Few-Body Systems

, 59:7 | Cite as

Approximate Treatment of the Dirac Equation with Hyperbolic Potential Function

  • Aysen Durmus


The time independent Dirac equation is solved analytically for equal scalar and vector hyperbolic potential function in the presence of Greene and Aldrich approximation scheme. The bound state energy equation and spinor wave functions expressed by the hypergeometric function have been obtained in detail with asymptotic iteration approach. In order to indicate the accuracy of this different approach proposed to solve second order linear differential equations, we present that in the non-relativistic limit, analytical solutions of the Dirac equation converge to those of the Schrödinger one. We introduce numerical results of the theoretical analysis for hyperbolic potential function. Bound states corresponding to arbitrary values of n and l are reported for potential parameters covering a wide range of interaction. Also, we investigate relativistic vibrational energy spectra of alkali metal diatomic molecules in the different electronic states. It is observed that theoretical vibrational energy values are consistent with experimental Rydberg–Klein–Rees (RKR) results and vibrational energies of NaK, \(K_2\) and KRb diatomic molecules interacting with hyperbolic potential smoothly converge to the experimental dissociation limit \(D_e=2508cm^{-1}\), \(254cm^{-1}\) and \(4221cm^{-1}\), respectively.


Dirac theory Greene and Aldrich approximation Asymptotic iteration method Hyperbolic potential Diatomic molecules 


