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

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.