Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 20

The rotation matrices D(j)(α,β,γ) are unitary. Thus, their matrix elements Dmm′(j)(α,β,γ) obey:

Dm′m(j)(−γ,−β,−α)=Dmm′(j)*(α,β,γ),(1.238)

∑mDmm′(j)*(α,β,γ)Dmm″(j)(α,β,γ)=δm′m″,(1.239)

∑mDm′m(j)*(α,β,γ)Dm″m(j)(α,β,γ)=δm′m″.(1.240)

The reduced rotation matrices d(j)(β) are real. Thus, their matrix elements, dmm′(j)(β), from equation (1.238), obey:

dm′m(j)(−β)=dmm′(j)(β).(1.241)

From the general expression for the matrix elements of dmm′(j)(β), equation (1.167), it follows that

(−1)m′−md−m′,−m(j)(β)=dm′m(j)(β)=(−1)m′−mdmm′(j)(β),(1.242)

and hence

Dm′m(j)(α,β,γ)=(−1)m′−mD−m′,−m(j)*(α,β,γ).(1.243)