130_notes.dvi

equation as we had for 1D odd solution to square well.

E

-V^0

0 -V 0 +π^2 h^2 /8ma^2

f

s if

15.6 Partial Wave Analysis of Scattering*

We can take a quick look atscattering from a potential in 3DWe assume thatV= 0 far from
the origin so the incoming and outgoing waves can be written in terms of our solutions for a constant
potential.

In fact, anincoming plane wave along thezdirectioncan be expanded in Bessel functions.

eikz=eikrcosθ=

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓjℓ(kr)Yℓ 0

Each angular momentum (ℓ) term is called apartial wave. The scattering for each partial wave
can be computed independently.

For largerthe Bessel function becomes

jℓ(ρ)→−

1

2 ikr

(

e−i(kr−ℓπ/2)−ei(kr−ℓπ/2)

)

,

so our plane wave becomes

eikz→−

∑∞

ℓ=0

√

4 π(2ℓ+ 1)iℓ

1

2 ikr

(

e−i(kr−ℓπ/2)−ei(kr−ℓπ/2)

)

Yℓ 0

The scattering potential will modify the plane wave, particularly theoutgoing part. To maintain
the outgoing flux equal to the incoming flux, the most the scattering can do ischange the relative
phase of the incoming an outgoing waves.

Rℓ(r)→−

1

2 ikr

(

e−i(kr−ℓπ/2)−e^2 iδℓ(k)ei(kr−ℓπ/2)

)

=

sin(kr−ℓπ/2 +δℓ(k))

kr

eiδℓ(k)