Solving the time-dependent Schrödinger equation using complex-coordinate contours

The complex-coordinates approach in scattering theory is extended to the time-dependent Schr\"odinger equation by using Simon's exterior scaling contour, and it is shown to cause outgoing wave packets to vanish for large distances in collision problems. Numerical applications are given for one- and two-dimensional problems where the complex-coordinates method is shown to eliminate reflections of outgoing wave packets from the boundaries of the coordinate space grid used in the calculations. The extension of the exterior scaling approach to the time-dependent description of a charged particle in dc and ac electromagnetic fields is given, and numerical results on these problems are also presented.