PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
Symmetry and Flavour, Ch Isospin as a group: SU(2), Clebsch-Gordan ratios and G-parity
Problem
Starting from ∣23,21⟩=32∣1,0⟩∣21,21⟩+31∣1,1⟩∣21,−21⟩, apply the lowering operator T−=T−(1)+T−(2) once more, using T−∣I,M⟩=I(I+1)−M(M−1)∣I,M−1⟩ on each factor, to obtain ∣23,−21⟩ as a combination of ∣1,−1⟩∣21,21⟩ and ∣1,0⟩∣21,−21⟩. Then write down the orthogonal, normalised combination ∣21,−21⟩ of the same two basis states, fixing its overall sign by the Condon-Shortley convention, which takes the coefficient of the term with the larger first-factor I3 to be positive.