  Time-dependent quantum mechanics provides an intuitive picture of particle
propagation in external fields. Semiclassical methods link the classical
trajectories of particles with their quantum mechanical propagation. Many
analytical results and a variety of numerical methods have been developed to
solve the time-dependent Schroedinger equation. The time-dependent methods work
for nearly arbitrarily shaped potentials, including sources and sinks via
complex-valued potentials. Many quantities are measured at fixed energy, which
is seemingly not well suited for a time-dependent formulation. Very few methods
exist to obtain the energy-dependent Green function for complicated potentials
without resorting to ensemble averages or using certain lead-in arrangements.
Here, we demonstrate in detail a time-dependent approach, which can accurately
and effectively construct the energy-dependent Green function for very general
potentials. The applications of the method are numerous, including chemical,
mesoscopic, and atomic physics.
