Now we need to put the states in using an identity, then use the commutator withHto change~x
to~p.
1 =∑
j〈i|j〉 〈j|i〉i ̄hǫˆ·ˆǫ′ =∑
j[(ˆǫ·~x)ij(ˆǫ′·~p)ji−(ˆǫ′·~p)ij(ˆǫ·~x)ji][H,~x] =̄h
im~p
̄h
im(ˆǫ·~p)ij = (ˆǫ·[H,~x])ij= ̄hωij(ˆǫ·~x)ij(ˆǫ·~x)ij =−i
mωij(ˆǫ·~p)iji ̄hǫˆ·ˆǫ′ =∑
j[
−i
mωij(ˆǫ·~p)ij(ˆǫ′·p)ji−−i
mωji(ˆǫ′·~p)ij(ˆǫ·~p)ji]
=
∑
j[
−i
mωij(ˆǫ·~p)ij(ˆǫ′·p)ji+
−i
mωij(ˆǫ′·~p)ij(ˆǫ·~p)ji]
=
∑
j−i
mωij[(ˆǫ·~p)ij(ˆǫ′·p)ji+ (ˆǫ′·~p)ij(ˆǫ·~p)ji]ǫˆ·ˆǫ′ =− 1
m ̄h∑
j1
ωij[(ˆǫ′·~p)ij(ˆǫ·~p)ji+ (ˆǫ·~p)ij(ˆǫ′·~p)ji](Reminder:ωij=Ei−h ̄Ejis just a number. (ˆǫ·~p)ij=〈i|ˆǫ·~p|j〉is a matrix element between states.)
We may nowcombine the termsfor elastic scattering.
dσelas
dΩ=
(
e^2
4 πmc^2) 2
∣ ∣ ∣ ∣ ∣ ∣
δiiˆǫ·ˆǫ′−1
m ̄h∑
j[
〈i|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉
ωji−ω+
〈i|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉
ωji+ω]
∣ ∣ ∣ ∣ ∣ ∣
2δiiǫˆ·ˆǫ′ =− 1
m ̄h∑
j[
〈i|ˆǫ′·~p|j〉〈j|ˆǫ·~p)|i〉
ωij+
〈i|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉
ωij]
1
ωij+
1
ωji±ω=
ωji±ω+ωij
ωij(ωji±ω)=
∓ω
ωji(ωji±ω)dσelas
dΩ=
(
e^2
4 πmc^2) 2 (
1
m ̄h) 2
∣ ∣ ∣ ∣ ∣ ∣
∑
j[
ω〈i|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉
ωji(ωji−ω)−
ω〈i|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉
ωji(ωji+ω)]
∣ ∣ ∣ ∣ ∣ ∣
2This is anice symmetric form for elastic scattering. If computation of the matrix elements is
planned, it useful to again use the commutator to change~pinto~x.
dσelas
dΩ=
(
e^2
4 πmc^2) (^2) (
mω
̄h
) 2
∣
∣
∣∣
∣
∣
∑
jωji[
〈i|ǫˆ′·~x|j〉 〈j|ˆǫ·~x|i〉
ωji−ω−
〈i|ˆǫ·~x|j〉 〈j|ˆǫ′·~x|i〉
ωji+ω]
∣
∣
∣∣
∣
∣
2