It is widely used in all fields of physics. However, how accurate is it? It seems that many people use it without estimating its accuracy. When does it break down? Any example?
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It is based on the time-dependent perturbation theory. So, surely it is not an exactly result. As it is a perturbative result, generally it will break down if the perturbation (the coupling in the theory) is too strong. The fermi golden rule is widely used in atomic physics to calculate the life time of an excited state and the result is reasonably good, simply because the perturbation (the electron-photon coupling) is weak enough---it is $1/\alpha^2$ weaker than the electron-nucleus interaction.
Here is a good pedagogical paper on Fermi golden rule with a beautiful model:
Fermi's golden rule: its derivation and breakdown by an ideal model
They discussed the breakdown of the Fermi golden rule, and surprisingly, they showed that the rule can break down not only because of a strong perturbation, but also because of quantum revival. In other words, it can break down in the long time limit (beyond the so-called Heisenberg time) even though the perturbation approximation is still valid. It is purely a coherent effect in time.
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