Infinite plane wave evolution in a 1-D square quantum barrier

We analytically compute the time evolution of an initial infinite plane wave in the presence of a 1-dimensional square quantum barrier. This calculation generalizes the analysis of the shutter problem and sets the basis for the calculation of the transmission of general wave packets, aiming to work out the explicit contribution of the resonant (Gamow) states. The method relies mainly on the analytical properties of the Green function. The role of separate boundary conditions on the Green function and on the evolution equation is highlighted. As in previous works on related problems, only the determination of the resonant momenta requires numerical methods.