Solutions of Schrödinger Equation with Generalized Inverted Hyperbolic Potential

Author(s)
Akpan N. Ikot^{*},
Eno J. Ibanga,
Oladunjoye A. Awoga,
Louis E. Akpabio,
Akaninyene D. Antia

Affiliation(s)

Theoretical Physics Group, Department of Physics, University of Uyo-Nigeria, Uyo, Nigeria.

Theoretical Physics Group, Department of Physics, University of Uyo-Nigeria, Uyo, Nigeria.

ABSTRACT

The bound state solutions of the Schr?dinger equation with generalized inverted hyperbolic potential using the Nikiforov-Uvarov method are reported. We obtain the energy spectrum and the wave functions with this potential for arbitrary*l*-state. It is shown that the results of this potential reduced to the standard potentials—Rosen-Morse, Poschl-Teller and Scarf potential as special cases. We also discussed the energy equation and the wave function for these special cases.

The bound state solutions of the Schr?dinger equation with generalized inverted hyperbolic potential using the Nikiforov-Uvarov method are reported. We obtain the energy spectrum and the wave functions with this potential for arbitrary

Cite this paper

A. Ikot, E. Ibanga, O. Awoga, L. Akpabio and A. Antia, "Solutions of Schrödinger Equation with Generalized Inverted Hyperbolic Potential,"*Journal of Modern Physics*, Vol. 3 No. 12, 2012, pp. 1849-1855. doi: 10.4236/jmp.2012.312232.

A. Ikot, E. Ibanga, O. Awoga, L. Akpabio and A. Antia, "Solutions of Schrödinger Equation with Generalized Inverted Hyperbolic Potential,"

References

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[2] Y. F. Cheng and T. Q. Dai, “Exact Solutions of the Klein-Gordon Equation with a Ring-Shaped Modified Kratzer Potential,” Chinese Journal of Physics, Vol. 45, No. 5, 2007, p. 480.

[3] C. B. Compean and M. Kirchbach, “Trigonometric Quark Confinement Potential of QCD Traits,” The European Physical Journal A, Vol. 33, No. 1, 2007, pp. 1-4. doi:10.1140/epja/i2007-10444-0

[4] C. B. Compean and M. Kirchbach, “The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and Its Exact Solutions,” Journal of Physics A, Vol. 39, No. 3, 2006, p. 547. doi:10.1088/0305-4470/39/3/007

[5] A. Contreras-Astorga and D. J. Fernandez, “Supersymmetric Partners of the Trigonometric Poschl-Teller Potentials,” Journal of Physics A, Vol. 41, No. 47, 2008, Article ID: 475303.

[6] F. Cooper, A. Khare and U. Sukhature, “Supersymmetry and Quantum Mechanics,” Physics Reports, Vol. 251, No. 5-6, 1990, pp. 267-385. doi:10.1016/0370-1573(94)00080-M

[7] J. J. Diaz, J. Negro, I. M. Nieto and O. Rosas, “The Supersymmetric Modified Poschl-Teller and Delta-Well Potentials,” Journal of Physics A, Vol. 32, No. 48, 1999, p. 8447. doi:10.1088/0305-4470/32/48/308

[8] D. J. Fernandez, “Supersymmetric Quantum Mechanics,” arxiv: 0910.0192v1, 2009.

[9] R. I. Greene and C. Aldrich, “Variational Wave Function for a Screened Coulomb Potential,” Journal of Physics A, Vol. 14, No. 6, 1976, pp. 2363-2366.

[10] A. S. Halberg, “Quasi-Exact Solvability of a Hyperbolic Intermolecular Potential Induced by an Effective Mass Step,” International Journal Mathematics and Mathematical Sciences, Vol. 2011 Article ID: 358198.

[11] S. D. Hernandez and D. J. Fernandez, “Rosen-Morse Potentials and Its Supersymmetric Partners,” International Journal of Theoretical Physics, Vol. 50, 2011, pp. 1993-2001.

