A study of fractional Schrodinger equation composed of Jumarie fractional derivative

In this paper we have derived the fractional-order Schrodinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrodinger equation is obtained in terms of Mittag-Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrodinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero - rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Delta x)(alpha) (and (Delta t)(alpha)) with 0 < alpha < 1; called as 'fractional differentials'. For arbitrarily small Delta x and Delta t (tending towards zero), these 'fractional' differentials are greater than Delta x (and Delta t), i.e. (Delta x)(alpha) > Delta x and (Delta t)(alpha) > Delta t. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.