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Infinite Square-Well, Trigonometric Pöschl-Teller and Other Potential Wells with a Moving Barrier

  • Alonso Contreras-AstorgaEmail author
  • Véronique Hussin
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Using mainly two techniques, a point transformation and a time dependent supersymmetry, we construct in sequence several quantum infinite potential wells with a moving barrier. We depart from the well-known system of a one-dimensional particle in a box. With a point transformation, an infinite square-well potential with a moving barrier is generated. Using time dependent supersymmetry, the latter leads to a trigonometric Pöschl-Teller potential with a moving barrier. Finally, a confluent time dependent supersymmetry transformation is implemented to generate new infinite potential wells, all of them with a moving barrier. For all systems, solutions of the corresponding time dependent Schrödinger equation fulfilling boundary conditions are presented in a closed form.

Keywords

Infinite square-well potential Pöschl-Teller potential Supersymmetry Point transformation Moving barrier 

Notes

Acknowledgements

This work has been supported in part by research grants from Natural sciences and engineering research council of Canada (NSERC). ACA would like to thank the Centre de Recherches Mathématiques for kind hospitality.

References

  1. 1.
    J. Crank, Free and Moving Boundary Problems (Clarendon, Oxford, 1984)zbMATHGoogle Scholar
  2. 2.
    E. Fermi, On the origin of the cosmic radiation. Phys. Rev. 75, 1169–1174 (1949)ADSCrossRefGoogle Scholar
  3. 3.
    S.M. Ulam, On some statistical properties of dynamical systems, in Proceedings Fourth Berkeley Symposium on Mathematical Statistics and Problem, vol. 3 (University of California Press, Berkeley, CA, 1961), pp. 315–320Google Scholar
  4. 4.
    J.R. Ray, Exact solutions to the time-dependent Schrödinger equation. Phys. Rev. A 26, 729–733 (1982)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    G.W. Bluman, On mapping linear partial differential equations to constant coefficient equations. SIAM J. Appl. Math. 43, 1259–1273 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Cooper, A. Khare, U. Sukhatme, Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–385 (1995)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    D.J. Fernández C., Supersymmetric quantum mechanics. AIP Conf. Proc. 1287, 3–36 (2010)Google Scholar
  8. 8.
    V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)CrossRefGoogle Scholar
  9. 9.
    V.G. Bagrov, B.F. Samsonov, L.A. Shekoyan, Darboux transformation for the nonsteady Schrödinger equation. Russ. Phys. J. 38, 706–712 (1995)CrossRefGoogle Scholar
  10. 10.
    A. Contreras-Astorga, A time-dependent anharmonic oscillator. IOP Conf. Series J. Phys. Conf. Ser. 839, 012019 (2017)CrossRefGoogle Scholar
  11. 11.
    K. Zelaya, O. Rosas-Ortiz, Exactly solvable time-dependent oscillator-like potentials generated by Darboux transformations. IOP Conf. Series J. Phys. Conf. Ser. 839, 012018 (2017)CrossRefGoogle Scholar
  12. 12.
    F. Finkel, A. González-López, N. Kamran, M.A. Rodríguez, On form-preserving transformations for the time-dependent Schrödinger equation. J. Math. Phys. 40, 3268–3274 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, vol. I (Wiley/Hermann, Paris, 1977)zbMATHGoogle Scholar
  14. 14.
    R. Shankar, Principles of Quantum Mechanics (Plenum, New York, 1994)CrossRefGoogle Scholar
  15. 15.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics Non-Relativistic Theory (Pergamon, Exeter, 1991)zbMATHGoogle Scholar
  16. 16.
    A. Schulze-Halberg, B. Roy, Time dependent potentials associated with exceptional orthogonal polynomials. J. Math. Phys. 55, 123506 (2014)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    S.W. Doescher, M.H. Rice, Infinite square-well potential with a moving wall. Am. J. Phys. 37, 1246–1249 (1969)ADSCrossRefGoogle Scholar
  18. 18.
    D.N. Pinder, The contracting square quantum well. Am. J. Phys. 58, 54–58 (1990)ADSCrossRefGoogle Scholar
  19. 19.
    T.K. Jana, P. Roy, A class of exactly solvable Schrödinger equation with moving boundary condition. Phys. Lett. A 372, 2368–2373 (2008)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    M.L. Glasser, J. Mateo, J. Negro, L.M. Nieto, Quantum infinite square well with an oscillating wall. Chaos, Solitons Fractals 41, 2067–2074 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    O. Fojón, M. Gadella, L.P. Lara, The quantum square well with moving boundaries: a numerical analysis. Comput. Math. Appl. 59, 964–976 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Contreras-Astorga, D.J. Fernández C., Supersymmetric partners of the trigonometric Pöschl–teller potentials. J. Phys. A Math. Theor. 41, 475303 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    D.J. Fernández C., E. Salinas-Hernández, The confluent algorithm in second-order supersymmetric quantum mechanics. J. Phys. A Math. Theor. 36, 2537–2543 (2003)Google Scholar
  24. 24.
    D.J. Fernández, V. Hussin, O. Rosas-Ortiz, Coherent states for Hamiltonians generated by supersymmetry. J. Phys. A Math. Theor. 40, 6491 (2007)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    M.-A. Fiset, V. Hussin, Supersymmetric infinite wells and coherent states. J. Phys. Conf. Ser. 624, 012016 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Cátedras CONACYT—Departamento de Física, CinvestavCiudad de MéxicoMexico
  2. 2.Department of PhysicsIndiana University NorthwestGaryUSA
  3. 3.Centre de Recherches Mathématiques & Departement de Mathématiques et de StatistiqueUniversité de MontréalMontréalCanada

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