Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 21

The rotation of 〈jm∣ involves an important phase factor. From the rotation of ∣jm〉 by D(j)(α,β,γ):

D(α,β,γ)∣jm〉=∑m′Dm′m(j)(α,β,γ)∣jm′〉,(1.244)

∴〈jm∣D†(α,β,γ)=∑m′Dm′m(j)*(α,β,γ)〈jm′∣.(1.245)

Then, from the complex conjugate of equation (1.243):

〈jm∣D†(α,β,γ)=∑m′(−1)m′−mD−m′,−m(j)(α,β,γ)〈jm′∣,(1.246)

and replacing −m′↔m′, −m↔m, and noting that the sum is over m′=−j,−j+1,…,j−1,j and so is unaffected,

∴〈j,−m∣D†(α,β,γ)=∑m′(−1)−m′+mDm′m(j)(α,β,γ)〈j,−m′∣,(1.247)

∴(−1)−m〈j,−m∣D†(α,β,γ)=∑m′(−1)−m′Dm′m(j)(α,β,γ)〈j,−m′∣,(1.248)

i.e. (−1)−m〈j,−m∣ transforms like ∣jm〉. It is conventional to multiply both sides of equation (1.248) by (−1)j and then (−1)j−m〈j,−m∣ transforms like ∣jm〉, and the phase is real.