Next Article in Journal
New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties
Next Article in Special Issue
Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations
Previous Article in Journal
Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

Article

# Fractional Schrödinger Equation in the Presence of the Linear Potential

by * and
Institut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr. 12, D-89081 Ulm, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Rui A. C. Ferreira
Mathematics 2016, 4(2), 31; https://doi.org/10.3390/math4020031
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)
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
Show Figures

Graphical abstract

MDPI and ACS Style

Liemert, A.; Kienle, A. Fractional Schrödinger Equation in the Presence of the Linear Potential. Mathematics 2016, 4, 31. https://doi.org/10.3390/math4020031

AMA Style

Liemert A, Kienle A. Fractional Schrödinger Equation in the Presence of the Linear Potential. Mathematics. 2016; 4(2):31. https://doi.org/10.3390/math4020031

Chicago/Turabian Style

Liemert, André, and Alwin Kienle. 2016. "Fractional Schrödinger Equation in the Presence of the Linear Potential" Mathematics 4, no. 2: 31. https://doi.org/10.3390/math4020031

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

1