My recent study Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight binding model motivates me to ask this question:

Is there any example in which the Fermi golden rule gives a quantitatively exact prediction of the transition rate?

Note that the Fermi golden rule is just a first order perturbation result. In a generic problem, it should not be exact. I mean, the transition probability $p(t)$ should be a linear function of time for $t_{c1} <t < t_{c2}$, where $t_{c1}$ and $t_{c2}$ are two characteristic times, but the slope is not that predicted by Fermi golden rule.

  • $\begingroup$ Related: physics.stackexchange.com/q/1766/2451 and links therein. $\endgroup$ – Qmechanic Aug 6 '14 at 13:12
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    $\begingroup$ @Qmechanic I couldn't stop myself from googling that. You should get a badge when you get to 1k! $\endgroup$ – Emilio Pisanty Aug 6 '14 at 13:35
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    $\begingroup$ @Emilio Pisanty: Ha-ha. $\endgroup$ – Qmechanic Aug 6 '14 at 13:37
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    $\begingroup$ Are you willing to accept answers about when first-order time-dependent perturbation theory is exact? It seems very unlikely to me that Fermi's golden rule, by which I mean the $t\to\infty$ limit of first-order perturbation theory, is ever exact for all times. If you measure a system very quickly after applying a perturbation I wouldn't expect the energy expectation value with respect to the bare Hamiltonian of the initial and final states to be the same. $\endgroup$ – Mark Mitchison Aug 6 '14 at 14:53

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