Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section B / B8

Write $\gamma =ge/(2m)$, retaining the printed magnetic moment convention. The displayed two-dimensional matrices are Pauli matrices, and the spin Hamiltonian operator isFor an x-directed field let $\omega =\gamma B_x/2$. The Schrödinger equation gives $i{\dot{c}}_{+}=-\omega c_{-}$ and $i{\dot{c}}_{-}=-\omega c_{+}$. With $c_{+}(0)=1,c_{-}(0)=0$, these equations yieldEquivalently, $U(t)=e^{i\omega t{\sigma }_x}=I\cos \omega t+i{\sigma }_x\sin \omega t$. If $e$ denotes a positive elementary charge magnitude, a free electron is conventionally assigned a minus sign in its magnetic moment; that alternative reverses the rotation sense. The expressions here consistently use the moment specified in the paper.

Put $c=\cos \omega t$, $s=\sin \omega t$. The real part of the off-diagonal coherence vanishes:This also follows from $[H,S_x]=0$ and its zero initial expectation. Individual measurements still give $\pm \hslash /2$, each with probability $1/2$.

The populations giveTogether with $⟨S_y⟩=(\hslash /2)\sin (\Omega t)$, this describes Larmor precession of a spin initially perpendicular to the field. The positive angular frequency is $|\Omega |$; the sign specifies the rotation sense through $d⟨\mathbf{S}⟩/dt=\gamma ⟨\mathbf{S}⟩\times \mathbf{B}$. The factor of two between the spinor amplitudes and the measured spin is essential. The model with quantum coherence predicts no decay; spin relaxation or dephasing in a real solid can damp the oscillation.