Few-Body Systems

, Volume 57, Issue 9, pp 807–822 | Cite as

Bound and Scattering State of Position Dependent Mass Klein–Gordon Equation with Hulthen Plus Deformed-Type Hyperbolic Potential

  • A. N. Ikot
  • H. P. Obong
  • T. M. Abbey
  • S. Zare
  • M. Ghafourian
  • H. Hassanabadi
Article

Abstract

In this article we use supersymmetry quantum mechanics and factorization methods to study the bound and scattering state of Klein–Gordon equation with deformed Hulthen plus deformed hyperbolical potential for arbitrary state in D-dimensions. The analytic relativistic bound state eigenvalues and the scattering phase factor are found in closed form. We report on the numerical results for the bound state energy in D-dimensions.

Keywords

Klein–Gordon equation Supersymmetric quantum mechanics Factorization method 

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References

  1. 1.
    Dong, S.H.: Wave equations in higher dimensions. Springer, Dordrecht (2011)Google Scholar
  2. 2.
    Ikot, A.N., Obong, H.P., Hassanabadi, H., Sahehi, N., Thomas, O.S.: Solutions of D-dimensional Klein–Gordon equation for multiparameter exponential-type potential using supersymmtric quantum mechanics. Indian J. Phys. 69, 649 (2015)ADSCrossRefGoogle Scholar
  3. 3.
    Lu, L.L., Yazarloo, B.H., Zarrinkamar, S., Liu, G., Hassanabadi, H.: Calculation of the oscillator strength for the Klein–Gordon equation with tietz potential. Few Body Syst. 53, 573–581 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Oyewumi, K.J., Akinpelu, F.O., Agboola, A.D.: Exactly complete solutions of the pseudoharmonic potential in N-dimensions. Int. J. Theor. Phys. 47, 1039 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dong, S.H., Chen, C.Y., Cassou, M.L.: Generalized hypervirial and Blanchard’s recurrence relations for radial matrix elements. J. Phys. B 38, 2211 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    Garcia, M.G., de Catro, A.S.: Scattering and bound states of spinless particles in a mixed vector–scalar smooth step potential. Ann. Phys. 324, 2372 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Alhaidari, A.D., Bahlouli, H., Al Hassan, A.: Dirac and Klein–Gordon equations with equal scalar and vector potentials. Phys. Lett. A 349, 87 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Qiang, W.C., Dong, S.H.: Analytical approximations to the l-wave solutions of the Klein–Gordon equation for a second Pöschl–Teller like potential. Phys. Lett. A 322, 285 (2006)Google Scholar
  9. 9.
    Sun, G.H., Dong, S.H.: Quantum information entropies of the eigenstates for a symmetrically trigonometric Rosen–Morse potential. Phys. Scr. 87, 045003 (2013)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Arda, A., Sever, R.: Step-up and Step-down operators of a two-term molecular potential via Nikiforov–Uvarov method. Few Body Syst. 55, 265 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    Akcay, H., Sever, R.: Analytical solutions of Schrodinger equation for the diatomic molecular potentials with any angular momentum. J. Math. Chem. 50, 1973 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ikot, A.N., Obong, H.P., Owate, I.O., Onjeaju, M.C., Hassanabadi, H.: Scattering state of Klein-Gordon particles by q-parameter hyperbolic Poschl-Teller potential. Adv. High Energy. Phys. 2015, 1–7 (2015)CrossRefGoogle Scholar
  13. 13.
    Nikiforov, A.F., Uvarov, V.B.: Special functions of mathematical physics. Birkhauser, Basel (1988)Google Scholar
  14. 14.
    Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry and quantum mechanics. Phys. Rep. 251, 267 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ciftci, H., Hall, R.L., Saad, N.: Construction of exact solutions to eigenvalue problems by the asymptotic iteration method. J. Phys. A Math. Gen. 38, 1147 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ma, Z.Q., Gonzalez-Cisneros, A., Xu, B.W., Dong, S.H.: Energy spectrum of the trigonometric Rosen–Morse potential using an improved quantization rule. Phys. Lett. A 371, 180 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ikot, A.N., Maghsoodi, E., Zarrinkamar, S., Hassanabadi, H.: Supersymmetry quantum mechanics to Dirac equation with a modified Yukawa potential and a Yukawa tensor term. Indian J. Phys. 88, 283 (2014)CrossRefGoogle Scholar
  18. 18.
    Liu, J.Y., Zhang, L.H., Jia, C.S.: Calculation of the interaction potential energy curve and vibrational levels for the \( a^{3} \Sigma_{u} + \text{state}\,\text{of}\, ^{7}Li_{2}\, molecule\). Phys. Lett. A 377, 1444 (2013)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Xie, X.J., Jia, C.S.: Solutions of the Klein–Gordon equation with the Morse potential energy model in higher spatial dimensions. Phys. Scr. 90, 035207 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Nieto, M.M.: Hydrogen atom and relativistic pi-mesic atom in N-space dimensions. Am. J. Phys. 47, 1067 (1979)ADSCrossRefGoogle Scholar
  21. 21.
    Dong, S.H., Gu, X.Y., Ma, Z.Q., Yu, J.: The Klein–Gordon equation whit the coulomb potential in D dimensions. Int. J. Mod. Phys. E 12, 555–565 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    Ma, Z.Q., Dong, S.H., Gu, X.Y., Yu, J., Lozada-Cassou, M.: The Klein–Gordon equation with a coulomb plus scalar potential in D-dimensions. Int. J. Mod. Phys. E 13, 597–610 (2004)ADSCrossRefGoogle Scholar
  23. 23.
    Qiang, W.C., Dong, S.H.: Analytical approximations to the l-wave solutions of the Klein–Gordon equation for a second Pöoschl–Teller like potential. Phys. Lett. A 372, 4789 (2008)ADSCrossRefMATHGoogle Scholar
  24. 24.
    Dong, S.H., Dong, S.H., Bahl Ouli, H., Bezerra, V.B.: Algebraic approach to the Klein–Gordon equation with hyperbolic scarf potential. Int. J. Mod. Phys. E 20, 55 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    Isonguyo, C.N., Ituen, I.B., Ikot, A.N., Hassanabadi, H.: Solution of Klein Gordon equation for some diatomic molecules with new generalized morse-like potential using SUSYQM. Bull. Korean Chem. Soc. 35, 3443 (2014)CrossRefGoogle Scholar
  26. 26.
    Ikot, A.N., Ita, B.I., Awoga, O.A.: Exact solutions of the Klein–Gordon equation with Hylleraas potential. Few Body Syst. 53, 539 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    Antia, A.D., Ikot, A.N., Hassanabadi, H., Maghsoodi, E.: Bound state solutions of Klein-Gordon equation with Mobius square plus Yukawa potentials. Indian J. Phys. 87, 1133 (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Kratzer, A.: Die ultraroten Rotationsspektren der Halogenwasserstoffe. Z. Phys. 3, 289 (1920)ADSCrossRefGoogle Scholar
  29. 29.
    Tezcan, C., Sever, R.: Exact solutions of the Schrödinger equation with position-dependent effective mass via general point canonical transformation. J. Math. Chem. 42, 387 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhang, L.H., Li, X.P., Jia, C.S.: Approximate solutions of the Schrödinger equation with the generalized Morse potential model including the centrifugal term. Int. J. Quantum Chem. 111, 1870 (2011)CrossRefGoogle Scholar
  31. 31.
    Sever, R., Tezcan, C., Aktas, M., Yesiltas, O.: Exact solution of Schrodinger equation for pseudoharmonic potential. J. Math. Chem. 43, 845 (2007)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ikot, A.N., Hassanabadi, H., Obong, H.P., Chad-Umoren, Y.E., Isonguyo, C.N., Yazarloo, B.H.: Approximate solutions of Klein–Gordon equation with improved Manning–Rosen potential in D-dimensions using SUSYQM. Chin. Phys. B 23, 120303 (2014)CrossRefGoogle Scholar
  33. 33.
