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Quantum Mechanics and Control Using Fractional Calculus: A Study of the Shutter Problem for Fractional Quantum Fields

by 1,2,3,4,5,†
1
Science Foundation Ireland, Three Park Place, Hatch Street Upper, Saint Kevin’s, D02 FX65 Dublin, Ireland
2
Centre for Advanced Studies, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland
3
Department of Computer Science, University of Western Cape, Robert Sobukwe Road, Bellville, Cape Town 7535, South Africa
4
Faculty of Arts, Science and Technology, Wrexham Glyndŵr University of Wales, Mold Road, Wrexham LL11 2AW, UK
5
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, University Road, Westville, Durban 3629, South Africa
Current address: School of Electrical and Electronic Engineering, Technological University Dublin, Grangegorman, D07 EWV4 Dublin, Ireland.
Appl. Mech. 2022, 3(2), 413-463; https://doi.org/10.3390/applmech3020026
Received: 7 February 2022 / Revised: 9 March 2022 / Accepted: 16 March 2022 / Published: 12 April 2022
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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. View Full-Text
Keywords: fractional quantum mechanics; quantum shutter problem; diffraction in time; fractional diffusion equation; fractional Schrödinger equation; fractional Klein–Gordon equation fractional quantum mechanics; quantum shutter problem; diffraction in time; fractional diffusion equation; fractional Schrödinger equation; fractional Klein–Gordon equation
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MDPI and ACS Style

Blackledge, J. Quantum Mechanics and Control Using Fractional Calculus: A Study of the Shutter Problem for Fractional Quantum Fields. Appl. Mech. 2022, 3, 413-463. https://doi.org/10.3390/applmech3020026

AMA Style

Blackledge J. Quantum Mechanics and Control Using Fractional Calculus: A Study of the Shutter Problem for Fractional Quantum Fields. Applied Mechanics. 2022; 3(2):413-463. https://doi.org/10.3390/applmech3020026

Chicago/Turabian Style

Blackledge, Jonathan. 2022. "Quantum Mechanics and Control Using Fractional Calculus: A Study of the Shutter Problem for Fractional Quantum Fields" Applied Mechanics 3, no. 2: 413-463. https://doi.org/10.3390/applmech3020026

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