  One of the motivations for using fractional calculus in physical systems is
due to fact that many times, in the space and time variables we are dealing
which exhibit coarse-grained phenomena, meaning that infinitesimal quantities
cannot be placed arbitrarily to zero-rather they are non-zero with a minimum
length. Especially when we are dealing in microscopic to mesoscopic level of
systems. Meaning if we denote x the point in space and t as point in time; then
the differentials dx (and dt) cannot be taken to limit zero, rather it has
spread. A way to take this into account is to use infinitesimal quantities as
(\Deltax)^\alpha (and (\Deltat)^\alpha) with 0<\alpha<1, which for very-very
small \Deltax (and \Deltat); that is trending towards zero, these 'fractional'
differentials are greater that \Deltax (and \Deltat). That is
(\Deltax)^\alpha>\Deltax. This way defining the differentials-or rather
fractional differentials makes us to use fractional derivatives in the study of
dynamic systems. In fractional calculus the fractional order trigonometric
functions play important role. The Mittag-Leffler function which plays
important role in the field of fractional calculus; and the fractional order
trigonometric functions are defined using this Mittag-Leffler function. In this
paper we established the fractional order Schrodinger equation-composed via
Jumarie fractional derivative; and its solution in terms of Mittag-Leffler
function with complex arguments and derive some properties of the fractional
Schrodinger equation that are studied for the case of particle in one
dimensional infinite potential well.