    Hassanabadi, H., Yazarloo, B.H.: Bound and scattering states of spinless particles under the generalized Pöschl-Teller potential. Indian J. Phys. 87, 1017 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    You, Y., Lu, F.L., Sun, D.S., Chen, C.Y.: Improved analytical approximations to the scattering solutions of the Schrödinger equation with a hyperbolical potential. Commun. Theor. Phys. 62, 315 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ikot, A.N., Obong, H.P., Olisa, J.D., Hassanabadi, H.: Scattering state of coupled Hulthen–Woods–Saxon potentials for the Duffin–Kemmer–Petiau equation with Pekeris approximation for the centrifugal term. Z. Naturforsch 70, 185 (2015)ADSGoogle Scholar
  36. 36.
    Arda, A., Sever, R.: Effective-mass Klein–Gordon–Yukawa problem for bound and scattering states. J. Math. Phys. 52, 092101 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Chen, C.Y., Lu, F.L., Sun, D.S.: Relativistic scattering states of coulomb potential plus a new ring-shaped potential. Commun. Theor. Phys. 45, 889 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    You, Y., Lu, F.L., Sun, D.S., Chen, C.Y., Dong, S.H.: Solutions of the second Pöschl-Teller potential solved by an improved scheme to the centrifugal term. Few Body Syst. 54, 2125 (2013)ADSCrossRefGoogle Scholar
  39. 39.
    Xu, Y., He, S., Jia, C.S.: Approximate analytical solutions of the Dirac equation with the Pöschl–Teller potential including the spin–orbit coupling term. J. Phys. A Math. Gen. 41, 255302 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wei, G.F., Dong, S.H.: A novel algebraic approach to spin symmetry for Dirac equation with scalar and vector second Pöschl–Teller potentials. Eur. Phys. J. A 43, 185 (2010)ADSCrossRefGoogle Scholar
  41. 41.
    Ikot, A.N., Awoga, O.A., Hassanabadi, H., Maghsoodi, E.: Analytical approximate solution of Schrödinger equation in D dimensions with quadratic exponential-type potential for arbitrary l-State. Commun. Theor. Phys. 61, 457 (2014)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Dong, S.H.: Relativistic treatment of spinless particles subject to a rotating Deng–Fan oscillator. Commun. Theor. Phys. 55, 969 (2011)ADSCrossRefMATHGoogle Scholar
  43. 43.
    Jia, C.S., Chen, T., Cui, L.G.: Approximate analytical solutions of the Dirac equation with the generalized Pöschl–Teller potential including the pseudo-centrifugal term. Phys. Lett. A 373, 1621–1626 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Wang, Z., Long, Z.W., Long, C.Y., Wang, L.Z.: Analytical solutions of position-dependent mass Klein–Gordon equation for unequal scalar and vector Yukawa potentials. Indian J. Phys. 89, 1059 (2012)ADSCrossRefGoogle Scholar
  45. 45.
    Ikot, A.N., Awoga, O.A., Antia, A.D., Hassanabadi, H., Maghsoodi, E.: Approximate solutions of D-dimensional Klein–Gordon equation with modified Hylleraas potential. Few Body syst. 54, 2041 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    Alhaidari, A.: Solutions of the nonrelativistic wave equation with position-dependent effective mass. Phys. Rev. A 66, 042116 (2002)ADSCrossRefGoogle Scholar
  47. 47.
    Xiang, J.G.: Non-hypergeometric type of polynomials and solutions of Schrödinger equation with position-dependent mass. Commun. Theor. Phys. 56, 235 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    de Souza, A., Almeida, C.A.S.: Exact solvability of potentials with spatially dependent effective masses. Phys. Lett. A 275, 25 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Jia, C.S., d Souza Dutra, A.: Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass. Ann. Phys. 323, 566–579 (2008)Google Scholar
  50. 50.
