104

[14] D. Agboola, “Dirac Equation with Spin Symmetry for Modified Poschl-Teller Potential in D Dimensions,” Pramana journal of Physics, Vol. 76, No. 6, 2011, pp. 875-885.

[15] S. D. Bosanac, “Solution of Dirac Equation for a Step Potential and Klein Paradox,” Journal of Physics A: Ma thematics and Theory, Vol. 40, No. 30, 2007, pp. 8991-9001. http://dx.doi.org/10.1088/1751-8113/40/30/021

[16] M. Hamzavi and M. Amirfakhrian, “Dirac Equation for a Spherically Pseudoharmonic Oscillatory Ring-Shaped Po tential,” International Journal of Physical Sciences, Vol. 6, No. 15, 2011, pp. 3803-3807.

[17] C. Berkdemir and Y. F. Cheng, “On the Exact Solutions of the Dirac Equation with a Novel Angle-Dependent Po tential,” Physica Scripta, Vol. 79, No. 3, 2009, pp. 035003-035010.

http://dx.doi.org/10.1088/0031-8949/79/03/035003

[18] C. Y. Chen, “Exact Solutions of the Dirac Equation with Scalar and Vector Hartmann Potentials,” Physics Letters A, Vol. 339, No. 3-5, 2005, pp. 283-287.

http://dx.doi.org/10.1016/j.physleta.2005.03.031

[19] F. Zhou, Y. Wu and J. Y. Guo,” Solutions of the Dirac Equation for Makarov Potential with Pseudospin Symme try,” Communications in Theoretical Physics, Vol. 52, No. 5, 2009, pp. 813-816.

http://dx.doi.org/10.1088/0253-6102/52/5/09

[20] E. Maghsoodi, H. Hassanabadi and S. Zarrinkamar, “Ex act Solutions of the Dirac Equation with Poschl-Teller Double-Ring Shaped Coulomb Potential via the Nikiforov-Uvarov Method,” Chinese Physics B, Vol. 22, No. 3, 2013, pp. 030302-1-030302-5.

[21] I. O. Akpan, A. D. Antia and A. N. Ikot, “Bound State Solutions of the Klein-Gordon Equation with q-Deformed Equal Scalar and Vector Eckart Potential Using a Newly Improved Approximation Scheme,” High Energy Physics, Vol. 2012, 2012, Article ID: 798209. http://dx.doi.org/10.5402/2012/798209

[22] M. N. Berberan-Santos, E. N. Bodunov and L. Pogliani, “Classical and Quantum Study of the Motion of a Particle in a Gravitational Field,” Journal of Mathematical Chem istry, Vol. 37, No. 2, 2005, pp. 101-105. http://dx.doi.org/10.1007/s10910-004-1443-y

[23] K. J. Oyewumi and C. O. Akoshile, “Bound State Solutions of the Dirac-Rosen-Morse Potential with Spin and Pseudospin Symmetry,” European Physics Journal A, Vol. 45, No. 3, 2010, pp. 311-318. http://dx.doi.org/10.1140/epja/i2010-11007-0

[24] W. A. Yahaya, K. J. Oyewumi, C. O. Akoshile and T. T. Ibrahim, “Bound State Solutions of the Relativistic Dirac Equation with Equal Scalar and Vector Eckart Potential Using Nikiforov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 3, 2010, pp. 27-34.

[25] 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, 2006, pp. 87-96.

http://dx.doi.org/10.1016/j.physleta.2005.09.008

[26] C. Berkdemir, A. Berkdemir and R. Sever, “Systematic Approach to the Exact Solution of the Dirac Equation for a Special Form of Woods-Saxon Potential,” Physics A: Mathematics and General, Vol. 39, No. 43, 2006, pp. 13455-13470. http://dx.doi.org/10.1088/0305-4470/39/43/005

[27] B. I. Ita and A. I. Ikeuba, “Solutions of the Klein-Gordon Equation with Quantum Mechanical Gravitational Poten tial,” Open Journal of Microphysics, Accepted for Publi cation, 2013.

[28] A. F. Nikiforov and A. B. Uvarov, “Special functions of Mathematical Physics,” Birkhauser, Basle, 1988. http://dx.doi.org/10.1007/978-1-4757-1595-8

[29] M. Hamzavi and A. A. Rajabi, “Exact S-Wave Solution of the Trigonometric Poschl-Teller Potential,” Interna tional Journal of Quantum Chemistry, Vol. 112, No. 6, 2012, pp. 1592-1597.

http://dx.doi.org/10.1002/qua.23166

[30] 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,” An nalen der Physik, Vol. 16, No. 3, 2007, pp. 218-232.

[31] N. Candemir, “Pseudospin Symmetry in Trigonometric Poschl-Teller Potential,” International Journal of Modern Physics E, Vol. 21, No. 6, 2010, Article ID: 1250060.

