Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 33B / b / i / Solution

Take $|\uparrow ⟩$ to be the initial ${\sigma }_z$ eigenstate and put $T=L/v$. The unperturbed level separation is $\hslash \omega =2\mu B$, while the perturbation $V=-\mu b{\sigma }_x$ has transition matrix element of magnitude $\mu b$. Part (a) therefore gives