Half-turn phase of an SU(2) multiplet
= Half-turn phase of an SU(2) multiplet
{title2=$\alpha_M=(-1)^{I-M}$}
With positive ladder coefficients, the second-axis half-turn is $e^{-i\pi I_2}|I,M\rangle=(-1)^{I-M}|I,-M\rangle$. Conjugation sends $I_\pm$ to $-I_\mp$, making consecutive phases differ by a minus sign. The highest-state rotation formula, expanded through its last nonzero lowering term, fixes the highest-state phase to one.