# A first principles derivation of sponteneous emission probability

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Take a hydrogen in empty space, whose electron is at (say) one above ground state.

Question: Is there a way to compute exactly the probability that the electron moves to ground state in the next t seconds?

Clarification: I'm not specifying how to mathematically model this setup because I don't know of a model in which the probability can be computed exactly. Specifiying a model is part of the answer. For instance, the model may ignore photon reabsorbtion if the affect on the probability is less than experimental error (although in this example I doubt it is the case).

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I've asked this question before in real life and on here, and not really had a satisfactory answer. To avoid this happening again, I'll make precise what I'm looking/not looking for:

1. An answer which *exactly* matches with experiment. QFT/QED is meant to make extremely precise predictions, and this example is surely simple enough that this maxim holds.
2. So no approximations should be made which affect the result more than experimental error (which in this case is extremely small).
3. Expressing the probability in terms of a sum over Feynman diagrams, but not actually computing it, is fine (so long as e.g. a computer could compute the integrals).
4. I'm *not* looking for a restatement of the question. Often people repeat back the question but rephrased in terms of Hilbert spaces etc. yet avoid the actual content of the question (which in that language is the computation of the interaction Hamiltonian).
5. Often (e.g. Schwartz's book chapter 1.3) some easy partial results are computed, but the basic question above is avoided  (e.g. Scwartz doesn't compute $|\mathcal{M}_0|$, among other reservations I have).
6. So if you'd like to provide a reference, please make sure it answers the question in the way indicated above.

To summarise: I'm looking for an exact answer which produces a number. Thank you!

When you prepare the initial excited state from the ground one, it takes some time and brings uncertainty to the final state - it may be a superposition of many excited states rather than one excited state.

A transition to the ground state takes too some time depending on the excited level width. It means again a superposition of states with photons of different frequencies/energies being emitted. Theoretically, the full transition only happens at $t\to\infty$.

I think your problem setup is unrealistic: "exactness" interferes with "uncertainties".

Thank you @VladimirKalitvianski, that was useful. I suppose my question should be given a known initial wavefunction, what is the probability that, observed after $t$ seconds, the electron will be in the ground state (e.g. having emitted a photon)?''. Is that question alright, and do you have an idea how to answer it as in the body of the question?

Your question about an exact number for a given initil state is still meaningless, because for one atomic transition the result is rather arbitrary. You must repeat the decay experiment many-many times to converge to some exact probability. With a small number of experiments the fluctuations are significant. You see, in QM there is no certaity if a limited number of experiments are carried out..

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