Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 33A / c / Solution

For a harmonic perturbation and continuum detuning $\Omega(k)$, first-order amplitudes contain
$$
\frac{\sin(\Omega(k)t/2)}{\Omega(k)/2}.
$$
The distributional limit
$$
\frac{\sin^2(\Omega t/2)}{(\Omega/2)^2}
\longrightarrow 2\pi t\,\delta(\Omega)
$$
selects energy-conserving continuum states and produces a transition probability linear in time.


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 Incoming links: Continuum transition probability at long times