130_notes.dvi

point. Diagrams (a) and (b) represent two second order terms. In diagram (a) the initial state
photon is absorbed at timet 1 , leaving the atom in an intermediate state which may or may not be
the same as the initial (or final) atomic state. This intermediate state has no photons in the field. In
diagram (b), the atom emits the final state photon at timet 1 , leaving the atom in some intermediate
state. The intermediate state|I〉includes two photons in the field for this diagram. At timet 2 the
atom absorbs the initial state photon.

Looking again at the formula for the second order scattering amplitude, note that we integrate over
the timest 1 andt 2 and thatt 1 < t 2. For diagram (a), the annihilation operatorak,αis active at time
t 1 and the creation operator is active at timet 2. For diagram (b) its just the opposite. The second
order formula above contains four terms as written. Thea†aandaa†terms are the ones described
by the diagram. Theaaanda†a†terms will clearly give zero. Note that we are just picking the
terms that will survive the calculation, not changing any formulas.

Now, reduce to the two nonzero terms. The operators just give afactor of 1 and make the photon
states work out. If|j〉is the intermediate atomic state, thesecond order term reduces to.

c(2)n;~k′ˆǫ(α′)(t) =

−e^2

2 V m^2 ̄h

√

ω′ω

∑

j

∫t

0

dt 2

∫t^2

0

dt 1

[

ei(ω

′+ωnj)t 2

〈n|ˆǫ(α

′)

·~p|j〉〈j|ˆǫ(α)·~p|i〉ei(ωji−ω)t^1

+ ei(ωnj−ω)t^2 〈n|ˆǫ(α)·~p|j〉〈j|ǫˆ(α)

′

·~p|i〉ei(ω

′+ωji)t 1 ]

c(2)n;~k′ˆǫ′(t) =

−e^2

2 V m^2 ̄h

√

ω′ω

∑

j

∫t

0

dt 2

[

ei(ω

′+ωnj)t 2

〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉

[

ei(ωji−ω)t^1

i(ωji−ω)

]t 2

0

+ ei(ωnj−ω)t^2 〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

[

ei(ω

′+ωji)t 1

i(ω′+ωji)

]t 2

0





c(2)n;~k′ˆǫ′(t) =

−e^2

2 V m^2 ̄h

√

ω′ω

∑

j

∫t

0

dt 2

[

ei(ω

′+ωnj)t 2

〈n|ǫˆ′·~p|j〉〈j|ˆǫ·~p|i〉

[

ei(ωji−ω)t^2 − 1

i(ωji−ω)

]

+ ei(ωnj−ω)t^2 〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

[

ei(ω

′+ωji)t 2

− 1

i(ω′+ωji)

]]

The−1 terms coming from the integration overt 1 can be dropped. We can anticipate that the
integral overt 2 will eventually give us a delta function of energy conservation, goingto infinity when
energy is conserved and going to zero when it is not. Those−1 terms can never go to infinity and
can therefore be neglected. When the energy conservation is satisfied, those terms are negligible and
when it is not, the whole thing goes to zero.

c(2)n;~k′ˆǫ′(t) =

−e^2

2 V m^2 ̄h

√

ω′ω

∑

j

∫t

0

dt 2

[

ei(ωni+ω

′−ω)t 2

〈n|ǫˆ′·~p|j〉〈j|ˆǫ·~p|i〉

[

1

i(ωji−ω)

]

+ ei(ωni+ω

′−ω)t 2

〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉

[

1

i(ω′+ωji)

]]

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|ˆǫ′·~p|i〉

ω′+ωji

]