Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V0tanh(ax)

Abstract.

In this work we report the semi-exact solutions of the position-dependent mass Schrödinger equation (PDMSE) with a class of hyperbolic potential \( V_{0}\tanh(a x)\). The terminology of semi-exact solutions arises from the fact that the wave functions of this quantum system can be expressed by confluent Heun functions, but the energy levels cannot be obtained analytically. The potential \( V(z)\) obtained by some canonical transformations essentially keeps invariant when the potential parameter v is replaced by - v . The properties of the wave functions depending on v are also illustrated graphically. We find that the quasi-symmetric and quasi-antisymmetric wave functions that appear only for very small v are violated completely when v becomes large. This arises from the fact that the parity, which is almost a defined symmetry for very small v, is completely violated for large v. We also notice that the energy level \( \varepsilon_{1}\) decreases and the \( \varepsilon_{6-8}\) ones increase with the increasing potential parameter v, respectively, while the \( \varepsilon_{2-5}\) ones first increase and then decrease with increasing v.

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

References

  1. 1

    L.I. Schiff, Quantum Mechanics, 3rd edition (New York, McGraw-Hill Book Co., 1955)

  2. 2

    L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 3rd edition (Pergamon, New York, 1977)

  3. 3

    D. ter Haar, Problems in Quantum Mechanics, 3rd edition (Pion Ltd., London, 1975)

  4. 4

    F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    S.H. Dong, Factorization Method in Quantum Mechanics (Springer, Kluwer Academic Publisher, 2007)

  6. 6

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

  7. 7

    B.J. Falaye, S.M. Ikhdair, M. Hamzavi, Few-Body Syst. 56, 63 (2015)

    ADS  Article  Google Scholar 

  8. 8

    Z.Q. Ma, B.W. Xu, Europhys. Lett. 69, 685 (2005)

    ADS  Article  Google Scholar 

  9. 9

    W.C. Qiang, S.H. Dong, EPL 89, 10003 (2010)

    ADS  Article  Google Scholar 

  10. 10

    L. Dekar, L. Chetouani, T.F. Hammann, J. Math. Phys. 39, 2551 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11

    L. Dekar, L. Chetouani, T.F. Hammann, Phys. Rev. A 59, 107 (1999)

    ADS  Article  Google Scholar 

  12. 12

    J.R. Barker, J.R. Watling, Microelectron. Eng. 63, 97 (2002)

    Article  Google Scholar 

  13. 13

    A.R. Plastino, A. Rigo, M. Casas, A. Plastino, Phys. Rev. A 60, 4318 (1999)

    ADS  Article  Google Scholar 

  14. 14

    A. de Souza Dutra, C.A.S. Almeida, Phys. Lett. A 275, 25 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15

    A.D. Alhaidari, Phys. Rev. A 66, 042116 (2002)

    ADS  Article  Google Scholar 

  16. 16

    G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructure, Les Editions de Physique (Les Ulis, 1988)

  17. 17

    M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandez, J. Navarro, Phys. Rev. B 56, 8997 (1997)

    ADS  Article  Google Scholar 

  18. 18

    L. Serra, E. Lipparini, Europhys. Lett. 40, 667 (1997)

    ADS  Article  Google Scholar 

  19. 19

    O. Von Roos, Phys. Rev. B 27, 7547 (1983)

    ADS  Article  Google Scholar 

  20. 20

    O. Von Roos, H. Mavromatis, Phys. Rev. B 31, 2294 (1985)

    ADS  Article  Google Scholar 

  21. 21

    J. Yu, S.H. Dong, Phys. Lett. A 325, 194 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22

    J. Yu, S.H. Dong, G.H. Sun, Phys. Lett. A 322, 290 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23

    M.S. Cunha, H.R. Christiansen, Commun. Theor. Phys. 60, 642 (2013)

    MathSciNet  Article  Google Scholar 

  24. 24

    A. Ronveaux (Editor), Heun's Differential Equations (Oxford University Press, Oxford, 1995)

  25. 25

    C.A. Downing, J. Math. Phys. 54, 072101 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26

    P.P. Fiziev, J. Phys. A: Math. Theor. 43, 035203 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27

    R. Hartmann, M.E. Portnoi, Phys. Rev. A 89, 012101 (2014)

    ADS  Article  Google Scholar 

  28. 28

    G. Yanez-Navarro, G.H. Sun, T. Dytrich, K.D. Launey, S.H. Dong, J.P. Draayer, Ann. Phys. 348, 153 (2014)

    ADS  Article  Google Scholar 

  29. 29

    G.H. Sun, S.H. Dong, K.D. Launey, T. Dytrych, J.P. Draayer, Int. J. Quantum. Chem. 115, 891 (2015)

    Article  Google Scholar 

  30. 30

    H. Panahiy, Z. Bakhshi, Acta Phys. Pol. B 41, 11 (2010)

    Google Scholar 

  31. 31

    B. Bagchi, P. Gorain, C. Quesne, R. Roychoudhury, Mod. Phys. Lett. A 19, 2765 (2004)

    ADS  Article  Google Scholar 

  32. 32

    M.S. Cunha, private communication

  33. 33

    S.H. Dong, Wave Equations in Higher Dimensiones (Springer, Netherlands, 2011)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shishan Dong.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dong, S., Sun, G., Falaye, B.J. et al. Semi-exact solutions to position-dependent mass Schrödinger problem with a class of hyperbolic potential V0tanh(ax). Eur. Phys. J. Plus 131, 176 (2016). https://doi.org/10.1140/epjp/i2016-16176-5

Download citation