# 5.6. Summary¶

In this chapter, we have analysed scattering of particles by a potential
localised in a finite region around some point, which we take as the
origin. The starting point of the quantum mechanical analysis is the
wave function far from the scattering centre, which reads:
The first term represents the incoming wave (the wave vector
is usually taken along the -axis) and the second
term represents the scattered wave whose amplitude can be measured as a
function of and at a detector -- this amplitude is
given as the *differential cross section*:
The
shape of the function is determined by the
interaction potential , which is often taken to be
spherically symmetric: . The total cross section is
the integral of the differential cross section:

The calculation of from the scattering potential
can be performed quite easily provided the interaction potential is
weak. In that case, we can use the exact Green's function solution:
and its *first Born approximation*:
From this last equation, the form of
can readily be derived:
We usually set . For a
spherically symmetric potential, the integral depends only on the length
of this vector. This length is given as
where is the length of the wave
vector of the in- and outgoing waves, and is the scattering
angle.

The Born approximation yields the *exact* result for scattering off a
Coulomb potential (the Rutherford formula):

We have derived the *optical theorem* as a consequence of the
conservation of matter:

We then have approached the quantum scattering problem from a very
different angle: Partial Wave Analysis. In this formalism, we expans the
incoming wave in terms of different , and for each of these, we
separately calculate the scattering waves and their contribution to the
scattering cross sections. The central quantity which contains the
information concerning the scattering of a wave with definite , is
the *phase shift*, . This can be found from solving the radial
Schrödinger equation with the scattering potential , and then
identifying the radial part opf the solution for large as a shifted
spherical Bessel function:
From these
we obtain:
For the total cross
section we find, using the orthonormality relations of the Legendre
polynomials: