Principle of stationary phase for propagating wave packets in the unidimensional scattering problem

Abstract

We point out some incompatibilities which appear when one applies the stationary phase method for deriving phase times to obtain the spatial localization of wave packets scattered by a unidimensional potential barrier. We concentrate on the above barrier diffusion problem where the wave packet collision implies the possibility of multiple reflected and transmitted wave packets, which, depending on the boundary conditions, can overlap or stand in relative separation in space. We demonstrate that the indiscriminate use of the method for such a particular configuration leads to paradoxical results for which the correct interpretation, confirmed by analytical/numerical calculations, imposes the necessity of the appearance of multiple peaks as a consequence of multiple reflections by the barrier steps.