As a result, the productk·x∼αis small compare to 1, and the exponential may be approximated
by 1, at the same order of approximation that we use when we neglected two photon exchanges.
The resulting formula for the total rate is,
Γ =
e^2
8 π^2 m(
E(0)i −Ef(0))∑α∫
dΩ∣∣
∣∣〈ψf|ε∗α(k)·p|ψi〉∣∣
∣∣2
(20.15)Finally, we shall perform the summation over polarizations, and the integral over directions, as
follows,
∑
α∫
dΩ∣∣
∣∣〈ψf|ε∗α(k)·p|ψi〉∣∣
∣∣2
=∑
α∫
dΩε∗a(k)iεa(k)j〈ψf|pi|ψi〉〈ψf|pj|ψi〉∗ (20.16)Letn=k/|k|, so thatn^2 = 1, then we have
∑
αε∗a(k)iεa(k)j=δij−ninj (20.17)Note that this formula is in agreement with the fact thatεα(k) is transverse tokand that this
space is 2-dimensional. The integration over Ω may be evaluated as follows. The the result of
the integral must be, by construction, a rank 2 symmetric tensor ini,jwhich is rotation invariant
(since we integrate over all directionsnwith a rotation invariant measure). There is only one such
tensor, namelyδijup to an overall multiplicative factorℓ, so that
∫
dΩ(
δij−ninj)
=ℓδij (20.18)The value ofℓis determined by taking the trace overi,j, which gives 8π= 3ℓ. Putting all together,
we find,
Γ =
e^2
6 πm^2(
E(0)i −E(0)f)∣∣
∣∣〈ψf|p|ψi〉∣∣
∣∣2
(20.19)Finally, it is customary to recast the atomic matrix elementin terms of the position operatorx
instead of momentum. This may be achieved by noticing that
i ̄h
m
p= [x,Hm] (20.20)so that
Γ =
e^2
6 π ̄h^2(
E(0)i −Ef(0)) 3 ∣∣
∣∣〈ψf|x|ψi〉∣∣
∣∣2
(20.21)