Exact solutions of the radial Schrödinger equation for some physical potentials

Abstract

By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    L.I. Schiff: Quantum Mechanics, 3rd ed., McGraw-Hill Book Co., New York, 1955.

    MATH  Google Scholar 

  2. [2a]

    L.D. Landau and E.M. Lifshitz: Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, New York, 1977

    Google Scholar 

  3. [2b]

    E.T. Whittaker and G.N. Watson: Modern Analysis, 4th ed., Cambridge University Press, London, 1927.

    MATH  Google Scholar 

  4. [3]

    R.L. Liboff: Introductory Quantum Mechanics, 4th ed., Addison Wesley, San Francisco, CA, 2003.

    Google Scholar 

  5. [4]

    M.M. Nieto: “Hydrogen atom and relativistic pi-mesic atom in N-space dimension”, Am. J. Phys., Vol. 47, (1979), pp. 1067–1072.

    Article  ADS  MathSciNet  Google Scholar 

  6. [5]

    S.M. Ikhdair and R. Sever: “Exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential”, J. Mol. Struc.-Theochem, Vol. 806, (2007), pp. 155–158.

    Article  Google Scholar 

  7. [6]

    S.M. Ikhdair and R. Sever: “Exact polynomial solutions of the Mie-type potential in the N-dimensional Schrödinger equation”, Preprint: arXiv:quant-ph/0611065.

  8. [7]

    M. Sage and J. Goodisman: “Improving on the conventional presentation of molecular vibrations: Advantages of the pseudoharmonic potential and the direct construction of potential energy curves”, Am. J. Phys., Vol. 53, (1985), pp. 350–355.

    Article  ADS  Google Scholar 

  9. [8a]

    F. Cooper, A. Khare and U. Sukhatme: “Supersymmetry and quantum mechanics and large-N expansions ”, Phys. Rep., Vol. 251, (1995), pp. 267–385

    Article  MathSciNet  Google Scholar 

  10. [8b]

    T.D. Imbo and U.P. Sukhatme: “Supersymmetric quantum mechanics”, Phys. Rev. Lett., Vol. 54, (1985), pp. 2184–2187.

    Article  ADS  Google Scholar 

  11. [9]

    Z.-Q. Ma and B.-W. Xu: “Quantum correction in exact quantization rules”, Europhys. Lett., Vol. 69, (2005), pp. 685–691.

    Article  ADS  Google Scholar 

  12. [10]

    S.-H. Dong, C.-Y. Chen and M. Lozada-Casson: “Generalized hypervirial and Balanchard’s recurrence relations for radial matrix elements”, J. Phys. B: At. Mol. Opt. Phys., Vol. 38, (2005), pp. 2211–2220.

    Article  ADS  Google Scholar 

  13. [11]

    S.-H. Dong, D. Morales and J. Garc’ia-Ravelo: “Exact quantization rule and its applications to physical potentials”, Int. J. Mod. Phys. E, Vol. 16, (2007), pp. 189–198.

    Article  ADS  Google Scholar 

  14. [12]

    W.-C. Qiang and S.-H. Dong: “Arbitrary l-state solutions of the rotating Morse potential through the exact quantization rule method”, Phys. Lett. A, Vol. 363, (2007), pp. 169–176.

    Article  ADS  MathSciNet  Google Scholar 

  15. [13]

    A.F. Nikiforov and V.B. Uvarov: Special Functions of Mathematical Physics, Birkhauser, Basel, 1988.

    MATH  Google Scholar 

  16. [14]

    G. Sezgo: Orthogonal Polynomials, American Mathematical Society, New York, 1959.

    Google Scholar 

  17. [15]

    S.M. Ikhdair and R. Sever: “Exact polynomial solution of PT/non-PT-symmetric and non-Hermitian modified Woods-Saxon potential by the Nikiforov-Uvarov method”, Preprint: arXiv:quant-ph/0507272; S.M. Ikhdair and R. Sever: “Polynomial solution of non-central potentials”, Preprint: arXiv:quant-ph/0702186.

  18. [16]

    S.M. Ikhdair and R. Sever: “Exact solution of the Klein-Gordon equation for the PTsymmetric generalized Woods-Saxon potential by the Nikiforov-Uvarov method”, Ann. Phys. (Leipzig), Vol. 16, (2007), pp. 218–232.

    MATH  Article  MathSciNet  Google Scholar 

  19. [17]

    S.M. Ikhdair and R. Sever: “Approximate eigenvalue and eigenfunction solutions for the generalized Hulthén potential with any angular momentum”, Preprint: arXiv:quant-ph/0508009.

