  A fast and stable method is formulated to compute the time evolution of a
wavefunction by numerically solving the time-dependent Schr{\"o}dinger
equation. This method is a real space/real time evolution method implemented by
several computational techniques such as Suzuki's exponential product, Cayley's
form, the finite differential method and an operator named adhesive operator.
This method conserves the norm of the wavefunction, manages periodic conditions
and adaptive mesh refinement technique, and is suitable for vector- and
parallel-type supercomputers. Applying this method to some simple electron
dynamics, we confirmed the efficiency and accuracy of the method for simulating
fast time-dependent quantum phenomena.
