# Continuous Eigenvalues and Collision Theory

### From FSUPhysicsWiki

In this chapter we shall investigate problems connected with a particle which, coming from infinity, encounters or "collides with" some atomic system and, after being scattered through a certain angle, goes off to infinity again. The atomic system which does the scattering is called the *scatterer*. We thus have a dynamical system composed of an incident particle and a scatterer interacting with each other, which we must deal with according to the laws of quantum mechanics, and for which we must, in particular, calculate the probability of scattering through any given angle. The scatterer is usually assumed to be of infinite mass and to be at rest throughout the scattering process. The problem was first solved by Max Born.

We must take into account the possibility that the scatterer, considered as a system by itself, may have a number of different stationary states and that if it is initially in one of these states when the particle arrives from infinity, it may be left in a different one when the particle goes off to infinity again. The colliding particle may thus induce transitions in the scatterer. However, to calculate the transition probability, we need to use perturbation theory, which is one the topics in *Quantum Mechanics B*. Hence in this chapter we will deal with simple scattering problems (i.e. processes involving no transitions) only.