  1. 1.
    P.A.M. Dirac, The quantum theory of the electron. Proc. R. Soc. A 117, 610624 (1928)CrossRefGoogle Scholar
  2. 2.
    P.A.M. Dirac, A theory of electrons and protons. Proc. R. Soc. A 126, 360365 (1930)ADSCrossRefGoogle Scholar
  3. 3.
    P.A.M. Dirac, Principles of Quantum Mechanics (Oxford University Press, London, 1958)zbMATHGoogle Scholar
  4. 4.
    W.L. Chen, G.F. Wei, Spin symmetry in the relativistic modified Rosen–Morse potential with the approximate centrifugal term. Chin. Phys. B 20, 062101, 5 (2011)ADSGoogle Scholar
  5. 5.
    G.F. Wei, S.H. Dong, A novel algebraic approach to spin symmetry for Dirac equation with scalar and vector second Pöschl–Teller potentials. Eur. Phys. J. A 43, 185–190 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    P. Zhang, H.C. Long, C.S. Jia, Solutions of the Dirac equation with the Morse potential energy model in higher spatial dimensions. Eur. Phys. J. Plus 131, 117–118 (2016)CrossRefGoogle Scholar
  7. 7.
    G.F. Wei, S.H. Dong, Pseudospin symmetry in the relativistic Manning–Rosen potential including a Pekeris-type approximation to the pseudo-centrifugal term. Phys. Lett. B 686, 288292 (2010)CrossRefGoogle Scholar
  8. 8.
    C.L. Pekeris, The rotation-vibration coupling in diatomic molecules. Phys. Rev. 45, 98–103 (1934)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    R.L. Greene, C. Aldrich, Variational wave functions for a screened Coulomb potential. Phys. Rev. A 14, 2363–2366 (1976)ADSCrossRefGoogle Scholar
  10. 10.
    W.C. Qiang, J.Y. Wu, S.H. Dong, The Eckart-like potential studied by a new approximate scheme to the centrifugal term. Phys. Scr. 79, 065011 (2009)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    S.H. Dong, W.C. Qiang, G.H. Sun, V.B. Bezerra, Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential. J. Phys. A Math. Theor. 40, 1053510540 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    G.F. Wei, Z.Z. Zhen, S.H. Dong, The relativistic bound and scattering states of the Manning–Rosen potential with an improved new approximate scheme to the centrifugal term. Cent. Eur. J. Phys. 7, 175–183 (2009)Google Scholar
  13. 13.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34, 57–64 (1929)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Hulthén, ber die Eigenlsungen der Schrödinger-Gleichung des Deuterons. Ark. Mat. Astron. Fys. A 28, 5 (1942)MathSciNetGoogle Scholar
  16. 16.
    N. Rosen, P.M. Morse, On the vibrations of polyatomic molecules. Phys. Rev. 42, 210–217 (1932)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Eckart, The penetration of a potential barrier by electrons. Phys. Rev. 35, 1303–1309 (1930)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    C. Manning, N. Rosen, Potential functions for vibration of diatomic molecules. Phys. Rev. 44, 951–954 (1933)CrossRefGoogle Scholar
  19. 19.
    F. Büyükkilic, E. Egrifes, D. Demirhan, Solution of the Schrödinger equation for two different molecular potentials by the Nikiforov–Uvarov method. Theor. Chem. Acc. 98, 192–196 (1997)zbMATHGoogle Scholar
  20. 20.
    H. Egrifes, D. Demirhan, F. Büyükkilic, Exact solutions of the Schrödinger equation for the deformed hyperbolic potential well and the deformed four-parameter exponential type potential. Phys. Lett. A 275, 229–237 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Garcia-Martinez, J. Garcia-Ravelo, J. Morales, J.J. Pena, Exactly solvable Schrödinger equation for a class of multiparameter exponential-type potentials. Int. J. Quantum Chem. 112, 195–200 (2002)CrossRefGoogle Scholar
  22. 22.
    C.S. Jia, Y. Li, Y. Sun, J.Y. Liu, L.T. Sun, Bound states of the five-parameter exponential-type potential model. Phys. Lett. A 311, 115–125 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Q.B. Yang, Deformed symmetrical double-well potential. Acta Photon. Sin 32, 882–884 (2003)Google Scholar
  24. 24.
    A. Arai, Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl. 158, 63–79 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    X.Q. Zhao, C.S. Jia, Q.B. Yang, Bound states of relativistic particles in the generalized symmetrical double-well potential. Phys. Lett. A 337, 189–196 (2005)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    G.F. Wei, S.H. Dong, Spin symmetry in the relativistic symmetrical well potential including a proper approximation to the spinorbit coupling term. Phys. Scr. 81, 035009 (2010)ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    G.F. Wei, S.H. Dong, Algebraic approach to pseudospin symmetry for the Dirac equation with scalar and vector modified Pöschl–Teller potentials. EPL 87, 40004 (2009)ADSCrossRefGoogle Scholar
  28. 28.
    N. Candemir, Klein–Gordon particles in symmetrical well potential. App. Math. Comp. 274, 531–538 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    G.F. Wei, W.L. Chen, S.H. Dong, The arbitrary l continuum states of the hyperbolic molecular potential. Phys. Lett. A 378, 2367–2370 (2014)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    G.F. Wei, W.L. Chen, Arbitrary l-wave bound states of the Schrödinger equation for the hyperbolical molecular potential. Int. J. Quantum Chem. 114, 1602–1606 (2014)CrossRefGoogle Scholar
  31. 31.
    H. Ciftci, R.L. Hall, N. Saad, Asymptotic iteration method for eigenvalue problems. J. Phys. A Math. Gen. 36, 11807–11816 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    H. Ciftci, R.L. Hall, N. Saad, Construction of exact solutions to the eigenvalue problems by the asymptotic iteration method. J. Phys. A Math. Gen. 38, 1147–1155 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    H. Ciftci, R.L. Hall, N. Saad, Iterative solutions to the Dirac equation. Phys. Rev. A 72, 022101 (2005)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    A.D. Alhaidari, H. Bahlouli, A. Al-Hasan, Dirac and Klein–Gordon equations with equal scalar and vector potentials. Phys. Lett. A 349, 87–97 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    W. Lucha, F.F. Schöberl, Solving the Schrödinger equation for bound states with mathematica 3.0. Int. J. Mod. Phys. C 10, 607–619 (1999)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    R. Krems, W.C. Stwalley, B. Friedrich, Cold Molecules: Theory, Experiment, Applications (Taylor and Francis, New York, 2009)Google Scholar
  37. 37.
    J. Banerjee, D. Rahmlow, R. Carollo, M. Bellos, E.E. Eyler, P.L. Gould, W.C. Stwalley, Direct photoassociative formation of ultracold KRb molecules in the lowest vibrational levels of the electronic ground state. Phys. Rev. A 86, 053428 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    L. Li, A.M. Lyyra, W.T. Luh, W.C. Stwalley, Observation of the \({ }^{39}K_2\) \({}c^3\Sigma ^{+}\) state by perturbation facilitated opticaloptical double resonance resolved fluorescence spectroscopy. J. Chem. Phys. 93, 8452–8463 (1990)ADSCrossRefGoogle Scholar
  39. 39.
    R. Ferber et al., The \({}c^3\Sigma ^{+}\), \({}b^3\Pi \), and \({}a^3\Sigma ^{+}\) states of NaK revisited. J. Chem. Phys. 112, 5740–5750 (2000)ADSCrossRefGoogle Scholar
  40. 40.
    N. Okada, S. Kasahara, T. Ebi, M. Baba, H. Kato, Opticaloptical double resonance polarization spectroscopy of the \({}b^1\Pi \) state of \({ }^{39}K\,{ }^{85}Rb\). J. Chem. Phys. 105, 3458–3465 (1996)ADSCrossRefGoogle Scholar
  41. 41.
    Y.P. Varshni, Comparative study of potential energy functions for diatomic molecules. Rev. Mod. Phys. 29, 664–682 (1957)ADSCrossRefGoogle Scholar
  42. 42.
    J.J. Peña, G. Ovando, J. Morales, J. García-Ravelo, Non-deformed singular and non-singular exponential-type potentials. J. Mol. Model 23, 265 (2017)CrossRefGoogle Scholar
  43. 43.
    Q. Fan, W. Sun, Studies on the full vibrational spectra and molecular dissociation energies for some diatomic electronic states. Spectrochim. Acta A 72, 298–305 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    Q. Fan, W. Sun, H. Feng, Studies on the full vibrational energies and dissociation energies of some heteronuclear diatomic molecules. Spectrochim. Acta A 74, 911–916 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsErciyes UniversityKayseriTurkey

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