Here 0< ηℓ<1, with 0 represent complete absorption of the partial wave and 1 representing purely
elastic scattering. An interesting example of the effect of absorption (or inelastic production of
another state) is theblack disk. The disk has a definite radiusaand absorbs partial waves for
ℓ < ka. If one works out this problem, one finds that there is an inelastic scattering cross section
ofσinel=πa^2. Somewhat surprisingly the total elastic scattering cross sectionisσelas=πa^2. The
disk absorbs part of the beam and there is also diffraction around the sharp edges. That is, the
removal of the outgoing spherical partial waves modifies the planewave to include scattered waves.

For high energies relative to the inverse range of the potential, a partial wave analysis is not helpful
and it is far better to use perturbation theory. TheBorn approximationis valid for high energy
and weak potentials. If the potential is weak, only one or two termsin the perturbation series need
be calculated.

If we work in the usual center of mass system, we have a problem with one particle scattering in a
potential. The incoming plane wave can be written as

ψi(~r) =

1

√

V

ei

~ki·~x

.

The scattered plane wave is

ψf(~r) =

1

√

V

ei

~kf·~x

.

We can use Fermi’s golden rule to calculate the transition rate to firstorder in perturbation theory.

Ri→f=

2 π

̄h

∫

V d^3 ~kf

(2π)^3

|〈ψf|V(~r)|ψi〉|^2 δ(Ef−Ei)

The delta function expresses energy conservation for elastic scattering which we are assuming at
this point. If inelastic scattering is to be calculated, the energy of the atomic state changes and
that change should be included in the delta function and the change inthe atomic state should be
included in the matrix element.

The elastic scattering matrix element is

〈ψf|V(~r)|ψi〉=

1

V

∫

d^3 ~re−i

~kf·~x

V(~r)ei

~ki·~x

=

1

V

∫

d^3 ~re−i

∆~·~x

V(~r) =

1

V

V ̃(∆)~

where∆ =~ ~kf−~ki. We notice that this is just proportional to the Fourier Transformof the potential.

Assuming for now non-relativistic final state particles we calculate

Ri→f =

2 π

̄h

∫

V dΩfkf^2 dkf

(2π)^3

1

V^2

∣

∣

∣V ̃(∆)~

∣

∣

∣

2

δ

(

̄h^2 k^2 f

2 μ

−Ei

)

=

2 π

̄h

1

(2π)^3 V

∫

dΩfk^2 f

∣

∣

∣V ̃(∆)~

∣

∣

∣

(^2) μ
̄h^2 kf

1

4 π^2 ̄h^3 V

∫

dΩfμkf

∣

∣

∣V ̃(∆)~

∣

∣

∣

2

We now need to convert this transition rate to a cross section. Ourwave functions are normalize
to one particle per unit volume and we should modify that so that there is a flux of one particle