Skip to main content
SpringerLink
Log in

Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrödinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthën–Yukawa inversely quadratic potential, Hulthën plus inversely quadratic exponential Mie-type potential and Hulthën plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for \(CO, NO, I_2\), HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. S. Dong, G.-H. Sun, S.H. Dong, Arbitrary l-wave solutions of the Schröodinger equation for the screen Coulomb potential. Int. J. Mod. Phys. E 22(6), 1350036 (2013)

    ADS  Google Scholar 

  2. A. Ghoshal, Y.K. Ho, Ground states of helium in exponential-cosine-screened Coulomb potentials. J. Phys. B 42, 075002 (2009)

    ADS  Google Scholar 

  3. A. Soylu, Plasma screening effects on the energies of hydrogen atom. Phys. Plasmas 19, 072701 (2012)

    ADS  Google Scholar 

  4. A.N. Ikot, W. Azogor, U.S. Okorie, F.E. Bazuaye, M.C. Onjeaju, C.A. Onate, E.O. Chukwuoch, Exact and Poisson summation thermodynamic properties for diatomic molecules with shifted Tietz potential. Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01375-0

    Article  Google Scholar 

  5. B.I. Ita, P. Ekuri, Bound state solutions of Schrödinger equation for a more general exponential screened Coulomb potential via Nikiforov–Uvarov method. Ecleticalquimica 35, 103–107 (2010)

    Google Scholar 

  6. S.M. Ikhdair, R. Sever, Bound states of a more general exponential screened Coulomb potential. J. Math. Chem. 41, 343–353 (2006)

    MathSciNet  MATH  Google Scholar 

  7. B.I. Ita et al., Bound state solutions of Schrödinger equation for a more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov–Uvarov method. J. Quantum Inf. Sci. 8, 24–45 (2018)

    Google Scholar 

  8. C.C. Gerry, B.T. James, A large-N phase integral approximation for Coulomb type system using SO (2, 1) coherent states. J. Phys. A 19 (1986)

  9. A. Soylu, O. Bayrak, I. Boztosun, Exact solutions of the Klein–Gordon equation with equal scalar and vector Rosen–Morse type potential. Chin. Phys. Lett. 25, 2754–2757 (2008)

    ADS  Google Scholar 

  10. B.I. Ita, A.I. Ikeuba, Solutions to the Schrödinger equation with inversely quadratic Yukawa plus inversely quadratic Hellmann potential using Nikiforov–Uvarov method. J. Atomic Mol. Phys. Article ID 582610 (2013)

  11. A.K. Roy, Critical parameters and spherical confinment of H atom in screened Coulomb potential. Int. J. Quantum Chem. 116, 953–960 (2016)

    Google Scholar 

  12. C.A. Onate, J.O. Ojonubah, Eigensolutions of the Schrödinger equation with a class of Yukawa potential via Supersymmetric approach. J. Theor. Appl. Phys. 10, 21–26 (2016)

    ADS  Google Scholar 

  13. A.N. Ikot, H.P. Obon, T.M. Abbey, J.D. Olisa, Approximate analytical solutions of the Klein–Gordon equation with an exponential-type potential. Sae Mulli New Phys. 65, 825–836 (2015)

    Google Scholar 

  14. K.R. Purohit, R.H. Parmar, A.K. Rai, Eigensolution and various properties of the screened cosine Kratzer potential in D dimensions via relativistic and non-relativistic treatment. Eur. Phys. J. Plus. 135, 286 (2020)

    Google Scholar 

  15. A.N. Ikot, U.S. Okorie, R. Sever, G.J. Rampho, Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential. Eur. Phys. J. Plus. 134, 386 (2019)

    Google Scholar 

  16. I.B. Okon, O. Popoola, C.N. Isonguyo, A.D. Antia, Solutions of Schrödinger and Klein-Gordon equations with Hulthen plus inversely quadratic exponential Mie-Type potential. Phys. Sci. Int. J. 19(2), 1–27 (2018)

    Google Scholar 

  17. U.A. Deta, A. Suparmi, C. Cari, Approximate solution of Schrödinger equation in D-Dimensions for scarf hyperbolic plus non-central Poschl–Teller potential using Nikiforov–Uvarov method. J. Phys. Conf. Ser. 539, 012018 (2014)