[12] S. Ikhdair and R. Sever, “Polynomial Solutions of Non-Central Potentials,” International Journal of Theoretical Physics, Vol. 46, No. 10, 2002, pp. 2384-2395. doi:10.1007/s10773-007-9356-8

[13] A. N. Ikot and L. E. Akpabio, “Approximate Solution of Schrodinger Equation with Rosen-Morse Potential Including a Centrifugal Term,” Applied Physics Research, Vol. 2, No. 2, 2010, p. 202.

[14] K. J. Oyewumi, E. O. Akinpelu and A. D. Agboola, “Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions,” International Journal of Theoretical Physics, Vol. 47, No. 4, 2008, pp. 1039-1057. doi:10.1007/s10773-007-9532-x

[15] R. Sever, C. Tezcan, O. Yesiltas and M. Bucurget, “Exact Solution of Effective Mass Schrodinger Equation for the Hulthen Potential,” International Journal of Theoretical Physics, Vol. 47, No. 9, 2008, pp. 2243-2248. doi:10.1007/s10773-008-9656-7

[16] G. F. Wei, X. Y. Liu and W. L. Chen, “The Relativistic Scattering States of the Hulthen Potential with an Improved New Approximation Scheme to the Centrifugal Term,” International Journal of Theoretical Physics, Vol. 48, No. 6, 2009, pp. 1649-1658. doi:10.1007/s10773-009-9937-9

[17] H. X. Quan, L. Guang, W. Z. Min, N. L. Bin and M. Yan, “Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator,” Communications in Theoretical Physics, Vol. 53, No. 2, 2010, p. 242. doi:10.1088/0253-6102/53/2/07

[18] A. N. Ikot, A. D. Antia, L. E. Akpabio and J. A. Obu, “Analytic Solutions of Schrodinger Equation with Two-Dimensional Harmonic Potential in Cartesian and Polar Coordinates via Nikiforov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 6, No. 2, 2011, pp. 65-76.

[19] A. N. Ikot, L. E. Akpabio and E. J. Uwah, “Bound State Solution of the Klein-Gordon Equation with Hulthen Potential,” Electronic Journal of Theoretical Physics, Vol. 8, No. 25, 2011, pp. 225-232.

[20] A. F. Nikiforov and U. B. Uvarov, “Special Functions of Mathematical Physics,” Birkhausa, Basel, 1988.

[21] A. N. Ikot, L. E. Akpabio and J. A. Obu, “Exact Solutions of Schrodinger Equation with Five-Parameter Potential,” Journal of Vectorial Relativity, Vol. 6, No. 1, 2011, p. 1.

[22] S. Meyur and S. Debnath, “Lie Algebra Approach to Non-Hermitian Hamiltonians,” Bulgarian Journal of Physics, Vol. 36, No. 2, 2009, pp. 77-87.

[23] S. Meyur, “Algebraic Aspect for Two Solvable Potentials,” Electronic Journal of Theoretical Physics, Vol. 8, No. 25, 2011, pp. 217-224.

[24] S. G. Roy, J. Choudhury, N. K. Sarkar, S. R. Karumuri and R. Bhattacharjee, “A Lie Algebra Approach to the Schrodinger Equation for Bound States of Poschl-Teller Potential,” Electronic Journal of Theoretical Physics, Vol. 7, No. 24, 2010, pp. 235-240.

[25] C. S. Jia, T. Chen and L. G. Cui, “Approximate Analytic Solutions of the Dirac Equation with the Generalized Poschl-Teller Potential Including the Pseudo-Spin-Centrifugal Term,” Physics Letters A, Vol. 373, 2009, pp. 1621-1626. doi:10.1016/j.physleta.2009.03.006

[26] K. J. Oyewumi and C. O. Akoshile, “Bound State Solutions of the Dirac-Rosen-Morse Potential with Spin and Pseudospin Symmetry,” The European Physical Journal A, Vol. 45, 2010, pp. 311-318. doi:10.1140/epja/i2010-11007-0

[27] K. J. Oyewumi, “Analytic Solution of the Kratzer-Feus Potential in an Arbitrary Number of Dimensions,” Foundations of Physics Letters, Vol. 18, No. 75, 2005.

