•• The general proof of the selection rule (11.46) is beyond our scope, but you can prove...

•• The general proof of the selection rule (11.46) is beyond our scope, but you can prove it in a few special cases, (a) It is a fact that the wave function ψnlm(r) = Rnl(r)Өlm(θ)eimϕ satisfies

[This property is often described by saying that the wave function has parity (−1)l.] Use the angular functions listed in Table 8.1 to prove (11.51) for all wave functions with l = 0,1, or 2. (b) The probability of a radiative transition (n, l, m → n′,l′, m′) is given by (11.45), which now takes the form

(This is for radiation polarized with ℰ in the x direction. For isotropic unpolarized radiation, we must average over this and the two corresponding expressions with x replaced by y and by z.) Use the property (11.51) to prove that the transition probability (11.52) is zero if l′ − l (whether the integrand is x, y, or z). This proves that transitions with Δl = 0 are forbidden — a particular case of (11.46).