    Google Scholar 

  18. A.N. Ikot, E.J. Ibanga, O.A. Awoga, L.E. Akpabio, A.D. Antia, Solutions of Schrödinger equation with generalized inverted hyperbolic potential. J. Mod. Phys. 3, 1849–1855 (2011)

    Google Scholar 

  19. A.D. Antia, Analytical solutions of Schrödinger equation with generalized hyperbolic potential using Nikiforov–Uvarov method. Afr. Rev. Phys. 6, 0026 (2011)

    Google Scholar 

  20. A.D. Antia, Solutions of nonrelativistic Schrödinger equation with scarf II plus Rosen–Morse II potential via ansaltz method. Am. J. Phys. Chem. 4(5), 38 (2015)

    Google Scholar 

  21. A.S. Halberg, Quasi-exact solvability of a hyperbolic intermolecular potential induced by an effectivemass step. Int. J. Math. Math. Sci. Article ID 358198 (2011)

  22. S.M. Ikhdair, Bound state of the Klein–Gordon for exponential-type potential in D-dimensions. J. Quantum Inf. Sci. 1, 73–86 (2011)

    Google Scholar 

  23. F.-K. Wen, Z.-Y. Yang, C. Liu, W.-L. Yang, Y.-Z. Zhang, Exact polynomial solutions of Schrödinger equation with various hyperbolic potentials. Commun. Theor. Phys. 62(2), 153 (2014)

    ADS  MATH  Google Scholar 

  24. A. Onate, J.O. Ojonubah, Relativistic and nonrelativistic solutions of the generalized Pöschl–Teller and hyperbolical potentials with some thermodynamic properties. Int. J. Mod. Phys. E 24(03), 1550020 (2015)

    ADS  Google Scholar 

  25. T. Das, Analytical approximate bound state solution of Schrödinger equation in D-dimensions with a new mixed class of potential for arbitrary ‘ state via asymptotic iteration method. Chin. J. Phys. 24(5), 850–858 (2016)

    Google Scholar 

  26. R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14, 2363 (1976)

    ADS  Google Scholar 

  27. C.O. Edet, U.S. Okorie, A.T. Ngiangia, A.N. Ikot, Bound state solutions of the Schrödinger equation for the modified Kratzer potential plus screened Coulomb potential. Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01477-9

    Article  Google Scholar 

  28. A. Kratzer, Z. Phys. 3, 289 (1920)

    ADS  Google Scholar 

  29. C.A. Onate, O. Adebimpe, A.F. Lukman, I.J. Adama, E.O. Davids, Approximate eigensolutions of the attractive potential via parametric Nikiforov-Uvarov method. Heliyon 4, e00977 (2018)

    Google Scholar 

  30. A.N. Ikot, O.A. Awoga, H. Hassanabadi, E. Maghsoodi, Analytical approximate solution of Schrödinger equation in D Dimensions with quadratic exponential-type potential for arbitrary state. Commun. Theor. Phys.61(4) (2014)

  31. A. Arda, R. Sever, C. Tezcan, Approximate pseudospin and spin solutions of the Dirac equation for a class of exponential potentials. Chin. J. Phys. 48, 27 (2010)

    MathSciNet  Google Scholar 

  32. X. Zou, L.Z. Yi, C.S. Jia, Bound states of the Dirac equation with vector and scalar Eckart potentials. Phys. Lett. A 346, 54 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  33. M. Eshghi, M. Hamzavi, Spin symmetry in Dirac-attractive radial problem and tensor potential. Commun. Theor. Phys. 57, 355–360 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  34. B.I. Ita, H. Louis, T.O. Magu, N.A. Nzeata-Ibe, Bound state solutions of the Schrödinger equation with Manning-Rosen plus a class of Yukawa potential using Pekeris-like approximation of the Coulomb term and parametric Nikiforov–Uvarov method. World Sci. News 70, 367–385 (2017)

    Google Scholar 

  35. B.I. Ita, H. Louis, O.U. Akakuru, N.A. Nzeata-Ibe, A.I. Ikeuba, T.O. Magu, P.I. Amos, C.O. Edet, Approximate solution to the Schrödinger Equation with Manning–Rosen plus a class of Yukawa potential via WKBJ approximation method. Bulg. J. 45, 323–333 (2018)