  20. [18]

    S.M. Ikhdair and R. Sever: “A perturbative treatment for the energy levels of neutral atoms”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 6465–6476.

    MATH  Article  ADS  Google Scholar 

  21. [19]

    S.M. Ikhdair and R. Sever: “Bound energy for the exponential-cosine-screened Coulomb potential”, Preprint: arXiv:quant-ph/0604073.

  22. [20]

    S.M. Ikhdair and R. Sever: “Bound states of a more general exponential screened Coulomb potential”, Preprint: arXiv:quant-ph/0604078.

  23. [21]

    S.M. Ikhdair and R. Sever: “A perturbative treatment for the bound states of the Hellmann potential”, J. Mol. Struc.-Theochem, Vol. 809, (2007), pp. 103–113.

    Article  Google Scholar 

  24. [22]

    O. Bayrak, I. Boztosun and H. Ciftci: “Exact analytical solutions to the Kratzer potential by the asymptotic iteration method”, Int. J. Quantum Chem., Vol. 107, (2007), pp. 540–544.

    Article  ADS  Google Scholar 

  25. [23]

    R.L. Hall and N. Saad: “Smooth transformations of Kratzer’s potential in N dimensions”, J. Chem. Phys., Vol. 109, (1998), pp. 2983–2986.

    Article  ADS  Google Scholar 

  26. [24]

    M.R. Setare and E. Karimi: “Algebraic approach to the Kratzer potential”, Phys. Scr., Vol. 75, (2007), pp. 90–93.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  27. [25a]

    S.M. Ikhdair and R. Sever: “Heavy-quark bound states in potentials with the Bethe-Salpeter equation“, Z. Phys. C, Vol. 56, (1992), pp. 155–160

    Article  ADS  Google Scholar 

  28. [25b]

    S.M. Ikhdair and R. Sever: “Bethe-Salpeter equation for non-self-conjugate mesons in a power-law potential”, Z. Phys. C, Vol. 58, (1993), pp. 153–157

    Article  ADS  Google Scholar 

  29. [25c]

    S.M. Ikhdair and R. Sever: “Bound state enrgies for the exponential cosine screened Coulomb potential”, Z. Phys. D, Vol. 28, (1993), pp. 1–5

    Google Scholar 

  30. [25d]

    S.M. Ikhdair and R. Sever: “Solution of the Bethe-Salpeter equation with the shifted 1/N expansion technique”, Hadronic J., Vol. 15, (1992), pp. 389–403

    Google Scholar 

  31. [25e]

    S.M. Ikhdair and R. Sever: “Bc meson spectrum and hyperfine splittingsin the shifted large-N expansion technique”, Int. J. Mod. Phys. A, Vol. 18, (2003), pp. 4215–4231

    MATH  Article  ADS  Google Scholar 

  32. [25f]

    S.M. Ikhdair and R. Sever: “Spectroscopy of Bc meson in the semi-relativistic quark model using the shifted large-N expansion method”, Int. J. Mod. Phys. A, Vol. 19, (2004), pp. 1771–1791

  33. [25g]

    S.M. Ikhdair and R. Sever: “Bc and heavy meson spectroscopy in the local approximation of the Schrödinger equation with relativistic kinematics”, Int. J. Mod. Phys. A, Vol. 20, (2005), pp. 4035–4054

    Article  ADS  Google Scholar 

  34. [25h]

    S.M. Ikhdair and R. Sever: “Mass spectra of heavy quarkonia and Bc decay constant for static scalar-vector interactions with relativistic kinematics”, Int. J. Mod. Phys. A, Vol. 20, (2005), pp. 6509–6531

    MATH  Article  ADS  Google Scholar 

  35. [25i]

    S.M. Ikhdair and R. Sever: “Bound energy masses of mesons containing the fourth generation and iso-singlet quarks”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 2191–2199

    Article  ADS  Google Scholar 

  36. [25j]

    S.M. Ikhdair and R. Sever: “A systematic study on non-relativistic quarkonium interaction”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 3989–4002

  37. [25k]

    S.M. Ikhdair, O. Mustafa and R. Sever: “Light and heavy meson spectra in the shifted 1/N expansion method”, Tr. J. Phys., Vol. 16, (1992), pp. 510–518

    Google Scholar 

  38. [25l]

    S.M. Ikhdair, O. Mustafa and R. Sever: “Solution of Dirac equation for vector and scalar potentials and some applications” Hadronic J., Vol. 16, (1993), pp. 57–74.