    Yu, J., Dong, S.H., Sun, G.H.: Series solutions of the Schrödinger equation with position-dependent mass for the morse potential. Phys. Lett. A 322, 290 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Yu, J., Dong, S.H.: Exactly solvable potentials for the Schrödinger equation with spatially dependent mass. Phys. Lett. A 325, 194 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Chen, X.Y., Chen, T., Jia, C.S.: Solutions of the Klein–Gordon equation with the improved Manning–Rosen potential energy model in D dimensions. Eur. Phys. J. Plus 129, 75 (2014)CrossRefGoogle Scholar
  53. 53.
    Tan, M.S., He, S., Jia, C.S.: Molecular spinless energies of the improved Rosen–Morse potential energy model in D dimensions. Eur. Phys. J. Plus 129, 264 (2014)CrossRefGoogle Scholar
  54. 54.
    Jiang, L., Yi, L.Z., Jia, C.S.: Exact solutions of the Schrödinger equation with position-dependent mass for some Hermitian and non-Hermitian potentials. Phys. Lett. A 345, 279 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Panahi, H., Bakhshi, Z.: Solvable potentials with position-dependent effective mass and constant mass Schrödinger equation. Acta Phys. Polo. B 4111, (2010)Google Scholar
  56. 56.
    Kurniawan, A., Suparmi, A., Cari, C.: Approximate analytical solution of the Dirac equation with q-deformed hyperbolic Pöschl–Teller potential and trigonometric Scarf II non-central potential. Chin. Phys. B 24, 030302 (2015)CrossRefGoogle Scholar
  57. 57.
    Hassanabadi, H., Zarrinkamar, S., Rahimov, H.: Approximate solution of D-dimensional Klein–Gordon equation with Hulthén-type potential via SUSYQM. Commun. Theor. Phys. 56, 423 (2011)ADSCrossRefMATHGoogle Scholar
  58. 58.
    Chen, C.Y., Liu, L.F., Yuan, Y.: Scattering states of modified Pöschlben–Teller potential in D-dimension. Chin. Phys. B 21, 030302 (2012)ADSCrossRefGoogle Scholar
  59. 59.
    Feng, W.G., Li, C.W., Ying, W.H., Yuan, L.Y.: The scattering states of the generalized Hulthén potential with an improved new approximate scheme for the centrifugal term. Chin. Phys. B 18, 3663 (2009)ADSCrossRefGoogle Scholar
  60. 60.
    Ikot, A.N., Hassanabdi, H., Maghsoodi, E., Yazarloo, B.H.: Yukawa potential under relativistic spin and pseudospin symmetries with three tensor interactions. Eur. J. Phys. Plus 129, 218 (2014)CrossRefGoogle Scholar
  61. 61.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Dover, New York (1965)MATHGoogle Scholar
  62. 62.
    Chen, C.Y., Sun, D.S., Liu, C.L., Lu, F.L.: Scattering states of n-dimensional hydrogen atom. Acta Phys. Sin. 52, 781 (2003)Google Scholar
  63. 63.
    Chen, C.Y., Sun, D.S., Liu, C.L., Lu, F.L.: Approximate analytical solutions for scattering states of D-dimensional Hulthén potentials. Commun. Theor. Phys. 55, 399 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Chen, C.Y., Sun, D.S., Lu, F.L.: Scattering states of the Klein-Gordon equation with coulomb-like scalar plus vector potentials in arbitrary dimension. Phys. Lett. A 330, 424 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Chen, C.Y., Lu, F.L., Sun, D.S.: Exact solutions of scattering states for the s-wave Schrödinger equation with the manning-rosen potential. Acta Phys. Sin. 56, 0204 (2007)MathSciNetMATHGoogle Scholar
  66. 66.
    Woods, R.D., Saxon, D.S.: Diffuse surface optical model for nucleon–nuclei scattering. Phys. Rev. 95, 577 (1964)ADSCrossRefGoogle Scholar
  67. 67.
    Ikot, A.N., Ibanga, E.J., Hassanabadi, H.: Scattering state of the multiparameter potential with an improved approximation for the centrifugal term in D-dimensions. Int. J. Quantum Chem. 116, 81 (2016)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  • A. N. Ikot
    • 1
  • H. P. Obong
    • 1
  • T. M. Abbey
    • 1
  • S. Zare
    • 2
  • M. Ghafourian
    • 2
  • H. Hassanabadi
    • 3
  1. 1.Theoretical Physics Group, Department of PhysicsUniversity of Port HarcourtPort HarcourtNigeria
  2. 2.Department of Basic SciencesIslamic Azad University, North Tehran BranchTehranIran
  3. 3.Physics DepartmentShadrood University of TechnologyShahroodIran

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