130_notes.dvi

c(1)n;~k′ˆǫ(α′)(t) =

1

i ̄h

e^2

2 mc^2

1

V

̄hc^2

√

ω′ω

ǫ(μα)ǫ(α

′)

μ δni

∫t

0

eiωnit

′

e−i(ω−ω

′)t

dt′

=

e^2

2 imV

√

ω′ω

ˆǫ(α)·ˆǫ(α

′)

δni

∫t

0

ei(ωni+ω

′−ω)t′

dt′

Recall that the absolute square of the time integral will turn into 2πtδ(ωni+ω′−ω). We will carry
along the integral for now, since we are not yet ready to square it.

Now we very carefully put the interaction term into the formula forsecond order time dependent

perturbation theory, again usingei
~k·~x
≈1. Our notation is that theintermediate state of
atom and field is called|I〉=|j,n~k,α,n~k′,α′〉wherejrepresents the state of the atom and we
may have zero or two photons, as indicated in the diagram.

V = −

e

mc

A~·~p=−e

mc

1

√

V

∑

~kα

√

̄hc^2

2 ω

ˆǫ(α)·~p

(

ak,αe−iωt+a†k,αeiωt

)

c(2)n (t) =

− 1

̄h^2

∑

j,~k,α

∫t

0

dt 2 VnI(t 2 )eiωnjt^2

∫t^2

0

dt 1 eiωjit^1 VIi(t 1 )

c
(2)
n;~k′ˆǫ(α′)(t) =

−e^2

m^2 c^2 ̄h^2

∑

I

1

V

̄hc^2

2

√

ω′ω

∫t

0

dt 2 〈n;~k′ˆǫ(α

′)

|(ˆǫ(α)ak,αe−iωt^2 + ˆǫ(α

′)

a†k′,α′eiω

′t 2

)·~p|I〉eiωnjt^2

×

∫t^2

0

dt 1 eiωjit^1 〈I|(ˆǫ(α)ak,αe−iωt^1 + ˆǫ(α

′)

a†k′,α′eiω

′t 1

)·~p|i;~kˆǫ(α)〉

We can understand this formula as a second order transition from state|i〉to state|n〉through all
possible intermediate states. The transition from the initial state to the intermediate state takes
place at timet 1. The transition from the intermediate state to the final state takes place at timet 2.

Thespace-time diagrambelow shows the three terms incn(t) Time is assumed to run upward in
the diagrams.

Diagram (c) represents theA^2 term in which one photon is absorbed and one emitted at the same