[28] W. A. Yahya, K. J. Oyewumi, C. O. Akoshile and T. T. Ibrahim, “Bound State Solutions of Relativistic Dirac Equation with Equal Scalar and Vector Eckart Potentials Using the Nikivorov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 5, No. 3, 2010, pp. 1-8.

[29] Y. Xu, S. He and C. S. Jia, “Approximate Analytical Solutions of the Klein-Gordon Equation with Poschl-Teller Potential Including the Centrifugal Term,” Physica Scripta, Vol. 81, No. 4, 2010, Article ID: 045001. doi:10.1088/0031-8949/81/04/045001

[1] O. M. Al-Dossary, “Morse Potential Eigen-Energies through the Asymptotic Iteration Method,” International Journal of Quantum Chemistry, Vol. 107, No. 10, 2007, pp. 2040-2046. doi:10.1002/qua.21335

[2] Y. F. Cheng and T. Q. Dai, “Exact Solutions of the Klein-Gordon Equation with a Ring-Shaped Modified Kratzer Potential,” Chinese Journal of Physics, Vol. 45, No. 5, 2007, p. 480.

[3] C. B. Compean and M. Kirchbach, “Trigonometric Quark Confinement Potential of QCD Traits,” The European Physical Journal A, Vol. 33, No. 1, 2007, pp. 1-4. doi:10.1140/epja/i2007-10444-0

[4] C. B. Compean and M. Kirchbach, “The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and Its Exact Solutions,” Journal of Physics A, Vol. 39, No. 3, 2006, p. 547. doi:10.1088/0305-4470/39/3/007

[5] A. Contreras-Astorga and D. J. Fernandez, “Supersymmetric Partners of the Trigonometric Poschl-Teller Potentials,” Journal of Physics A, Vol. 41, No. 47, 2008, Article ID: 475303.

[6] F. Cooper, A. Khare and U. Sukhature, “Supersymmetry and Quantum Mechanics,” Physics Reports, Vol. 251, No. 5-6, 1990, pp. 267-385. doi:10.1016/0370-1573(94)00080-M

[7] J. J. Diaz, J. Negro, I. M. Nieto and O. Rosas, “The Supersymmetric Modified Poschl-Teller and Delta-Well Potentials,” Journal of Physics A, Vol. 32, No. 48, 1999, p. 8447. doi:10.1088/0305-4470/32/48/308

[8] D. J. Fernandez, “Supersymmetric Quantum Mechanics,” arxiv: 0910.0192v1, 2009.

[9] R. I. Greene and C. Aldrich, “Variational Wave Function for a Screened Coulomb Potential,” Journal of Physics A, Vol. 14, No. 6, 1976, pp. 2363-2366.

[10] A. S. Halberg, “Quasi-Exact Solvability of a Hyperbolic Intermolecular Potential Induced by an Effective Mass Step,” International Journal Mathematics and Mathematical Sciences, Vol. 2011 Article ID: 358198.

[11] S. D. Hernandez and D. J. Fernandez, “Rosen-Morse Potentials and Its Supersymmetric Partners,” International Journal of Theoretical Physics, Vol. 50, 2011, pp. 1993-2001.

[12] S. Ikhdair and R. Sever, “Polynomial Solutions of Non-Central Potentials,” International Journal of Theoretical Physics, Vol. 46, No. 10, 2002, pp. 2384-2395. doi:10.1007/s10773-007-9356-8

[13] A. N. Ikot and L. E. Akpabio, “Approximate Solution of Schrodinger Equation with Rosen-Morse Potential Including a Centrifugal Term,” Applied Physics Research, Vol. 2, No. 2, 2010, p. 202.

