Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 33B (Principles of Quantum Mechanics)

In the interaction picture, first-order time-dependent perturbation theory gives the transition amplitudeTaking its squared modulus and using $|e^{i\theta }-1{|}^2=4{\sin }^2(\theta /2)$ gives

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

During the field region the exact time-evolution operator isUsing the stated Pauli matrix exponential with $\Omega =\mu \sqrt{B^2+b^2}/\hslash$,Only the ${\sigma }_x$ term connects the two ${\sigma }_z$ eigenstates. ThusExpanding this expression for $b/B\to 0$ recovers part (i).