= Solution
For independent angular momenta $\mathbf J_1,\mathbf J_2$, set $\mathbf J=\mathbf J_1+\mathbf J_2$. Since different subsystems commute, $\mathbf J$ satisfies the same angular-momentum algebra. The product basis $|j_1m_1\rangle|j_2m_2\rangle$ diagonalizes $J_1^2,J_2^2,J_{1z},J_{2z}$; the coupled basis diagonalizes $J_1^2,J_2^2,J^2,J_z$. Its labels obey
$$
\boxed{j=|j_1-j_2|,|j_1-j_2|+1,\ldots,j_1+j_2,\qquad m=m_1+m_2}.
$$
The <Clebsch-Gordan coefficients> are the unitary change of basis between them. Start from the maximum-weight state $j=j_1+j_2,m=j$, apply $J_-=J_{1-}+J_{2-}$ with <wavefunction normalization>, and construct the remaining multiplets by <orthogonality>.
For two spin-one-half systems, the symmetric <triplet state> is $|\uparrow\uparrow\rangle$, $(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)/\sqrt2$, $|\downarrow\downarrow\rangle$, with $j=1$; the antisymmetric <singlet state> $(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt2$ has $j=0$. The dimension identity $\sum_j(2j+1)=(2j_1+1)(2j_2+1)$ checks completeness. Coupling orbital and intrinsic <spin> similarly gives total atomic <angular momentum> and organizes spin-orbit <energy> splittings. Exchange symmetry is an additional restriction when the constituents are identical.
Back to article page