    Google Scholar 

  36. I.B. Okon, O. Popoola, C.N. Isonguyo, Approximate solutions of Schrödinger equation with some diatomic molecular interactions using Nikiforov–Uvarov method. Advances in High Energy Physics 9671816 (2017)

  37. A. Murat, The energy eigenvalues of the exponential cosine screened Coulomb potential with magnetic field. Bitlis Eren Univ. J. Sci. Technol. 3(2), 32–38 (2013)

    Google Scholar 

  38. S.M. Ikhdair, R. Sever, Bound state energies for the exponential cosine screened Coulomb potential. Z. Phys. D 28, 1 (1993)

    ADS  Google Scholar 

  39. S.M. Ikhdair, R. Sever, Bound energy for the exponential-cosine-screened Coulomb potential. J. Math. Chem. 41, 329–341 (2007)

    MathSciNet  MATH  Google Scholar 

  40. M.K. Bahar, An alternative approach to solutions of the MGECSC potential in presence of external electricfield. Advances in High Energy Physics. 807417 (2015)

  41. I.B. Okon, O. Popoola, E.E. Ituen, Bound state solution to Schrödinger equation with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvaarov method. Int. J. Rec. Adv. Phys. (IJRAP) 5(2), 1–15 (2016)

    Google Scholar 

  42. S.H. Dong, W.C. Qiang, G.H. Sun, V.R. Bezerra, Analytical approximations to the \(\ell \) wave solutions of the Schrödinger equation with the Eckart potential. J. Phys. A Math. Theor. 40, 10535–10540 (2007)

    ADS  MATH  Google Scholar 

  43. K.J. Oyewunmi, B.J. Falaye, C.A. Onate, O.J. Oluwadare, W.A. Yahya, Thermodynamic properties and the approximate solutions of the (1993) equation with the shiftedDeng Fan potential model. Mol. Phys. 112(1), 127–141 (2014)

    ADS  Google Scholar 

  44. C.A. Onate, C.A. Ojonubah, Relativistic and nonrelativistic solutions of the generalized Pöschl–Teller and hyperbolical potentials with some thermodynamic properties. Int. J. Mod. Phys. E. 24 (2015)

  45. U.S. Okorie, A.N. Ikot, M.C. Onyeaju, E.O. Chukwuocha, Bound state solutions of Schrödinger equation with modified Mobius square potential (MMSP) and its thermodynamic properties. J. Mol. Mod. 24, 289 (2018)

    Google Scholar 

  46. A.N. Ikot, Thermodynamical properties of diatomic molecule with general molecular potential. Pramana. J. Phys. 90, 22 (2018). https://doi.org/10.1007/s12043-1007-151-0

    Article  ADS  Google Scholar 

  47. A.N. Ikot, U.S. Okorie, R. Sever, G.J. Rampho, Eigensolution, expectation values and thermodynamical properties of the screened Kratezar potential. Eur. Phys. J. Plus 134, 386 (2019)

    Google Scholar 

  48. A.D. Antia, I.E. Essien, E.B. Umoren, C.C. Eze, Approximate solutions of the nonrelativistic Schrödinger equation with inversely quadratic Yukawa plus Mobius square potential via parametric Nikiforov–Uvarov method. In: Advances in Physics Theories and Applications. vol. 44, pp. 1–13 (2015)

  49. A.D. Antia, I.O. Akpan, A.O. Akankpo, Relativistic treatment of spinless particles subject to modified Scarf II potential. Int. J. High Energy Phys. 2(4), 50–55 (2015)

    Google Scholar 

  50. H. Hassanabadi, S. Zarrinkamar, A.A. Rajabi, Exact solutions of D-dimensional Schrödinger equation for energy-dependent potential by Nikiforov–Uvarov method. Commun. Theor. Phys. 55(4), 541–544 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  51. W.C. Qiang, S.H. Dong, Analytical approximations to the solutions of the Manning–Rosen potential with centrifugal term. Phys. Lett. A 368, 13–17 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  52. A.N. Ikot, Z.E. Maghsoodi, S. Zarrinkamar, H. Hassanabadi, Solutions of Dirac equation in the presence o fmodified Tietz and modified Pöschl–Teller potentials plus a Coulomb-like tensor interaction using SUSYQM. Few-Body Syst. 54(11), 2027–2040 (2013)

    ADS  Google Scholar 

  53. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–365 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  54. R.H. Parmar, Construction of solvable non-central potential using vector superpotential: a new approach. Indian J. Phys. 93(9), 1163–1170 (2019)