[32] A. N. Ikot, “Solution of Dirac Equation with Generalized Hylleraas Potential,” Communications in Theoretical Physics, Vol. 59, No. 3, 2013, pp. 268-272. http://dx.doi.org/10.1088/0253-6102/59/3/04

[1] M. K. Bahar and F. Yasuk, “Fermionic Particles with Position-Dependent Mass in the Presence of Inversely Quadratic Yukawa Potential and Tensor Interaction,” Pramana Journal of Physics, Vol. 80, No. 2, 2013, pp. 187-197. http://dx.doi.org/10.1007/s12043-012-0483-2

[2] A. N. Ikot and E. Maghsoodi, “Relativistic Spin and Pseudospin Symmetries of Inversely Quadratic Yukawa Like plus Mobius Square Potentials Including a Cou lomb-Like Tensor Interaction,” Few-Body Systems, 2013. http://dx.doi.org/10.1007/s00601-013-0701-6

[3] K. V. Koshelev, “Another Method to Solve Dirac’s One Electron Equation Numerically,” arXiv:

0811.3912v1 [Physics.gen-ph], 2008.

[4] H. Eleuch and H. Bahlouli, “Analytical Solution to the Dirac Equation in 3 + 1 Space-Time Dimensions,” Ap plied Mathematics and Information Science, Vol. 6, 2012, pp. 153-156.

[5] H. Nakashima and H. Nakatsuji, “Relativistic Free Com plement Method for Correctly Solving the Dirac Equation with the Applications to Hydrogen Isoelectronic Atoms,” Theoretical Chemistry Account, Vol. 129, No. 3-5, 2011, pp. 567-574. http://dx.doi.org/10.1007/s00214-011-0899-7

[6] M. Eshghi, “Relativistic Problem by Choosing Spa tially-Dependent Mass Coupled With Tensor Potential,” World Applied Sciences Journal, Vol. 6, No. 3, 2012, pp. 415-420.

[7] M. Hamzavi, A. A. Rajabi and H. Hassanabi, “Exact Pseudospin Symmetry Solution of the Dirac Equation for Spatially-Dependent Mass Coulomb Potential Including Coulomb-Like Tensor Interaction via Asymptotic Itera tion Method,” Physics Letters A, Vol. 374, No. 42, pp. 4303-4307.

http://dx.doi.org/10.1016/j.physleta.2010.08.065

[8] M. Eshghi and H. Mehraban, “Eigen Spectra for q-De formed Hyperbolic Scarf Potential Including a Coulomb Like Tensor Interaction,” Journal of Scientific Research, Vol. 3, No. 2, 2011, pp. 239-247.

[9] H. Akcay and C. Tezcan, “Exact Solutions of the Dirac Equation with Harmonic Oscillator Potential Including a Coulomb-Like Tensor Interaction,” International Journal of Modern Physics C, Vol. 20, No. 6, 2010, pp. 931-940. http://dx.doi.org/10.1142/S0129183109014084

[10] H. Akcay, “Dirac Equation with Scalar and Vector Quad ratic Potentials and Coulomb-Like Tensor Potential,” Physics Letters A, Vol. 373, No. 6, 2009, pp. 616-620.

http://dx.doi.org/10.1016/j.physleta.2008.12.029

[11] M. Hamzavi, H. Hassanabadi and A. A. Rajabi, “Ap proximate Pseudospin Solutions of the Dirac Equation with the Eckart Potential Including a Coulomb-Like Ten sor Potential,” International Journal of Theoretical Phys ics, Vol. 50, No. 2, 2011, pp. 454-464. http://dx.doi.org/10.1007/s10773-010-0552-6

[12] O. Aydogdu and R. Sever, “Exact Pseduspin Symmetric Solution of the Dirac Equation for Pseudoharmonic Po tential in the Presence of Tensor Potential,” Few-Body Systems, vol. 47, No. 3, 2010, pp. 193-200. http://dx.doi.org/10.1007/s00601-010-0085-9

[13] S. M. Ikhdair and R. Sever, “Approximate Bound State Solutions of Dirac Equation with Hulthen Potential In cluding Coulomb-Like Tensor Potential,” Applied Mathe matics and Communication, Vol. 216, No. 3, 2010, pp. 911-923. http://dx.doi.org/10.1016/j.amc.2010.01.104

[14] D. Agboola, “Dirac Equation with Spin Symmetry for Modified Poschl-Teller Potential in D Dimensions,” Pramana journal of Physics, Vol. 76, No. 6, 2011, pp. 875-885.

