130_notes.dvi

As an example, this has been used to compute the cross section forscattering from a spherical
potential well (See section 15.7) assuming only theℓ= 0 phase shift was significant. By matching
the boundary conditions at the boundary of the spherical well, we determined the phase shift.

tanδ 0 =−

C

B

=

kcos(ka) sin(k′a)−k′cos(k′a) sin(ka)

ksin(ka) sin(k′a) +k′cos(k′a) cos(ka)

The differential cross section is
dσ
dΩ

→

sin^2 (δ 0 )
k^2
which will have zeros if
k′cot(k′a) =kcot(ka).

We can compute the total scattering cross section using the relation

∫

dΩPℓ(cosθ)Pℓ′(cosθ) =

4 π

2 ℓ+1δℓℓ′.

σtot =

∫

dΩ|f(θ,φ)|^2

=

∫

dΩ

[

1

k

∑

ℓ

(2ℓ+ 1)eiδℓ(k)sin(δℓ(k))Pℓ(cosθ)

][

1

k

∑′

ℓ

(2ℓ′+ 1)e−iδ

ℓ′(k)

sin(δℓ′(k))Pℓ′(cosθ)

]

=

4 π

k^2

∑

ℓ

(2ℓ+ 1) sin(δℓ(k))^2

It is interesting that we can relate the total cross section to the scattering amplitude atθ= 0, for
whichPℓ(1) = 1.

f(θ= 0,φ) =

1

k

∑

ℓ

(2ℓ+ 1)eiδℓ(k)sin(δℓ(k))

Im[f(θ= 0,φ)] =

1

k

∑

ℓ

(2ℓ+ 1) sin^2 (δℓ(k))

σtot =

4 π

k

Im[f(θ= 0,φ)]

The total cross section is related to the imaginary part of the forward elastic scattering amplitude.
This seemingly strange relation is known as theOptical Theorem. It can be understood in terms
of removal of flux from the incoming plane wave. Remember we have an incoming plane wave plus
scattered spherical waves. The total cross section corresponds to removal of flux from the plane wave.
The only way to do this is destructive interference with the scattered waves. Since the plane wave is
atθ= 0 it is only the scattered amplitude atθ= 0 that can interfere. It is therefore reasonable that
a relation like the Optical Theorem is correct, even when elastic and inelastic processes are possible.

We have not treated inelastic scattering. Inelastic scatteringcan be a complex and interesting
process. It was with high energy inelastic scattering of electrons from protons that the quark
structure of the proton was “seen”. In fact, the electrons appeared to be scattering from essentially
free quarks inside the proton. The proton was broken up into sometimes many particles in the
process but the data could be simply analyzed using the scatter electron. In a phase shift analysis,
inelastic scattering removes flux from the outgoing spherical waves.

lim

r→∞

ψ=−

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓ

1

2 ikr

(

e−i(kr−ℓπ/2)−ηℓ(k)e^2 iδℓ(k)ei(kr−ℓπ/2)

)

Yℓ 0