Symplectic Pseudospectral Time-Domain Scheme for Solving Time-Dependent Schrodinger Equation

TL;DR

This paper introduces a symplectic pseudospectral time-domain scheme for solving the Schrödinger equation, combining Fourier-based spatial derivatives with high-order symplectic integrators for accurate, energy-conserving long-term simulations.

Contribution

The paper develops a novel SPSTD scheme that uses Fourier transforms and symplectic integrators, offering improved accuracy and energy conservation over traditional methods.

Findings

The SPSTD scheme outperforms traditional PSTD and FDTD methods.

It provides infinite order spatial accuracy and energy conservation in time.

Numerical tests on quantum well and harmonic oscillator validate its effectiveness.

Abstract

A symplectic pseudospectral time-domain (SPSTD) scheme is developed to solve Schrodinger equation. Instead of spatial finite differences in conventional finite-difference time-domain (FDTD) method, the fast Fourier transform is used to calculate the spatial derivatives. In time domain, the scheme adopts high-order symplectic integrators to simulate time evolution of Schrodinger equation. A detailed numerical study on the eigenvalue problems of 1D quantum well and 3D harmonic oscillator is carried out. The simulation results strongly confirm the advantages of the SPSTD scheme over the traditional PSTD method and FDTD approach. Furthermore, by comparing to the traditional PSTD method and the non-symplectic Runge-Kutta (RK) method, the explicit SPSTD scheme which is an infinite order of accuracy in space domain and energy-conserving in time domain, is well suited for a long-term simulation.