=
e^2 (Ei−En)
2 π ̄h^2 m^2 c^3∑
λ∫
dΩγ∣
∣
∣
∣
im(En−Ei)
̄h〈φn|ˆǫ·~r|φi〉∣
∣
∣
∣
2=
αω^3 in
2 πc^2∑
λ∫
dΩγ|〈φn|ˆǫ·~r|φi〉|^2This is a useful version of thetotal decay rate formulato remember.
Γtot=
αωin^3
2 πc^2∑
λ∫
dΩγ|〈φn|ǫˆ·~r|φi〉|^2We proceed with the calculation to find the E1 selection rules.
=
αω^3 in
2 πc^2∑
λ∫
dΩγ∣
∣
∣
∣
∣
√
4 π
3〈
φn∣
∣
∣
∣ǫzY^10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
∣
∣
∣
∣φi〉∣∣
∣
∣
∣
2=
αω^3 in
2 πc^2∑
λ∫
dΩγ∣ ∣ ∣ ∣ ∣ ∣
√
4 π
3∫∞
0r^3 drR∗nnℓnRniℓi∫
dΩYℓ∗nmn(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
Yℓimi∣ ∣ ∣ ∣ ∣ ∣
2We will attempt to clearly separate the terms due to〈φn|ǫˆ·~r|φi〉for the sake of modularity of the
calculation.
The integral withthree spherical harmonicsin each term looks a bit difficult, but, we can use
aClebsch-Gordan series like the one in addition of angular momentumto help us solve
the problem. We will write the product of two spherical harmonics in terms of a sum of spherical
harmonics. Its very similar to adding the angular momentum from thetwoYs. Its the same
series as we had for addition of angular momentum (up to a constant). (Note that things
will be very simple if either the initial or the final state haveℓ= 0, a case we will work out below for
transitions to s states.) The general formula for rewriting the product of two spherical harmonics
(which are functions of the same coordinates) is
Yℓ 1 m 1 (θ,φ)Yℓ 2 m 2 (θ,φ) =
ℓ (^1) ∑+ℓ 2
ℓ=|ℓ 1 −ℓ 2 |
√
(2ℓ 1 + 1)(2ℓ 2 + 1)
4 π(2ℓ+ 1)〈ℓ 0 |ℓ 1 ℓ 200 〉〈ℓ(m 1 +m 2 )|ℓ 1 ℓ 2 m 1 m 2 〉Yℓ(m 1 +m 2 )(θ,φ)The square root and〈ℓ 0 |ℓ 1 ℓ 200 〉can be thought of as a normalization constant in an otherwise
normal Clebsch-Gordan series. (Note that the normal addition ofthe orbital angular momenta of
two particles would have product states of two spherical harmonics indifferent coordinates, the
coordinates of particle one and of particle two.) (The derivation of the above equation involves a
somewhat detailed study of the properties of rotation matrices and would take us pretty far off the
current track (See Merzbacher page 396).)
First add the angular momentum from the initial state (Yℓimi) and the photon (Y 1 m) using the
Clebsch-Gordan series, with the usual notation for theClebsch-Gordan coefficients〈ℓnmn|ℓi 1 mim〉.
Y 1 m(θ,φ)Yℓimi(θ,φ) =ℓ∑i+1ℓ=|ℓi− 1 |√
3(2ℓi+ 1)
4 π(2ℓ+ 1)〈ℓ 0 |ℓi 100 〉〈ℓ(m+mi)|ℓi 1 mim〉Yℓ(mi+m)(θ,φ)