∫
dΩYℓ∗nmnY 1 mYℓimi =√
3(2ℓi+ 1)
4 π(2ℓn+ 1)
〈ℓn 0 |ℓi 100 〉〈ℓnmn|ℓi 1 mim〉∫
dΩYℓ∗nmn(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Yℓimi=
√
3(2ℓi+ 1)
4 π(2ℓn+ 1)〈ℓn 0 |ℓi 100 〉(
ǫz〈ℓnmn|ℓi 1 mi 0 〉+−ǫx+iǫy
√
2〈ℓnmn|ℓi 1 mi 1 〉+ǫx+iǫy
√
2〈ℓnmn|ℓi 1 mi− 1 〉)
I remind you that the Clebsch-Gordan coefficients in these equations are just numbers which are
less than one. They can often be shown to be zero if the angular momentum doesn’t add up. The
equation we derive can be used to give us a great deal of information.
〈φn|ˆǫ·~r|φi〉 =√
(2ℓi+ 1)
(2ℓn+ 1)〈ℓn 0 |ℓi 100 〉∫∞
0r^3 drR∗nnℓnRniℓi(
ǫz〈ℓnmn|ℓi 1 mi 0 〉+−ǫx+iǫy
√
2〈ℓnmn|ℓi 1 mi 1 〉+ǫx+iǫy
√
2〈ℓnmn|ℓi 1 mi− 1 〉)
We know, from the addition of angular momentum, that adding angular momentum 1 toℓ 1 can only
give answers in the range|ℓ 1 − 1 |< ℓn< ℓ 1 + 1 so the change in inℓbetween the initial and final
state can only be ∆ℓ= 0,±1. For other values, all the Clebsch-Gordan coefficients above will be
zero.
We also know that theY 1 mare odd under parity so the other two spherical harmonics must have
opposite parity to each other implying thatℓn 6 =ℓi, therefore
∆ℓ=± 1.We also know from the addition of angular momentum that the z components just add like integers,
so the three Clebsch-Gordan coefficients allow
∆m= 0,± 1.We can also easily note that we have no operators which can change the spin here. So certainly
∆s= 0.We actually haven’t yet included the interaction between the spin andthe field in our calculation,
but, it is a small effect compared to the Electric Dipole term.
The above selection rules apply only for the Electric Dipole (E1) approximation. Higher order terms
in the expansion, like the Electric Quadrupole (E2) or the Magnetic Dipole (M1), allow other decays
but the rates are down by a factor ofα^2 or more. There is one absolute selection rule coming from
angular momentum conservation, since the photon is spin 1. Noj= 0toj= 0transitions in
any order of approximation.
As a summary of our calculations in the Electric Dipole approximation, lets write out the decay rate
formula.