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

Abstract

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 $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function $V(x)$ is taken as follows: ${V(x),=0 \mathrm{for} xl0 \mathrm{and}\mathrm{for} xga,}{={V}_{0}g0 \mathrm{for} 0lxla.}$ 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.