Advertisement

Exact solutions of a nonpolynomial oscillator related to isotonic oscillator

  • Qian Dong
  • Guo-Hua Sun
  • N. Saad
  • Shi-Hai DongEmail author
Regular Article

Abstract.

We find that the analytical solutions to a quantum system with a nonpolynomial oscillator potential related to isotonic oscillator are given by the confluent Heun functions \( H_{c}(\alpha, \beta, \gamma, \delta, \eta; z)\). The properties of the wave functions, which are strongly relevant for the potential parameters a and g, are illustrated. It is shown that the wave functions are shrunk to the origin for a given a when the potential parameter g increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter \( \vert g\vert\) increases. Moreover, the wave peaks of the even wave functions become sharper when the potential parameter \( a < 1\) decreases, but they become flat when the potential parameter \( a > 1\) increases. When the minimum value \( V_{\min}=-g/a^2\) tends to zero, this nonpolynomial oscillator reduces to a harmonic oscillator.

References

  1. 1.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics, (Non-Relativistic Theory), 3rd ed. (Pergamon, New York, 1977)Google Scholar
  2. 2.
    L.I. Schiff, Quantum Mechanics, 3rd ed. (New York, McGraw-Hill Book Co., 1955)Google Scholar
  3. 3.
    D. ter Haar, Problems in Quantum Mechanics, 3rd ed. (Pion Ltd, London, 1975)Google Scholar
  4. 4.
    F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    S.H. Dong, Factorization Method in Quantum Mechanics (Springer, Kluwer Academic Publisher, 2007)CrossRefGoogle Scholar
  6. 6.
    Z.Q. Ma, B.W. Xu, Europhys. Lett. 69, 685 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    Z.Q. Ma, A. Gonzalez-Cisneros, B.W. Xu, S.H. Dong, Phys. Lett. A 371, 180 (2007)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    X.Y. Gu, S.H. Dong, Z.Q. Ma, J. Phys. A 42, 035303 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    W.C. Qiang, S.H. Dong, EPL 89, 10003 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    H. Ciftci, R.L. Hall, N. Saad, J. Phys. A 36, 11807 (2003)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhäuser, Basel, 1988)CrossRefGoogle Scholar
  12. 12.
    H. Konwent, Phys. Lett. A 118, 467 (1986)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    H. Konwent, P. Machnikowski, A. Radosz, J. Phys. A 28, 3757 (1995)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    M.A. Reyes, E. Condori-Pozo, C. Villaseñor-Mora, arXiv:1806.03388 [hep-th]Google Scholar
  15. 15.
    Q.T. Xie, J. Phys. A 45, 175302 (2012)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    B.H. Chen, Y. Wu, Q.T. Xie, J. Phys. A 46, 035301 (2013)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Razavy, M. Razavy, Am. J. Phys. 48, 285 (1980)ADSCrossRefGoogle Scholar
  18. 18.
    M. Razavy, M. Razavy, Phys. Lett. A 82, 7 (1981)ADSCrossRefGoogle Scholar
  19. 19.
    A.E. Sitnitsky, Comput. Theor. Chem. 1138, 15 (2018)CrossRefGoogle Scholar
  20. 20.
    A.E. Sitnitsky, Vibrational Spectroscopy 93, 36 (2017)CrossRefGoogle Scholar
  21. 21.
    C.A. Downing, J. Math. Phys. 54, 072101 (2013)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    G.H. Sun, S.H. Dong, K.D. Launey, T. Dytrych, J.P. Draayer, Int. J. Quan. Chem. 115, 891 (2015)CrossRefGoogle Scholar
  23. 23.
    Q. Dong, F. Serrano, G.H. Sun, J. Jing, S.H. Dong, Adv. High Energy Phys. 2018, 9105825 (2018)Google Scholar
  24. 24.
    S. Dong, Q. Dong, G.H. Sun, S. Femmam, S.H. Dong, Adv. High Energy Phys. 2018, 5824271 (2018)Google Scholar
  25. 25.
    P.P. Fiziev, J. Phys. A 43, 035203 (2010)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    R. Hartmann, M.E. Portnoi, Phys. Rev. A 89, 012101 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    Q. Dong, G.H. Sun, J. Jing, S.H. Dong, Phys. Lett. A 383, 270 (2019)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Agboola, J. Math. Phys. 55, 052102 (2014)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    F.K. Wen, Z.Y. Yang, C. Liu, W.L. Yang, Y.Z. Zhang, Commun. Theor. Phys. 61, 153 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    A. Ronveaux (Editor), Heun’s Differential Equations (Oxford University Press, Oxford, 1995)Google Scholar
  31. 31.
    J.F. Cariñena, A.M. Perelomov, M.F. Rañada, M. Santander, J. Phys. A 41, 085301 (2008)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    J.M. Fellows, R.A. Smith, J. Phys. A 42, 335303 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    R.A. Kraenkel, M. Senthilvelan, J. Phys. A 42, 415303 (2009)MathSciNetCrossRefGoogle Scholar
  34. 34.
    J. Sesma, J. Phys. A 43, 185303 (2010)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    S. Dong, Q. Fang, B.J. Falaye, G.H. Sun, C. Yáñez-Márquez, S.H. Dong, Mod. Phys. Lett. A 31, 1650017 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    S. Dong, G.H. Sun, B.J. Falaye, S.H. Dong, Eur. Phys. J. Plus 131, 176 (2016)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratorio de Información Cuántica, CIDETECInstituto Politécnico Nacional, UPALM, CDMXMexico CityMexico
  2. 2.Catedrática CONACYT, CICInstituto Politécnico NacionalMexico CityMexico
  3. 3.Department of Mathematics and StatisticsUniversity of Prince Edward IslandCharlottetownCanada

Personalised recommendations