Volume 18, Issue 2
Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

Siwei Duo & Yanzhi Zhang

Commun. Comput. Phys., 18 (2015), pp. 321-350.

Published online: 2018-04

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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.

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@Article{CiCP-18-321, author = {}, title = {Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well}, journal = {Communications in Computational Physics}, year = {2018}, volume = {18}, number = {2}, pages = {321--350}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.300414.120215a}, url = {http://global-sci.org/intro/article_detail/cicp/11030.html} }
TY - JOUR T1 - Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well JO - Communications in Computational Physics VL - 2 SP - 321 EP - 350 PY - 2018 DA - 2018/04 SN - 18 DO - http://doi.org/10.4208/cicp.300414.120215a UR - https://global-sci.org/intro/article_detail/cicp/11030.html KW - AB -

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.

Siwei Duo & Yanzhi Zhang. (2019). Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well. Communications in Computational Physics. 18 (2). 321-350. doi:10.4208/cicp.300414.120215a
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