High-Order Symplectic FDTD Scheme for Solving Time-Dependent Schrödinger Equation

Using the three-order symplectic integrators and fourth-order collocated spatial differences, a high-order symplectic finite-difference time-domain (SFDTD) scheme is proposed to solve the time-dependent Schrödinger equation. First, the high-order symplectic framework for discretizing Schrödinger equation is described. Then the numerical stability and dispersion analyses are provided for the FDTD(2,2), higher-order FDTD(2,4) and SFDTD(3,4) schemes. Next, to implement the Dirichlet boundary condition encountered in the quantum eigenvalue problem, the image theory and one-sided difference technique are manipulated particularly for high-order collocated differences. Finally, a detailed numerical study on 1D and 2D quantum eigenvalue problems is carried out. The simulation results of quantum wells and harmonic oscillators strongly confirm the advantages of the SFDTD scheme over the traditional FDTD method and other high-order approaches. The explicit SFDTD scheme, which is high-order-accurate and energy-conserving, is well suited for a long-term simulation and can save computer resources with large time step and coarse spatial grids. Keywords—symplectic integrators; high-order collocated differences; Schrödinger equation; numerical stability and dispersion.