Addition of Angular Momenta

Real systems carry several angular momenta at once — orbital and spin of one particle, or the spins of several. Because the total angular momentum $\hat{\mathbf{J}}={\hat{\mathbf{J}}}_1+{\hat{\mathbf{J}}}_2$ generates rotations of the whole system, it is total $\hat{\mathbf{J}}$, not the individual pieces, that is conserved and that labels states in a rotationally invariant problem. Combining angular momenta — decomposing a tensor product of representations into irreducibles — is therefore a constant practical need, from spin–orbit coupling to the singlet state to the coupling of quark spins.

1. Two Bases

Given two angular momenta $j_1,j_2$ with spaces of dimension $(2j_1+1)$ and $(2j_2+1)$, the product space has dimension $(2j_1+1)(2j_2+1)$ and admits two natural bases:

• Uncoupled $|j_1,m_1⟩\otimes |j_2,m_2⟩$: simultaneous eigenstates of ${\hat{J}}_1^2,{\hat{J}}_{1z},{\hat{J}}_2^2,{\hat{J}}_{2z}$. Natural when the parts don't interact.
• Coupled $|j,m⟩$: simultaneous eigenstates of ${\hat{J}}_1^2,{\hat{J}}_2^2,{\hat{J}}^2,{\hat{J}}_z$ of the total $\hat{\mathbf{J}}$. Natural when a rotationally invariant interaction (e.g. ${\hat{\mathbf{J}}}_1\cdot {\hat{\mathbf{J}}}_2$) mixes the parts.

Note ${\hat{J}}_{1z}$ and ${\hat{J}}_z$ commute but ${\hat{J}}_{1z}$ and ${\hat{J}}^2$ do not, so the two bases are genuinely different and related by a unitary change of basis.

2. The Clebsch–Gordan Series

The allowed total-$j$ values follow from adding the maximal projections and stepping down. The result is the angular-momentum addition rule:

$$\boxed{j=|j_1-j_2|,|j_1-j_2|+1,\dots ,j_1+j_2,}$$

each value occurring once, with $m=m_1+m_2$. In representation-theory language this is the decomposition of a tensor product of $SU(2)$ irreps into a direct sum (the Clebsch–Gordan series):

$$D^{(j_1)}\otimes D^{(j_2)}=\underset{j=|j_1-j_2|}{\overset{j_1+j_2}{⨁}}{D}^{(j)}.$$

Dimensions check: ${\sum }_j(2j+1)=(2j_1+1)(2j_2+1)$.

Example — two spin-$\frac{1}{2}$: $\frac{1}{2}\otimes \frac{1}{2}=1\oplus 0$, i.e. a triplet ($j=1$, symmetric) and a singlet ($j=0$, antisymmetric). The singlet $\frac{1}{\sqrt{2}}(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)$ is the maximally entangled Bell state.

3. Clebsch–Gordan Coefficients

The change of basis is expanded with Clebsch–Gordan coefficients $⟨j_1{m}_1;j_2{m}_2|jm⟩$:

$$|jm⟩=\sum _{m_1+m_2=m}⟨j_1{m}_1;j_2{m}_2|jm⟩|j_1{m}_1⟩|j_2{m}_2⟩.$$

They are real (in the standard Condon–Shortley convention), vanish unless $m=m_1+m_2$ and $|j_1-j_2|\le j\le j_1+j_2$, and are computed by starting from the "stretched" state $|j_1+j_2,j_1+j_2⟩=|j_1{j}_1⟩|j_2{j}_2⟩$ and applying the lowering operator ${\hat{J}}_{-}={\hat{J}}_{1-}+{\hat{J}}_{2-}$ repeatedly, orthogonalizing between the towers.

4. Spin–Orbit Coupling in Practice

The hydrogen fine structure term $\propto \hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$ is not diagonal in the uncoupled $|m_{\ell },m_s⟩$ basis, but is diagonal in the coupled $|j,m_j⟩$ basis, because

$$\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}=\frac{1}{2}({\hat{J}}^2-{\hat{L}}^2-{\hat{S}}^2)\Rightarrow ⟨\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}⟩=\frac{{\hslash }^2}{2}[j(j+1)-\ell (\ell +1)-s(s+1)].$$

For an electron ($s=\frac{1}{2}$) with $\ell >0$ this splits each level into $j=\ell \pm \frac{1}{2}$ — the fine-structure doublets. Coupling angular momenta is exactly the step that makes such perturbations tractable.

5. The Wigner–Eckart Theorem

The most powerful payoff of this machinery. A spherical tensor operator ${\hat{T}}_q^{(k)}$ (a set transforming among themselves under rotation like angular momentum $k$) has matrix elements that factor into a Clebsch–Gordan coefficient and a single geometry-independent reduced matrix element:

$$⟨j^{\prime }{m}^{\prime }|{\hat{T}}_q^{(k)}|jm⟩=⟨jm;kq|j^{\prime }{m}^{\prime }⟩⟨j^{\prime }\Vert {\hat{T}}^{(k)}\Vert j⟩.$$

All the $m$-dependence is in the (known) Clebsch–Gordan factor; the physics sits in one number. This instantly yields selection rules ($m^{\prime }=m+q$, $|j-k|\le j^{\prime }\le j+k$) and the relative intensities of spectral lines within a multiplet — the reason dipole transitions obey $\Delta \ell =\pm 1,\Delta m=0,\pm 1$ (see time-dependent perturbation theory).

See also

• Orbital angular momentum and Spin — the pieces being added.
• Time-independent perturbation theory — spin–orbit coupling.
• Entanglement & Bell — the singlet state.
• math/group-theory/00-README.md — $SU(2)$ tensor-product decomposition.