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
Author to whom correspondence should be addressed.
Academic Editor: Rui A. C. Ferreira
Received: 24 March 2016 / Revised: 18 April 2016 / Accepted: 21 April 2016 / Published: 4 May 2016
In this paper, we consider the time-dependent Schrödinger equation:
with the Riesz space-fractional derivative of order
in the presence of the linear potential
. 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
, 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
including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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MDPI and ACS Style
Liemert, A.; Kienle, A. Fractional Schrödinger Equation in the Presence of the Linear Potential. Mathematics 2016, 4, 31.
Liemert A, Kienle A. Fractional Schrödinger Equation in the Presence of the Linear Potential. Mathematics. 2016; 4(2):31.
Liemert, André; Kienle, Alwin. 2016. "Fractional Schrödinger Equation in the Presence of the Linear Potential." Mathematics 4, no. 2: 31.
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