chapter  10
34 Pages

Scattering Theory

An important application of quantum mechanics that has been used extensively in particle physics is the method of scattering. In principle, one fires a particle at an unknown potential and observes the scattering of the projectile, and from the analysis of the scattering data, one is able to determine what the potential was that it was scattered from. The problem of finding the scattering amplitudes for a given potential is called scattering, and the problem of obtaining the potential from the scattering data is called inverse scattering. In one dimension, the scattering data consists of measuring the transmission and reflection coefficients as a function of k, where the incident particle is represented by a plane wave of the form exp(±ikx). Again, in principle, one needs the transmission amplitude a(k) and the reflection amplitude, b(k), for all k (or all incident energies) in order to reconstruct the potential accurately. In one dimension, if one knows the a(k) and the b(k) for a sufficiently wide range of k, the potential can be reconstructed exactly. In three dimensions, which is the more typical case, the exact analysis is not available, but a series of approximations have been found that enable us to get a good idea of the potential the particle was scattered from. We will treat this more common problem first, beginning with a classical scattering problem, since the results are more useful and more easily obtained, and then treat the exact one-dimensional case which is more rigorous.