Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 34B / b / Solution

Now $g=1$, so $c=⟨\psi |\varphi ⟩$ is real andThe perturbation is diagonal in this basis:Hence the instantaneous eigenvalues are $1+2V(t)$ and $-1$ with time-independent eigenvectors, so Hamiltonians at different times commute. In units $\hslash =1$, putExpanding the initial state and evolving its two components givesTaking its overlap with $|\varphi ⟩$ and simplifying gives the exact transition probabilityWith explicit $\hslash$, each phase in the final sine is divided by $\hslash$.