Open AccessHigh-oder symplectic FDTD scheme for solving time-dependent Schrdinger equation
Author(s) -
Jing Shen,
Wei E. I. Sha,
Zhixiang Huang,
Mingsheng Chen,
Xiaoxia Wu
Publication year - 2012
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.199
H-Index - 47
ISSN - 1000-3290
DOI - 10.7498/aps.61.190202
Subject(s) - finite difference time domain method , symplectic geometry , symplectic integrator , discretization , stability (learning theory) , mathematics , integrator , dispersion (optics) , scheme (mathematics) , mathematical analysis , computer science , physics , quantum mechanics , symplectic manifold , voltage , machine learning
Using three-order symplectic integrators and fourth-order collocated spatial differences, a high-order symplectic finite-difference time-domain (SFDTD(3, 4)) scheme is proposed to solve the time-dependent Schrdinger equation. First, high-order symplectic framework for discretizing the Schrdinger equation is described. The numerical stability and dispersion analyses are provided for the FDTD(2, 2), FDTD(2, 4) and SFDTD(3, 4) schemes. The results are demonstrated in terms of theoretical analyses and numerical simulations. The spatial high-order collocated difference reduces the stability that can be improved by the high-order symplectic integrators. The SFDTD(3, 4) scheme and FDTD(2, 4) approach show better numerical dispersion than the traditional FDTD(2, 2) method. The simulation results of a two-dimensional quantum well and harmonic oscillator strongly confirm the advantages of the SFDTD(3, 4) scheme over the traditional FDTD(2, 2) method and other high-order approaches. The explicit SFDTD(3, 4) scheme, which is high-order-accurate and energy-conserving, is well suited for long-term simulation.