130_notes.dvi

Lets pick two transverse polarization vectors (to sum over) thatform a right handed system with
the direction of photon propagation.
ˆǫ(1)׈ǫ(2)=ˆk

The figure below shows the angles, basically picking the photon direction as the polar axis, and the
ˆǫ(1)direction as what is usually called the x-axis.

φ

θ

k

ε

ε

1

2

Θ

Θ 1

2

rni

The projection of the vector~rniinto the transverse plan gives a factor of sinθ. It is then easy to see
that

cos Θ 1 = sinθcosφ

cos Θ 2 = sinθsinφ

The sum of cos^2 Θ over the two polarizations then just gives sin^2 θ. Therefore the decay rate becomes

Γtot =

αωin^3

2 πc^2

∑

λ

∫

dΩγ|~rni|^2 cos^2 Θ

Γtot =

αωin^3

2 πc^2

|~rni|^2

∫

dΩγsin^2 θ

Γtot =

αωin^3

2 πc^2

|~rni|^22 π

∫

d(cosθ) sin^2 θ

Γtot =

αωin^3

2 πc^2

|~rni|^22 π

∫^1

− 1

d(cosθ)(1−cos^2 θ)θ