Highest-weight states in angular momentum addition
= Highest-weight states in angular momentum addition
With $J=j_1+j_2$, the maximal state is the product $|J,J\rangle=|j_1,j_1\rangle|j_2,j_2\rangle$. Lowering it once gives
$$
|J,J-1\rangle=
\sqrt{\frac{j_1}{J}}|j_1,j_1-1\rangle|j_2,j_2\rangle
+\sqrt{\frac{j_2}{J}}|j_1,j_1\rangle|j_2,j_2-1\rangle.
$$
The orthogonal combination is the highest-weight state $|J-1,J-1\rangle$.