Quantum shutter transient solutions and the delay time for the δ potential

The analytical solution to the time-dependent Schroedinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the δ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude