A Study of Fractional Schrodinger Equation-composed via Jumarie fractional derivative

TL;DR

This paper derives a fractional Schrödinger equation using Jumarie fractional derivatives, explores its solutions with Mittag-Leffler functions, and analyzes properties for a particle in a one-dimensional infinite potential well.

Contribution

The paper introduces a fractional Schrödinger equation based on Jumarie fractional derivatives and studies its solutions and properties in quantum systems.

Findings

Solution expressed in Mittag-Leffler functions with complex arguments.

Properties of the fractional Schrödinger equation analyzed for a particle in a potential well.

Highlights the role of fractional calculus in quantum mechanics.

Abstract

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.