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Time Fractional Schrodinger Equation Revisited

Author

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  • B. N. Narahari Achar
  • Bradley T. Yale
  • John W. Hanneken

Abstract

The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle” obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have “fractional” dimensions, chosen such that the “energy” has the correct dimension [ML2/T2]. The action S is defined as a fractional time integral of the Lagrangian, and a “fractional Planck constant” is introduced. The TFSE corresponds to a “subdiffusion” equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber’s work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a “free particle” is obtained. The total probability for a “free” particle is shown to go to zero in the limit of infinite time, in contrast with Naber’s result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.

Suggested Citation

  • B. N. Narahari Achar & Bradley T. Yale & John W. Hanneken, 2013. "Time Fractional Schrodinger Equation Revisited," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlamp:v:2013:y:2013:i:1:n:290216
    DOI: 10.1155/2013/290216
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    References listed on IDEAS

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    1. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    2. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    3. De Paoli, A.L. & Rocca, M.C., 2013. "The fractionary Schrödinger equation, Green functions and ultradistributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 111-122.
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    Cited by:

    1. Lorin, Emmanuel & Nhan, Howl, 2025. "Data-driven fractional algebraic system solver," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 170-182.

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