130_notes.dvi

×

∫t

0

dt 2 ei(ωni+ω

′−ω)t 2

We have calculated all the amplitudes. Thefirst order and second order amplitudes should
be combined, then squared.

cn(t) = c(1)n (t) +c(2)n (t)

c(1)n;~k′ˆǫ′(t) =

e^2

2 iV m

√

ω′ω

ˆǫ·ˆǫ′δni

∫t

0

ei(ωni+ω

′−ω)t′

dt′

c(2)n;~k′ˆǫ′(t) =

−e^2

2 iV m^2 ̄h

√

ω′ω

∑

j

[

〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ǫˆ′·~p|i〉

ω′+ωji

]∫t

0

dt 2 ei(ωni+ω

′−ω)t 2

cn;~k′ˆǫ′(t) =



δniˆǫ·ˆǫ′−^1

m ̄h

∑

j

[

〈n|ˆǫ′·~p|j〉〈j|ǫˆ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

ω′+ωji

]





×

e^2

2 iV m

√

ω′ω

∫t

0

dt 2 ei(ωni+ω

′−ω)t 2

|c(t)|^2 =

∣ ∣ ∣ ∣ ∣ ∣

δniˆǫ·ˆǫ′−

1

m ̄h

∑

j

[

〈n|ǫˆ′·~p|j〉〈j|ˆǫ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ǫˆ′·~p|i〉

ω′+ωji

]

∣ ∣ ∣ ∣ ∣ ∣

2

×

e^4

4 V^2 m^2 ω′ω

∣ ∣ ∣ ∣ ∣ ∣

∫t

0

dt 2 ei(ωni+ω

′−ω)t 2

∣ ∣ ∣ ∣ ∣ ∣

2

|c(t)|^2 =

∣ ∣ ∣ ∣ ∣ ∣

δniˆǫ·ˆǫ′−

1

m ̄h

∑

j

[

〈n|ǫˆ′·~p|j〉〈j|ˆǫ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ǫˆ′·~p|i〉

ω′+ωji

]

∣ ∣ ∣ ∣ ∣ ∣

2

×

e^4

4 V^2 m^2 ω′ω

2 πtδ(ωni+ω′−ω)

Γ =

∫

V d^3 k′

(2π)^3

∣ ∣ ∣ ∣ ∣ ∣

δniˆǫ·ˆǫ′−

1

m ̄h

∑

j

[

〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

ω′+ωji

]

∣ ∣ ∣ ∣ ∣ ∣

2

×

e^4

4 V^2 m^2 ω′ω

2 πδ(ωni+ω′−ω)

Γ =

∫

V ω′^2 dω′dΩ

(2πc)^3

∣ ∣ ∣ ∣ ∣ ∣

δniˆǫ·ˆǫ′−

1

m ̄h

∑

j

[

〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉

ωji−ω

+

〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

ω′+ωji

]

∣ ∣ ∣ ∣ ∣ ∣

2

×

e^4

4 V^2 m^2 ω′ω

2 πδ(ωni+ω′−ω)