9. One-Dimensional Scattering Theory
In classical as well as in quantum mechanics the motion of particles in external fields falls into one of two categories: Either the motion stays within an approximately finite region for all times or the particle escapes towards infinity and behaves asymptotically like a free particle. Here we treat the latter case.
We only consider the simplest case of a scattering process in one dimension. A particle is shot towards a "target" which is represented by a force field with a finite range. As usual, we are interested in the temporal behavior of wave packets (and their Fourier transforms). As in the case of free particles, the wave packets are formed as superpositions of solutions with a well-defined energy.
These energy eigenfunctions are not square-integrable. Asymptotically they are made of pieces which represent incoming and outgoing plane waves to the left and to the right of the target.
A wave packet hitting a target usually gets dispersed into all directions. In one dimension, the particle can be either reflected or transmitted. The probabilities for these events are determined by the reflection and transmission coefficients. We use the energy representation derived in Chapter 1 to explain how these scattering coefficients determine the asymptotic behavior of the wave packets. In a few cases, explicit expressions for the scattering coefficients can be derived. The most common examples are rectangular steps, barriers, and wells.
Again we have numerous opportunities to point out differences between classical and quantum mechanics. A particularly striking phenomenon is the tunnel effect, the ability of a quantum particle to pass through a repulsive barrier, even if the energy is to low to allow a transition in classical mechanics. The tunnel effect is used in the scanning tunneling microscope to obtain images of solid surfaces with a resolution showing single atoms (see the gallery on the CD).