130_notes.dvi

34 Scattering of Photons

In the scattering of photons, for example from an atom, aninitial state photonwith wave-number
~kand polarization ˆǫis absorbed by the atomand a final state photonwith wave-number~k′and

polarization ˆǫ′is emitted. The atom may remain in the same state (elastic scattering) or it may
change to another state (inelastic). Any calculation we will do will usethematrix element of the
interaction Hamiltonian between initial and final states.

Hni = 〈n;~k′ˆǫ(α

′)

|Hint|i;~kǫˆ(α)〉

Hint = −

e

mc

A~(x)·~p+ e

2

2 mc^2

A~·A~

The scattering process clearly requires terms inHintthatannihilate one photon and create
another. The order does not matter. The e

2

2 mc^2

A~·A~is the square of the Fourier decomposition of

the radiation field so it contains terms likea†k′,α′ak,αandak,αa†k′,α′which are just what we want.

The−mceA~·~pterm has both creation and annihilation operators in it but not products of them.
It changes the number of photons by plus or minus one, not by zeroas required for the scattering
process. Nevertheless this part of the interaction could contribute in second order perturbation
theory, by absorbing one photon in a transition from the initial atomic state to an intermediate
state, then emitting another photon and making a transition to thefinal atomic state. While this
is higher order in perturbation theory, it is the same order in the electromagnetic coupling constant
e, which is what really counts when expanding in powers ofα. Therefore, we will need toconsider
the e

2
2 mc^2
A~·A~term in first order and the−e
mc
A~·~pterm in second orderperturbation theory
to get an orderαcalculation of the matrix element.

Start with the first order perturbation theory term. All the terms in the sum that do not annihi-
late the initial state photon and create the final state photon givezero. We will assume that the

wavelength of the photon’s is long compared to the size of the atom so thatei
~k·~r
≈1.

Aμ(x) =

1

√

V

∑

kα

√

̄hc^2

2 ω

ǫ(μα)

(

ak,α(0)eikρxρ+a†k,α(0)e−ikρxρ

)

e^2

2 mc^2

〈n;~k′ˆǫ(α

′)

|A~·A~|i;~kˆǫ(α)〉 =

e^2

2 mc^2

1

V

̄hc^2

2

√

ω′ω

ǫ(μα)ǫ(α

′)

μ 〈n;~k

′ˆǫ(α′)|

(

ak,αa†k′,α′+a†k′,α′ak,α

)

ei(kρ−k

ρ′)xρ

|i;~kˆǫ(α)〉

=

e^2

2 mc^2

1

V

̄hc^2

2

√

ω′ω

ǫ(μα)ǫ(α

′)

μ e

−i(ω−ω′)t〈n;~k′ˆǫ(α′)| 2 |i;k~′ˆǫ(α′)〉

=

e^2

2 mc^2

1

V

̄hc^2

2

√

ω′ω

ǫ(μα)ǫ(α

′)

μ e

−i(ω−ω′)t 2 〈n|i〉

=

e^2

2 mc^2

1

V

̄hc^2

√

ω′ω

ǫ(μα)ǫ(α

′)

μ e

−i(ω−ω′)tδni

This is the matrix elementHni(t). Theamplitude to be in the final state|n;~k′ˆǫ(α
′)
〉is given
by first order time dependent perturbation theory.

c(1)n (t) =

1

i ̄h

∫t

0

eiωnit

′

Hni(t′)dt′