    Google Scholar 

  39. [26]

    S. Özçelik and M. Şimşek: “Exact solutions of the radial Schr"odinger equation for inverse-power potentials”, Phys. Lett. A, Vol. 152, (1991), pp. 145–150.

    Article  MathSciNet  Google Scholar 

  40. [27]

    S.-H. Dong: “Schrödinger equation with the potential V (r) = Ar −4+Br −3+Cr −2+Dr −1”, Phys. Scr., Vol. 64, (2001), pp. 273–276; S.-H. Dong: “Exact solutions of the two-dimensional Schrödinger equation with certain central potentials”, Preprint: arXiv:quant-ph/0003100.

    MATH  Article  ADS  Google Scholar 

  41. [28]

    S.-H. Dong: “On the solutions of the Schrödinger equation with some anharmonic potentials”, Phys. Scr., Vol. 65, (2002), pp. 289–295.

    MATH  Article  ADS  Google Scholar 

  42. [29]

    S.M. Ikhdair and R. Sever: “On the solutions of the Schrödinger equation with some molecular potentials: Wave function ansatz”, Preprint: arXiv:quant-ph/0702052.

  43. [30]

    R.J. Le Roy and R.B. Bernstein: “Dissociation energy and long-range potential of diatomic molecules from vibration spacings of higher levels”, J. Chem. Phys., Vol. 52, (1970), pp. 3869–3879.

    Article  ADS  Google Scholar 

  44. [31]

    C. Berkdemir, A. Berkdemir and J. Han: “Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential”, Chem. Phys. Lett., Vol. 417, (2006), pp. 326–329.

    Article  Google Scholar 

  45. [32]

    A. Chatterjee: “Large N-expansion in Quantum mechanics, atomic physics and some O(N) invariant systems”, Phys. Rep., Vol. 186, (1990), pp. 249–370.

    Article  ADS  Google Scholar 

  46. [33]

    G. Esposito: “Complex parameters in quantum mechanics”, Found. Phys. Lett., Vol. 11, (1998), pp. 636–547.

    MathSciNet  Google Scholar 

  47. [34]

    G.A. Natanzon: “General properties of potentials for which the Schr"odinger equation can be solved by means of hypergeometric functions”, Theor. Math. Phys., Vol. 38, (1979), pp. 146–153.

    MATH  Article  MathSciNet  Google Scholar 

  48. [35a]

    G. Lévai: “A search for shape invariant solvable potentials”, J. Phys. A: Math. Gen., Vol. 22, (1989), pp. 689–702

    MATH  Article  ADS  Google Scholar 

  49. [35b]

    G. Lévai: “A class of exactly solvable potentials related to the Jacobi polynomials”; J. Phys. A: Math. Gen., Vol. 24, (1991), pp. 131–146.

    MATH  Article  ADS  Google Scholar 

  50. [36]

    K.J. Oyewumi and E.A. Bangudu: “Isotropic harmonic oscillator plus inverse quadratic potential in N-dimensional spaces”, Arab. J. Sci. Eng., Vol. 28, (2003), pp. 173–182.

    MathSciNet  Google Scholar 

  51. [37]

    P.M. Morse: “Diatomic molecules according to the wave mechanics. II. Vibrational levels”, Phys. Rev., Vol. 34, (1929), pp. 57–64

    Article  ADS  Google Scholar 

  52. [37b]

    N. Rosen and P.M. Morse: “On the vibrations of polyatomic molecules”, Phys. Rev., Vol. 42, (1932), pp 210–217.

    MATH  Article  ADS  Google Scholar 

  53. [38]

    K.J. Oyewumi: “Analytical solutions of the Kratzer-Fues potential in an arbitrary number of dimensions”, Found. Phys. Lett., Vol. 18, (2005), pp. 75–84.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sameer M. Ikhdair.

About this article

Cite this article

Ikhdair, S.M., Sever, R. Exact solutions of the radial Schrödinger equation for some physical potentials. centr.eur.j.phys. 5, 516–527 (2007). https://doi.org/10.2478/s11534-007-0022-9

Download citation

Keywords

  • Wavefunction ansatz
  • pseudoharmonic potential
  • Kratzer’s potential
  • bound-states
  • eigenvalues and eigenfunctions

PACS (2006)

  • 03.65.-w
  • 03.65.Fd
  • 03.65.Ge