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Jumarie fractional derivative; Mittag-Leffler function; fractional Schrödinger equation; fractional wave function

In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger 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 infinite simal 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 infinite simal 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 greaterthan $\Delta x$ (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.

JOYDIP BANERJEE¹ UTTAM GHOSH² SUSMITA SARKAR² SHANTANU DAS³

1. Uttar Buincha Kajal Hari Primary School, Fulia Buincha, Nadia 741 402, India
2. Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700 009, India
3. Reactor Control Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India

Published

16 April 2017

Manuscript received

4 April 2016

Manuscript revised

4 November 2016

Accepted

10 November 2016

Early published

Unedited version published online

Final version published online

27 March 2017