Spin on the Bloch Sphere

A spin-1/2 state is a unit Bloch vector $\vec S$. In a static field $B_0\hat z$ it precesses about $\hat z$ at the Larmor rate $\omega_0=\gamma B_0$ (the cone angle is fixed). Add a circularly polarized RF field of amplitude $B_1$ at frequency $\omega_{\mathrm{rf}}$ and the vector also nutates: on resonance ($\omega_{\mathrm{rf}}=\omega_0$) it sweeps pole to pole, so a $\pi$-pulse inverts $|0\rangle\to|1\rangle$ and a $\pi/2$-pulse builds an equal superposition. Off resonance the inversion is incomplete, capped at $\omega_1^2/(\omega_1^2+\Delta^2)$. The motion is the torque equation $\dot{\vec S}=\vec\Omega(t)\times\vec S$, integrated by exact rotations so $|\vec S|=1$ holds to machine precision.