A study of fractional Schrödinger equation composed of Jumarie fractional derivative
JOYDIP BANERJEE UTTAM GHOSH SUSMITA SARKAR SHANTANU DAS
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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 BANERJEE^{1} UTTAM GHOSH^{2} ^{} SUSMITA SARKAR^{2} SHANTANU DAS^{3}
Volume 94, 2020
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