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Mathematics 2016, 4(2), 31; doi:10.3390/math4020031

Fractional Schrödinger Equation in the Presence of the Linear Potential

Institut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr. 12, D-89081 Ulm, Germany
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Academic Editor: Rui A. C. Ferreira
Received: 24 March 2016 / Revised: 18 April 2016 / Accepted: 21 April 2016 / Published: 4 May 2016
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Abstract

In this paper, we consider the time-dependent Schrödinger equation: i ψ ( x , t ) t = 1 2 ( Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x R , t > 0 with the Riesz space-fractional derivative of order 0 < α 2 in the presence of the linear potential V ( x ) = β x . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1 , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential V ( ρ ) = F · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach. View Full-Text
Keywords: Riesz fractional derivative; Caputo fractional derivative; Mittag-Leffler matrix function; fractional Schrödinger equation Riesz fractional derivative; Caputo fractional derivative; Mittag-Leffler matrix function; fractional Schrödinger equation
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Liemert, A.; Kienle, A. Fractional Schrödinger Equation in the Presence of the Linear Potential. Mathematics 2016, 4, 31.

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