= Lowest-weight coupling of two angular momenta
{title2=$|s-1,-s+1\rangle=\sqrt{j_2/s}|a\rangle-\sqrt{j_1/s}|b\rangle$}
Put $s=j_1+j_2$, with both $j_i>0$. At magnetic quantum number $-s+1$, let $|a\rangle$ raise only the first factor of the lowest product state and $|b\rangle$ raise only the second. Applying total $J_+$ to the lowest product gives the normalized $J=s$ combination $\sqrt{j_1/s}|a\rangle+\sqrt{j_2/s}|b\rangle$. Its orthogonal combination is annihilated by total $J_-$ and therefore has $J=s-1$. This obtains a useful <Clebsch-Gordan coefficients> without a table.
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