    ADS  Google Scholar 

  55. S.M. Ikhdair, R. Sever, Exact solutionsof the modified Kratzer potential plus ring shape potential in the D dimensional Schrödinger equation by the Nikiforov–Uvarov method. Int. J. Mod. Phys. C 19(2), 221–235 (2008)

    ADS  MATH  Google Scholar 

  56. A.D. Antia, E.E. Ituen, H.P. Obong, C.N. Isonguyo, Analytical solutions of the modified coulomb potential using the factorization method. Int. J. Rec. Adv. Phys. 4(1), 55–65 (2015)

    Google Scholar 

  57. O. Bayrak, I. Boztosun, H. Ciftci, Exact analytical solutions to the Kratzer potential by the asymptotic iteration method. Int. J. Quantum Chem. 107, 540 (2007)

    ADS  Google Scholar 

  58. W.C. Qiang, S.H. Dong, Proper quantization rule. EPL (Eur. Lett.) 89, 10003 (2010)

    ADS  Google Scholar 

  59. Z.Q. Ma, B.W. Xu, Quantum correction in exact quantization rules. Eur. Phys. Lett. 69, 685–691 (2005)

    ADS  Google Scholar 

  60. A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhauser, Basel, 1988)

    MATH  Google Scholar 

  61. H. Karayer, D. Demirhan, F. Büyükkilic, A particular solution of Heun equation for Hulthen and Woods-Saxon potentials. Ann. Phys. 526, 11–12 (2014)

    MathSciNet  MATH  Google Scholar 

  62. R.H. Parmar, Generalized improved non-central potential and solution of Schrödinger equation with extended ring-shaped potential via Nikiforov–Uvarov method. Eur. Phys. J. Plus 134, 86 (2019)

    ADS  Google Scholar 

  63. C. Tezcan, U. Baskent, R. Sever, A general approach for the exact solution of the Schrödinger equation. Int. J. Theor. Phys. 48(2), 337–350 (2009)

    MATH  Google Scholar 

  64. J.F. Wang, X.L. Peng, L.H. Zhang, C.W. Wang, C.S. Jia, Entropy of gaseous boron monobromide. Chem. Phys. Lett. 686, 131 (2017)

    ADS  Google Scholar 

  65. M. Toutounji, A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators. Int. J. Quant. Chem 111, 1885 (2011)

    Google Scholar 

  66. A.N. Ikot, W. Azogor, U.S. Okorie, F. E. Bazuaye, M. C. Onjeaju, C. A. Onate, E. O. Chukwuocha, Exact and Poisson summation thermodynamic properties for diatomic molecules with shifted Tietz potential. Indian J. Phys. https://doi.org/10.1007/s12648-019-01375-0 (2019)

  67. M.L. Strekalov, An accurate closed-form expression for the partition function of Morse oscillators. Chem. Phys. Lett. 439, 209 (2007)

    ADS  Google Scholar 

  68. M.L. Strekalov, On the partition function of Morse oscillators. Chem. Phys. Lett. 393, 192 (2004)

    ADS  Google Scholar 

  69. P.M. Morse, H. Feshbash, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)

    Google Scholar 

  70. X.Q. Song, C.W. Wang, C.S. Jia, Thermodynamic properties for the sodium dimer. Chem. Phys. Lett. 673, 50 (2017)

    ADS  Google Scholar 

  71. O. Ebomwonyi, C.A. Onate, S.A. Ekong, M.C. Onyeaj, Thermodynamic Properties for the Carbon monoxide molecule under the influence of the Coulomb-Hulthen-Pöschl- Teller potential. J. Sci. Technol. Res. 1(1), 122–136 (2019)

    Google Scholar 

  72. S.M. Ikhdair, R. Sever, Approximate analytical solutions of the generalized Woods-Saxon potentials including the spin-orbit coupling term and spin symmetry. J. Math. Chem. 45, 1137 (2009)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Sir P T Science College, Modasa, 383315, India

    Rajendrasinh H. Parmar

Authors
  1. Rajendrasinh H. Parmar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajendrasinh H. Parmar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parmar, R.H. Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space. Few-Body Syst 61, 39 (2020). https://doi.org/10.1007/s00601-020-01572-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00601-020-01572-2

Search

Navigation