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Effective Potential from the Generalized Time-Dependent Schrödinger Equation

Author

Listed:
  • Trifce Sandev

    (Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
    Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia)

  • Irina Petreska

    (Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia)

  • Ervin K. Lenzi

    (Departamento de Fisica, Universidade Estadual de Ponta Grossa, Av. Carlos Cavalcanti 4748, 84030-900 Ponta Grossa, PR, Brazil)

Abstract

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.

Suggested Citation

  • Trifce Sandev & Irina Petreska & Ervin K. Lenzi, 2016. "Effective Potential from the Generalized Time-Dependent Schrödinger Equation," Mathematics, MDPI, vol. 4(4), pages 1-9, September.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:4:p:59-:d:79232
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    References listed on IDEAS

    as
    1. B. N. Narahari Achar & Bradley T. Yale & John W. Hanneken, 2013. "Time Fractional Schrodinger Equation Revisited," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-11, July.
    2. Sandev, Trifce & Tomovski, Živorad & Dubbeldam, Johan L.A., 2011. "Generalized Langevin equation with a three parameter Mittag-Leffler noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3627-3636.
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