Bring the chapter together: spin as intrinsic angular momentum, the Stern–Gerlach experiment, the Bloch sphere, the Pauli matrices, measurement along any axis, spinors and their sign, precession in a field and the two-level systems that are all spins in disguise. No worked example sits above the answers.

A pilot who knows one aircraft thoroughly can fly its whole family. Spin one-half is that aircraft: every two-level system in the rest of the course flies the same way.

$\hat{\mathbf{S}}=\tfrac{\hbar}{2}\vec{\sigma}$; every state is up along some direction; probabilities along a turned axis are $\cos^{2}\frac{\alpha}{2}$; rotations are $\cos\frac{\theta}{2}-i\,\mathbf{n}\cdot\vec{\sigma}\sin\frac{\theta}{2}$; a field along $z$ makes the spin precess.

• $S^{2}=\tfrac34\hbar^{2}$, $S_{z}=\pm\tfrac{\hbar}{2}$; $2s+1$ spots in a Stern–Gerlach magnet.
• $(\mathbf{a}\cdot\vec{\sigma})^{2}=|\mathbf{a}|^{2}$; energies of are .
• A full turn multiplies a spinor by $-1$; two full turns restore it.
• Precession: $\frac{d}{dt}\langle\hat{\mathbf{S}}\rangle=\vec{\omega}\times\langle\hat{\mathbf{S}}\rangle$, about 28 GHz per tesla for electrons.