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Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

Part of: General mathematical topics and methods in quantum theory Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems Applications to specific types of physical systems Eigenvalue problems

Published online by Cambridge University Press:  30 July 2015

Siwei Duo  and
Yanzhi Zhang
Siwei Duo
Affiliation:
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
Yanzhi Zhang*
Affiliation:
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
*
*Corresponding author. Email addresses: sddy9@mst.edu (S. Duo), zhangyanz@mst.edu (Y. Zhang)
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Abstract

In this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

Keywords

Fractional Schrödinger equationRiesz fractional Laplacianinfinite potential wellground statesfirst excited statesquadrature rule

MSC classification

Secondary: 35Q40: PDEs in connection with quantum mechanics 45C05: Eigenvalue problems 65M06: Finite difference methods 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 82D05: Gases
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 321 - 350
Copyright
Copyright © Global-Science Press 2015 

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