Bring the chapter together: the operators $\hat{\mathbf{L}}=\hat{\mathbf{r}}\times\hat{\mathbf{p}}$, the commutation relations, the ladder operators, the spectrum of $j$ and $m$, the matrices of angular momentum, orbital angular momentum and its whole numbers, spherical harmonics and rotation matrices. No worked example sits above the answers.

A musician who knows the scales, the chords and the key signatures can read any score at sight. The commutators, the ladder and the spectrum are the scales of angular momentum; spin, atoms and nuclei are the scores.

Everything follows from $[\hat{J}_{x},\hat{J}_{y}]=i\hbar\hat{J}_{z}$ and its cyclic partners: the ladder, the values $j(j+1)$ and $m$, the matrices, and the behaviour under rotations.

• $j$ is whole or half-integer; orbital $\ell$ is whole.
• $Y_{\ell m}\propto(\text{function of}\ \theta)\,e^{im\varphi}$, with parity $(-1)^{\ell}$; $Y_{00}=1/\sqrt{4\pi}$.
• $\langle J_{x}^{2}\rangle=\langle J_{y}^{2}\rangle=\tfrac{\hbar^{2}}{2}(j(j+1)-m^{2})$ in .
• Rotations keep $j$; about $z$ they give $e^{-im\varphi}$; a full turn gives $(-1)^{2j}$.