[14] K. J. Oyewumi, E. O. Akinpelu and A. D. Agboola, “Exactly Complete Solutions of the Pseudoharmonic Potential in N-Dimensions,” International Journal of Theoretical Physics, Vol. 47, No. 4, 2008, pp. 1039-1057. doi:10.1007/s10773-007-9532-x

[15] R. Sever, C. Tezcan, O. Yesiltas and M. Bucurget, “Exact Solution of Effective Mass Schrodinger Equation for the Hulthen Potential,” International Journal of Theoretical Physics, Vol. 47, No. 9, 2008, pp. 2243-2248. doi:10.1007/s10773-008-9656-7

[16] G. F. Wei, X. Y. Liu and W. L. Chen, “The Relativistic Scattering States of the Hulthen Potential with an Improved New Approximation Scheme to the Centrifugal Term,” International Journal of Theoretical Physics, Vol. 48, No. 6, 2009, pp. 1649-1658. doi:10.1007/s10773-009-9937-9

[17] H. X. Quan, L. Guang, W. Z. Min, N. L. Bin and M. Yan, “Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator,” Communications in Theoretical Physics, Vol. 53, No. 2, 2010, p. 242. doi:10.1088/0253-6102/53/2/07

[18] A. N. Ikot, A. D. Antia, L. E. Akpabio and J. A. Obu, “Analytic Solutions of Schrodinger Equation with Two-Dimensional Harmonic Potential in Cartesian and Polar Coordinates via Nikiforov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 6, No. 2, 2011, pp. 65-76.

[19] A. N. Ikot, L. E. Akpabio and E. J. Uwah, “Bound State Solution of the Klein-Gordon Equation with Hulthen Potential,” Electronic Journal of Theoretical Physics, Vol. 8, No. 25, 2011, pp. 225-232.

[20] A. F. Nikiforov and U. B. Uvarov, “Special Functions of Mathematical Physics,” Birkhausa, Basel, 1988.

[21] A. N. Ikot, L. E. Akpabio and J. A. Obu, “Exact Solutions of Schrodinger Equation with Five-Parameter Potential,” Journal of Vectorial Relativity, Vol. 6, No. 1, 2011, p. 1.

[22] S. Meyur and S. Debnath, “Lie Algebra Approach to Non-Hermitian Hamiltonians,” Bulgarian Journal of Physics, Vol. 36, No. 2, 2009, pp. 77-87.

[23] S. Meyur, “Algebraic Aspect for Two Solvable Potentials,” Electronic Journal of Theoretical Physics, Vol. 8, No. 25, 2011, pp. 217-224.

[24] S. G. Roy, J. Choudhury, N. K. Sarkar, S. R. Karumuri and R. Bhattacharjee, “A Lie Algebra Approach to the Schrodinger Equation for Bound States of Poschl-Teller Potential,” Electronic Journal of Theoretical Physics, Vol. 7, No. 24, 2010, pp. 235-240.

[25] C. S. Jia, T. Chen and L. G. Cui, “Approximate Analytic Solutions of the Dirac Equation with the Generalized Poschl-Teller Potential Including the Pseudo-Spin-Centrifugal Term,” Physics Letters A, Vol. 373, 2009, pp. 1621-1626. doi:10.1016/j.physleta.2009.03.006

[26] K. J. Oyewumi and C. O. Akoshile, “Bound State Solutions of the Dirac-Rosen-Morse Potential with Spin and Pseudospin Symmetry,” The European Physical Journal A, Vol. 45, 2010, pp. 311-318. doi:10.1140/epja/i2010-11007-0

[27] K. J. Oyewumi, “Analytic Solution of the Kratzer-Feus Potential in an Arbitrary Number of Dimensions,” Foundations of Physics Letters, Vol. 18, No. 75, 2005.

[28] W. A. Yahya, K. J. Oyewumi, C. O. Akoshile and T. T. Ibrahim, “Bound State Solutions of Relativistic Dirac Equation with Equal Scalar and Vector Eckart Potentials Using the Nikivorov-Uvarov Method,” Journal of Vectorial Relativity, Vol. 5, No. 3, 2010, pp. 1-8.

[29] Y. Xu, S. He and C. S. Jia, “Approximate Analytical Solutions of the Klein-Gordon Equation with Poschl-Teller Potential Including the Centrifugal Term,” Physica Scripta, Vol. 81, No. 4, 2010, Article ID: 045001. doi:10.1088/0031-8949/81/04/045001