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

• Published in: Applied Mechanics Vol. 3; no. 2; pp. 413 - 463
• Main Author: Blackledge, Jonathan
• Format: Journal Article
• Language: English
• Published: Zwijnaarde MDPI AG 01.06.2022
• Subjects: Calculus; diffraction in time; fractional diffusion equation; fractional Klein–Gordon equation; fractional quantum mechanics; fractional Schrödinger equation; Partial differential equations; Quantum physics; quantum shutter problem
• ISSN: 2673-3161; 2673-3161
• DOI: 10.3390/applmech3020026

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.