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The exact determination of the spontaneous emission coefficient for anexcited atom is an extremely complex problem so various approximations aretypically used. One of the most popular ones is the use of the dipoleapproximation of a two-level atom followed by rotating wave approximation(RWA). However, such an approach applied to the entire frequency spectrumresults in the appearance of divergent integrals in the derivations, which arenot treated rigorously in typical student textbooks. It is known from theliterature that the introduction of cutoff for frequencies, justified by thefinite size of the atom, may solve this problem. For didactic purposes, in thispaper, we introduce a mathematically simple cutoff, which allows for astraightforward yet mathematically consistent rederivation of theWeisskopf-Wigner spontaneous emission rate (up to the small correction) withinRWA. Importantly, this cutoff is not a mathematical trick to make calculationseasier but is related to a real feature of the physical system, the neglect ofwhich leads to inconsistency. More precise analysis demand going beyond RWA anddipole approximation.

Simple and consistent spontaneous emission rate derivation with a physically justified frequency cutoff