130_notes.dvi

Γtot =

αωin^3

2 πc^2

|~rni|^22 π

∫^1

− 1

dx(1−x^2 )

Γtot =

αωin^3

2 πc^2

|~rni|^22 π

[

x−

x^3

3

] 1

− 1

Γtot =

αωin^3

2 πc^2

|~rni|^22 π

[

2 −

2

3

]

Γtot =

αωin^3

2 πc^2

|~rni|^2

8 π

3

Γtot =

4 αωin^3

3 c^2

|~rni|^2

This is now a very nice and simple result for the total decay rate of a state, summed over photon
polarizations and integrated over photon direction.

Γtot=

4 αω^3 in

3 c^2

|~rni|^2

We still need to sum over the final atomic states as necessary. Forthe case of a transition in a
single electron atomψnℓm→ψn′ℓ′m′+γ, summed overm′, the properties of the Clebsch-Gordan
coefficients can be used to show (See Merzbacher, second edition,page 467).

Γtot=

4 αωin^3

3 c^2

{ ℓ+1

2 ℓ+1

ℓ

2 ℓ+1

}

∣ ∣ ∣ ∣ ∣ ∣

∫∞

0

R∗n′ℓ′Rnℓr^3 dr

∣ ∣ ∣ ∣ ∣ ∣

2

for ℓ′=

{

ℓ+ 1

ℓ− 1

The result is independent ofmas we would expect from rotational symmetry.

As a simple check, lets recompute the 2p to 1s decay rate for hydrogen. We must choose theℓ′=ℓ− 1
case andℓ= 1.

Γtot=

4 αωin^3

3 c^2

ℓ

2 ℓ+ 1

∣ ∣ ∣ ∣ ∣ ∣

∫∞

0

R∗ 10 R 21 r^3 dr

∣ ∣ ∣ ∣ ∣ ∣

2

=

4 αω^3 in

9 c^2

∣ ∣ ∣ ∣ ∣ ∣

∫∞

0

R∗ 10 R 21 r^3 dr

∣ ∣ ∣ ∣ ∣ ∣

2

This is the same result we got in the explicit calculation.

29.8 Angular Distributions

We may also deduce the angular distribution of photons from our calculation. Lets take the 2p to
1s calculation as an example. We had the equation for the decay rate.

Γtot=

αω^3 in

2 πc^2

∑

λ

∫

dΩγ

∣ ∣ ∣ ∣ ∣ ∣

√

4 π

3

∫∞

0

r^3 drR∗ 10 R 21

∫

dΩY 00 ∗

(

ǫzY 10 +

−ǫx+iǫy

√

2

Y 11 +

ǫx+iǫy

√

2

Y 1 − 1

)

Y 1 mi

∣ ∣ ∣ ∣ ∣ ∣

2