10. Scattering Theory
The basic idea behind scattering theory is simple: there’s an object that you want to understand. So you throw something at it. By analysing how that something bounces o↵, you can glean information about the object itself.
A very familiar example of scattering theory is called “looking at things”. In this section we’re going to explore what happens when you look at things by throwing a quantum particle at an object.
10.1 Scattering in One Dimension We start by considering a quantum particle moving along a line. The maths here will be simple, but the physics is su ciently interesting to exhibit many of the key ideas.
The object that we want to understand is some poten- V(x) tial V (x). Importantly, the potential is localised to some x region of space which means that V (x) 0asx . ! !±1 An example is shown to the right. We will need the poten- tial to fall-o↵to be suitably fast in what follows although, for now, we won’t be careful about what this means. A Figure 89: quantum particle moving along the line is governed by the Schr¨odinger equation, ~2 d2 + V (x) = E (10.1) 2m dx2 Solutions to this equation are energy eigenstates. They evolve in time as (x, t)= iEt/~ e (x). For any potential, there are essentially two di↵erent kinds of states that we’re interested in. Bound States are states that are localised in some region of space. The wavefunc- • tions are normalisable and have profiles that drop o↵exponentially far from the potential