Barrier Tunneling and Reflection in the Time and Energy Domains: The Battle of the Exponentials

The issue of quantum barrier crossing and reflection in the time domain is addressed. We find that (1) classically forbidden barrier tunneling and above-barrier reflection are well-defined and important processes in the time domain, (2) classically forbidden processes can overshadow allowed ones when both are present, and (3) classically allowed trajectories in the time domain are not, in general, sufficient to explain tunneling amplitudes in the energy domain. We also make clear the essential distinction of barriers which flatten out at large distance and those which do not.