%0 Journal Article
%T Time Fractional Schrodinger Equation Revisited
%A B. N. Narahari Achar
%A Bradley T. Yale
%A John W. Hanneken
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/290216
%X 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. 1. Introduction There has been an explosive research output in recent years in the application of methods of fractional calculus [1¨C13] to the study of quantum phenomena [14¨C42]. The well-known Schrodinger equation with a first-order derivative in time and second-order derivatives in space coordinates was given by Schrodinger as an Ansatz. The Schrodinger equation has been generalized to (i) a space fractional Schrodinger equation involving noninteger order space derivatives but retaining first-order time derivative [14¨C18], (ii) a time fractional Schrodinger equation involving non-integer order time derivative but retaining the second-order space derivatives [19], or (iii) more general fractional Schrodinger equation where both time and space derivatives are of non-integer order [20¨C26]. The fractional Schrodinger equation has also been obtained by using a fractional generalization of the Laplacian operator [20] and by using a fractional variation principle and a fractional Klein-Gordon equation [36]. In all these cases the fractional derivatives employed have been the regular fractional derivatives of the Riemann-Liouville type or the Caputo type (generally used in physical applications with initial conditions) which are both nonlocal in nature. The fractional derivative which is nonlocal by definition can be made ¡°local¡± by a limiting process as
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