**
Quantum Scattering
**

**Born approximation**:
incident particle is a plane wave and scattered amplitude is a
spherical wave.

In general, the differential cross-section can be written as

where f(**q**) is essentially the Fourier transform
of the scattering potential

If the scattering is spherically symmetric

If V(r)~1/r, then simple dimensional
counting requires f(q^{2})~1/q^{2}.

Rutherford scattering can be analysed either classically or quantum mechanically.

In general, if there is no range or size scale in a scattering problem, then dimensional analysis requires

Which is very different from the the differential cross section expected from billiard ball scattering (ds/dW µ constant).

For electromagnetic scattering of incident point particles, we can consider scattering from an extended target as point-point scattering integrated over the charge distribution of the target:

The **Form Factor
**is the Fourier transform of the target's
charge distribution:

and (ds/dW)_{R}
is the reference cross section for poin-point scattering (e.g.
For electron scattering this is usually Mott scattering, but Rutherford
scattering can be used in the low energy limit.)

** **

** **

Comparison of differential cross sections for scattering
a spinless point particle from either a Rutherford point particle
or particles with a size R=10/q_{max}.

** **

** **

**Electron-proton scattering**

F&H Figure 6.13: the proton is not a point electric charge, but has a size ~1fm.

F&H Figure 6.5: atomic nuclei have a size and a reasonable well defined edge.

F&H Figure 6.19:
protons do not have a well defined edge, but inside the proton
there seem to be point charge constituents.