When a quantum system contains two angular momenta $\mathbf J_1$ and $\mathbf J_2$, the total is $$\mathbf J=\mathbf J_1+\mathbf J_2.$$ A product basis $|j_1,m_1\rangle|j_2,m_2\rangle$ gives definite component values for each subsystem, while a coupled basis $|j,m\rangle$ gives definite total $J^2$ and $J_z$.

The allowed total quantum numbers are $$j=|j_1-j_2|,|j_1-j_2|+1,\ldots,j_1+j_2,$$ and $$m=m_1+m_2.$$

For two spin-$1/2$ particles, $j_1=j_2=1/2$, so the total spin can be $j=1$ or $j=0$. The triplet states are $$|1,1\rangle=|\uparrow\uparrow\rangle,$$ $$|1,0\rangle=\frac{1}{\sqrt2}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle),$$ $$|1,-1\rangle=|\downarrow\downarrow\rangle,$$ and the singlet is $$|0,0\rangle=\frac{1}{\sqrt2}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle).$$

The coefficients connecting product and coupled bases are Clebsch-Gordan coefficients. They encode how angular-momentum eigenstates combine rather than introducing new dynamics.

Angular-momentum addition is essential for combining orbital and spin angular momentum in atoms, coupling multiple particle spins, interpreting selection rules and constructing nuclear states.