  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.
