Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 22

The transformations of the Ylm(θ,ϕ) under rotation follow from equation (1.221)

〈nˆ∣lm〉=Ylm(θ,ϕ)=Ylm(nˆ),(1.249)

and

D(R)∣nˆ〉=∣n′ˆ〉;(1.250)

whence from

D(R−1)∣lm〉=∑m′∣lm′〉〈lm′∣D(R−1)∣lm〉,(1.251)

i.e.

D(R−1)∣lm〉=∑m′∣lm′〉Dm′m(l)(R−1),(1.252)

then

〈nˆ∣D(R−1)∣lm〉=∑m′〈nˆ∣lm′〉Dm′m(l)(R−1).(1.253)

But

〈nˆ∣D(R−1)=〈nˆ′∣,(1.254)

∴Ylm(nˆ′)=∑m′Ylm′(nˆ)Dmm′(l)*(R),(1.255)

or

Ylm(θR,ϕR)=∑m′Ylm′(θ,ϕ)Dmm′(l)*(R).(1.256)

Similarly, from equation (1.248)

(−1)−mYl,−m(θR,ϕR)=∑m′(−1)−m′Yl,−m′(θ,ϕ)Dmm′(l)*(R).(1.257)