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Reports on Mathematical Physics

Nonlocal Phenomena in Quantum Mechanics with Fractional Calculus

Kazım Gökhan Atman, Hüseyin Şirin

Published Oct 1, 2020• 28 citations
Publisher
DOI
10.1016/s0034-4877(20)30075-6

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Applied Mechanics•Apr 12, 2022•11 citations
We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.
Our research meticulously navigates the realms of Fractional Quantum Mechanics (FQM), focusing on a critical examination of both numerical and analytical methods that harness the potential of fractional calculus to illuminate the quantum world's complexities. By embarking on this scholarly journey, we aim to decode the intricate dynamics that fractional equations reveal about quantum systems, pushing the boundaries of conventional quantum mechanics. This comparative study meticulously evaluates the efficacy and insights provided by these two distinct approaches, highlighting their contributions to a deeper understanding of quantum phenomena. As we traverse through the layers of quantum dynamics, our work seeks to contribute significantly to the theoretical framework of FQM, offering innovative perspectives and methodologies. The essence of this research lies in its potential to forge new theoretical pathways and inspire further exploration in the quantum domain, leveraging the unique capabilities of fractional calculus. By enriching the scientific community with our findings, we aspire to open new horizons in the study of quantum mechanics, marking a step forward in the ongoing quest to unravel the mysteries of the quantum universe.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.