Note on the Transmission and Reflection of Wave Packets by Potential Barriers

In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V (x) exceeds the value of the total energy E, attention has been confined to wave functions of the form f (x, E) exp (-2h). In the present note wave packets are considered, instead of these trains of waves. The function V (x) is taken as follows: V (x), =0 for xa, ={V₀>0 for 0<x<a. } A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.