To scatter a photon the field must act twice, once to annihilate the initial state photon and once
to create the final state photon. Since the quantized field contains both creation and annihilation
operators,
Aμ(x) =1
√
V
∑
kα√
̄hc^2
2 ω
ǫ(μα)(
ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ)
either theA^2 term in first order, or theA~·~pterm in second order can contribute to scattering. Both
of these amplitudes are of ordere^2.
The matrix element of theA^2 term to go from a photon of wave vector~kand an atomic stateito a
scattered photon of wave vectork~′and an atomic statenis particularly simple since it contains no
atomic coordinates or momenta.
e^2
2 mc^2〈n;~k′ˆǫ(α′)
|A~·A~|i;~kˆǫ(α)〉 =
e^2
2 mc^21
V
̄hc^2
√
ω′ωǫ(μα)ǫ(α′)
μ e−i(ω−ω′)tδ
niThe second order terms can change atomic states because of the~poperator.
The cross section for photon scattering is then given by the
dσ
dΩ=
(
e^2
4 πmc^2) 2 (
ω′
ω)
∣ ∣ ∣ ∣ ∣ ∣
δniˆǫ·ˆǫ′−1
m ̄h∑
j[
〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉
ωji−ω+
〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉
ωji+ω′]
∣ ∣ ∣ ∣ ∣ ∣
2Kramers-Heisenberg Formula. The three terms come from the three Feynman diagrams that
contribute to the scattering to ordere^2.
This result can be specialized for the case of elastic scattering, withthe help of some commutators.
dσelas
dΩ=
(
e^2
4 πmc^2) (^2) (
mω
̄h
) 2
∣ ∣ ∣ ∣ ∣ ∣
∑
jωji[
〈i|ˆǫ′·~x|j〉 〈j|ˆǫ·~x|i〉
ωji−ω−
〈i|ˆǫ·~x|j〉 〈j|ˆǫ′·~x|i〉
ωji+ω]
∣ ∣ ∣ ∣ ∣ ∣
2Lord Rayleigh calculatedlow energy elastic scattering of lightfrom atoms using classical
electromagnetism. If the energy of the scattered photon is less than the energy needed to excite the
atom, then the cross section is proportional toω^4 , so that blue light scatters more than red light
does in the colorless gasses in our atmosphere.
If the energy of the scattered photon is much bigger than the binding energy of the atom,ω >> 1
eV. then the cross section approaches that forscattering from a free electron, Thomson
Scattering.
dσ
dΩ
=
(
e^2
4 πmc^2) 2
|ˆǫ·ˆǫ′|^2The scattering is roughly energy independent and the only angular dependence is on polarization.
Scattered light can be polarized even if incident light is not.
1.41 Electron Self Energy
Even in classical electromagnetism, if one can calculates the energyneeded to assemble an electron,
the result is infinite, yet electrons exist. The quantum self energy correction (See section 35) is also