130_notes.dvi

Theδℓ(k) is called thephase shiftfor the partial wave of angular momentumℓ. We can compute
thedifferential cross sectionfor scattering

dσ

dΩ

≡

scattered flux intodΩ

incident flux

in terms of the phase shifts.

dσ

dΩ

=

1

k^2











∑∞

ℓ=0

(2ℓ+ 1)eiδℓsin(δℓ)Pℓ(cosθ)











2

The phase shifts must be computed by actually solving the problem for the particular potential.

In fact, for low energy scattering and short range potentials, the first termℓ= 0 is often enough to
solve the problem.

e

ikz

V 0

1

1

2

2

3

3

4

4

only small l waves get near the

Only the lowℓpartial waves get close enough to the origin to be affected by the potential.

15.7 Scattering from a Spherical Well*

For the scattering problem, the energy is greater than zero. We must choose the Bessel function in
the region containing the origin.

Rℓ=Ajℓ(k′r)

k′=

√

2 μ(E+V 0 )

̄h^2

For larger, we can have a linear combination of functions.

Rℓ=Bjℓ(kr) +Cnℓ(kr)

k=

√

2 μE

̄h^2