Transient effects and reconstruction of the energy spectra in the time evolution of transmitted Gaussian wave packets

We derive an exact analytical solution to the time-dependent Schr\"odinger equation for transmission of a Gaussian wave packet through an arbitrary potential of finite range. We consider the situation where the initial Gaussian wave packet is sufficiently broad in momentum space to guarantee that the resonance structure of the system is included in the dynamical description. We demonstrate that the transmitted wave packet exhibits a transient behavior which at very large distances and long times may be written as the free evolving Gaussian wave packet solution times the transmission amplitude of the system and hence it reproduces the resonance spectra of the system. This is a novel result that predicts the ultimate fate of the transmitted Gaussian wave packet. We also prove that at a fixed distance and very long times the solution goes as $t^{-3/2}$ which extends to arbitrary finite range potentials previous analysis on this issue. Our results are exemplified for single and multibarrier systems.