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Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions

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Abstract

A one-dimensional non-Hermitian PT symmetric Hamiltonian, characterized by position-dependent masses, defines a Schrödinger equation in terms of a field Ψ(x, t). Based on an exact classical field theory, the necessity of an extra field Φ(x, t) (which satisfies a conjugate equation and in general different is from Ψ∗(x, t)) is shown. Simple applications are investigated by solving analytically both equations and it is shown that the effective masses proposed lead to a probability density characterized by a finite norm, typical of the physical situation that occurs with the concentration of electrons in some semiconductor heterojunctions. An extension to a three-dimensional space is also presented.

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References

  1. R.L. Liboff. Introductory Quantum Mehanics, 4th edn. (Addison Wesley, San Francisco, 2003)

    Google Scholar 

  2. C. Sulem, P.-L. Sulem. The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, New York, 1999)

    MATH  Google Scholar 

  3. A.C. Scott. The nonlinear universe (Springer, Berlin, 2007)

    Google Scholar 

  4. F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106 (2011)

  5. F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Europhys. Lett. 97, 41001 (2012)

    Article  ADS  Google Scholar 

  6. A.R. Plastino, C. Tsallis, J. Math. Phys. 54, 041505 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  7. M. A. Rego-Monteiro, F. D. Nobre, J. Math. Phys. 54, 103302 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  8. C. Tsallis. Introduction to nonextensive statistical mechanics (Springer, New York, 2009)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. J.-M. Lévy-Leblond, Phys. Rev. A. 52, 1845 (1995)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. S.-H. Dong, M. Lozada-Cassou, Phys. Lett. A. 337, 313 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. C. Quesne, J. Phys. A. 40, 13107 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. R.N. Costa Filho, M.P. Almeida, G.A. Farias, J.S. Jr. Andrade, Phys. Rev. A. 84, 050102(R) (2011)

    Article  ADS  Google Scholar 

  15. S.H. Mazharimousavi, Phys. Rev. A. 85, 034102 (2012)

    Article  ADS  Google Scholar 

  16. R.N. Costa Filho, G. Alencar, B.-S. Skagerstam, J.S. Andrade Jr., Europhys. Lett. 101, 10009 (2013)

    Article  ADS  Google Scholar 

  17. M.A. Rego-Monteiro, F.D. Nobre, Phys. Rev. A. 88, 032105 (2013)

    Article  ADS  Google Scholar 

  18. G. Bastard. Wave mechanics applied to semiconductor heterostructures (Les Editions de Physique Les Ulis, France, 1988)

    Google Scholar 

  19. C. Weisbuch, B. Vinter. Quantum semiconductor structures: Fundamentals and applications (Academic Press, New York, 1991)

    Google Scholar 

  20. C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)

    Article  ADS  Google Scholar 

  21. F.G. Scholtz, H.B. Geyer, F.J.W. Hahne, Ann. Phys. 213, 74 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. F. Cannata, G. Junker, J. Trost, Phys. Lett. A. 246, 219 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. C.M. Bender, D.C. Brody, H.F. Jones, Phys. Rev. Lett. 89, 270401 (2002)

    Article  MathSciNet  Google Scholar 

  25. A. Mostafazadeh, J. Math. Phys. 43, 2814 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. J. da Providência, N. Bebiano, J.P. da Providência, Braz. J. Phys. 41, 78 (2011)

    Article  ADS  Google Scholar 

  27. S. Dey, A. Fring, Phys. Rev. D. 86, 064038 (2012)

    Article  ADS  Google Scholar 

  28. E.P. Borges, Physica A. 340, 95 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  29. C.G. Bollini, J.J. Giambiagi, Braz. J. Phys. 17, 14 (1987)

    Google Scholar 

  30. J. Barcelos-Neto, N.R.F. Braga, Acta Phys. Pol. B. 20, 205 (1989)

    Google Scholar 

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Acknowledgments

We thank C. Tsallis for fruitful conversations. The partial financial supports from CNPq and FAPERJ (Brazilian agencies) are acknowledged.

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Authors and Affiliations

  1. Centro Brasileiro de Pesquisas Físicas, National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil

    F. D. Nobre & M. A. Rego-Monteiro

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  1. F. D. Nobre
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  2. M. A. Rego-Monteiro
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Correspondence to F. D. Nobre.

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Nobre, F.D., Rego-Monteiro, M.A. Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions. Braz J Phys 45, 79–88 (2015). https://doi.org/10.1007/s13538-014-0277-8

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  • DOI: https://doi.org/10.1007/s13538-014-0277-8

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