=
1
4 π1
3
∑
mi(
ǫ^2 zδmi 0 +1
2
(ǫ^2 x+ǫ^2 y)(δmi(−1)+δmi 1 ))
=
1
12 π(
ǫ^2 z+1
2
(ǫ^2 x+ǫ^2 y)(1 + 1))
=
1
12 π(
ǫ^2 z+ǫ^2 x+ǫ^2 y)
=
1
12 πOur result is independent of photon polarization since we assumed the initial state was unpolarized,
but, we must still sum over photon polarization. Lets assume that we are not interested in measuring
the photon’s polarization. The polarization vector is constrained tobe perpendicular to the photons
direction
ˆǫ·~kp= 0
so there are two linearly independent polarizations to sum over. Thisjust introduces a factor of two
as we sum over final polarization states.
The integral over photon direction clearly just gives a factor of 4πsince there is no direction depen-
dence left in the integrand (due to our assumption of an unpolarizedinitial state).
Γtot=
2 αω^3 in
3 c^2(2)(4π)∣
∣
∣
∣
∣
4
√
6
(
2
3
) 5
a 0∣
∣
∣
∣
∣
2
1
12 π=
4 αωin^3
9 c^2∣
∣
∣
∣
∣
4
√
6
(
2
3
) 5
a 0∣
∣
∣
∣
∣
229.7 General Unpolarized Initial State
If we are just interested in the total decay rate, we can go further. The decay rate should not depend
on the polarization of the initial state, based on the rotational symmetry of our theory. Usually
we only want the total decay rate to some final state so we sum over polarizations of the photon,
integrate over photon directions, and (eventually) sum over the differentmnof the final state atoms.
We begin with a simple version of the total decay rate formula in the E1approximation.
Γtot =αω^3 in
2 πc^2∑
λ∫
dΩγ|〈φn|ˆǫ·~r|φi〉|^2Γtot =αω^3 in
2 πc^2∑
λ∫
dΩγ|〈φn|~r|φi〉·ˆǫ|^2Γtot =αω^3 in
2 πc^2∑
λ∫
dΩγ|~rni·ˆǫ|^2Γtot =
αω^3 in
2 πc^2∑
λ∫
dΩγ|~rni|^2 cos^2 ΘWhere Θ is the angle between the matrix element of the position vector~rniand the polarization
vector ˆǫ. It is far easier to understand the sum over polarizations in terms of familiar vectors in
3-space than by using sums of Clebsch-Gordan coefficients.