[15] S. D. Bosanac, “Solution of Dirac Equation for a Step Potential and Klein Paradox,” Journal of Physics A: Ma thematics and Theory, Vol. 40, No. 30, 2007, pp. 8991-9001. http://dx.doi.org/10.1088/1751-8113/40/30/021

[16] M. Hamzavi and M. Amirfakhrian, “Dirac Equation for a Spherically Pseudoharmonic Oscillatory Ring-Shaped Po tential,” International Journal of Physical Sciences, Vol. 6, No. 15, 2011, pp. 3803-3807.

[17] C. Berkdemir and Y. F. Cheng, “On the Exact Solutions of the Dirac Equation with a Novel Angle-Dependent Po tential,” Physica Scripta, Vol. 79, No. 3, 2009, pp. 035003-035010.

http://dx.doi.org/10.1088/0031-8949/79/03/035003

[18] C. Y. Chen, “Exact Solutions of the Dirac Equation with Scalar and Vector Hartmann Potentials,” Physics Letters A, Vol. 339, No. 3-5, 2005, pp. 283-287.

http://dx.doi.org/10.1016/j.physleta.2005.03.031

[19] F. Zhou, Y. Wu and J. Y. Guo,” Solutions of the Dirac Equation for Makarov Potential with Pseudospin Symme try,” Communications in Theoretical Physics, Vol. 52, No. 5, 2009, pp. 813-816.

http://dx.doi.org/10.1088/0253-6102/52/5/09

[20] E. Maghsoodi, H. Hassanabadi and S. Zarrinkamar, “Ex act Solutions of the Dirac Equation with Poschl-Teller Double-Ring Shaped Coulomb Potential via the Nikiforov-Uvarov Method,” Chinese Physics B, Vol. 22, No. 3, 2013, pp. 030302-1-030302-5.

[21] I. O. Akpan, A. D. Antia and A. N. Ikot, “Bound State Solutions of the Klein-Gordon Equation with q-Deformed Equal Scalar and Vector Eckart Potential Using a Newly Improved Approximation Scheme,” High Energy Physics, Vol. 2012, 2012, Article ID: 798209. http://dx.doi.org/10.5402/2012/798209

[22] M. N. Berberan-Santos, E. N. Bodunov and L. Pogliani, “Classical and Quantum Study of the Motion of a Particle in a Gravitational Field,” Journal of Mathematical Chem istry, Vol. 37, No. 2, 2005, pp. 101-105. http://dx.doi.org/10.1007/s10910-004-1443-y

[23] K. J. Oyewumi and C. O. Akoshile, “Bound State Solutions of the Dirac-Rosen-Morse Potential with Spin and Pseudospin Symmetry,” European Physics Journal A, Vol. 45, No. 3, 2010, pp. 311-318. http://dx.doi.org/10.1140/epja/i2010-11007-0

[24] W. A. Yahaya, K. J. Oyewumi, C. O. Akoshile and T. T. Ibrahim, “Bound State Solutions of the Relativistic Dirac Equation with Equal Scalar and Vector Eckart Potential Using Nikiforov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 3, 2010, pp. 27-34.

[25] 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, 2006, pp. 87-96.

http://dx.doi.org/10.1016/j.physleta.2005.09.008

[26] C. Berkdemir, A. Berkdemir and R. Sever, “Systematic Approach to the Exact Solution of the Dirac Equation for a Special Form of Woods-Saxon Potential,” Physics A: Mathematics and General, Vol. 39, No. 43, 2006, pp. 13455-13470. http://dx.doi.org/10.1088/0305-4470/39/43/005

[27] B. I. Ita and A. I. Ikeuba, “Solutions of the Klein-Gordon Equation with Quantum Mechanical Gravitational Poten tial,” Open Journal of Microphysics, Accepted for Publi cation, 2013.

[28] A. F. Nikiforov and A. B. Uvarov, “Special functions of Mathematical Physics,” Birkhauser, Basle, 1988. http://dx.doi.org/10.1007/978-1-4757-1595-8

[29] M. Hamzavi and A. A. Rajabi, “Exact S-Wave Solution of the Trigonometric Poschl-Teller Potential,” Interna tional Journal of Quantum Chemistry, Vol. 112, No. 6, 2012, pp. 1592-1597.

http://dx.doi.org/10.1002/qua.23166

[30] 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,” An nalen der Physik, Vol. 16, No. 3, 2007, pp. 218-232.

[31] N. Candemir, “Pseudospin Symmetry in Trigonometric Poschl-Teller Potential,” International Journal of Modern Physics E, Vol. 21, No. 6, 2010, Article ID: 1250060.

[32] A. N. Ikot, “Solution of Dirac Equation with Generalized Hylleraas Potential,” Communications in Theoretical Physics, Vol. 59, No. 3, 2013, pp. 268-272. http://dx.doi.org/10.1088/0253-6102/59/3/04