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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 290216, 11 pages

http://dx.doi.org/10.1155/2013/290216

## Time Fractional Schrodinger Equation Revisited

Physics Department, University of Memphis, Memphis, TN 38152, USA

Received 29 April 2013; Revised 1 July 2013; Accepted 2 July 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 B. N. Narahari Achar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 . The action 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.