Scattering Theory
Michael Fowler
1/16/08
References:
Baym,
Lectures on Quantum Mechanics
, Chapter 9.
Sakurai,
Modern Quantum Mechanics
, Chapter 7.
Shankar,
Principles of Quantum Mechanics
, Chapter 19.
Introduction
Almost everything we know about nuclei and elementary particles has been discovered in
scattering experiments, from Rutherford’s surprise at finding that atoms have their mass and
positive charge concentrated in almost point-like nuclei, to the more recent discoveries, on a far
smaller length scale, that protons and neutrons are themselves made up of apparently point-like
quarks.
The simplest model of a scattering experiment is given by solving Schrödinger’s equation for a
plane wave impinging on a localized potential.
A potential
V
(
r
) might represent what a fast
electron encounters on striking an atom, or an alpha particle a nucleus.
Obviously, representing
any such system by a potential
is
only a beginning, but in certain energy ranges it is quite
reasonable, and we have to start somewhere!
The basic scenario is to shoot in a stream of particles, all at the same energy, and detect how
many are deflected into a battery of detectors which measure angles of deflection.
We assume
all the ingoing particles are represented by wavepackets of the same shape and size, so we should
solve Schrödinger’s time-dependent equation for such a wave packet and find the probability
amplitudes for outgoing waves in different directions at some later time after scattering has taken
place.
But we adopt a simpler approach: we assume the wavepacket has a well-defined energy
(and hence momentum), so it is
many
wavelengths long.
This means that during the scattering
process it looks a lot like a plane wave, and for a period of time the scattering is time
independent.
We assume, then, that the problem is well approximated by solving the time-
independent
Schrödinger equation with an ingoing plane wave. This is much easier!
All we can detect are outgoing waves far outside the region of scattering.
For an ingoing plane
wave
, the wavefunction
far away from the scattering region
must have the form
ik r
e
.
G
G
()
(, )
ikr
k
e
re
f
r
ψ
θϕ
=+
.
G
G
G
G
where
θ
,
φ
are measured with respect to the ingoing direction.
Note that the
scattering amplitude
( )
,
f
has the dimensions of length.
We don’t worry about overall normalization, because what is relevant is the
fraction
of the
incoming beam scattered in a particular direction, or, to be more precise, into a small solid angle
in the direction
θ
,
φ
.
The ingoing particle current (with the above normalization) is
d
Ω