  We study non-relativistic propagation of Gaussian wave packets in
one-dimensional Eckart potential, a barrier, or a well. In the picture used,
the transmitted wave packet results from interference between the copies of the
freely propagating state with different spatial shifts (delays), x', induced by
the scattering potential. The Uncertainty Principle precludes relating the
particle's final position to the delay experienced in the potential, except in
the classical limit. Beyond this limit, even defining an effective range of the
delay is shown to be an impracticable task, owing to the oscillatory nature of
the corresponding amplitude distribution. Our examples include the classically
allowed case, semiclassical tunnelling, delays induced in the presence of a
virtual state, and scattering by a low barrier. The properties of the amplitude
distribution of the delays, and its pole representation are studied in detail.
