Clebsch-Gordan decomposition

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spinThe change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

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