How quickly does an atomic electron decay? I've looked in lots of books on atomic physics, but I've not been able to find an answer to:

Question: Take a hydrogen in empty space whose electron is excited. How long will it take for the electron to emit a photon and move to a lower energy level?

For simplicity assume that the electron begins in the first excited state $|1\rangle$, and that the only other state is the ground state $|0\rangle$. So the question becomes about the precise form of the wavefunction $\psi(t)$ which begins at $|1\rangle$. I guess that it shows oscillatory behavior for small $t$ and looks something like $|0\rangle+e^{-t}|1\rangle$ for large $t$; is this correct? I am very interested in seeing a derivation.
Also,

Question: What is the experimental evidence that $|\psi(t)|$ takes the form of the answer to the above question?

 A: 
Take a hydrogen in empty space whose electron is excited. How long will it take for the electron to emit a photon and move to a lower energy level?

The decay is exponential (at least, with respect to the population in the excited state). If you want to find the lifetimes, the simplest way is to use the NIST Atomic Spectra Database: go to Lines and enter H I for neutral hydrogen; the transition rates are the $A_{ki}$, which for the $2p→2s$ transition in hydrogen equals $A_{ki} = 6.2649\times 10^8\:\rm s^{-1}$, corresponding to a lifetime of about $1.59\:\rm ns$.

I guess that it shows oscillatory behavior for small $t$ and looks something like $|0\rangle+e^{-t}|1\rangle$ for large $t$; is this correct?

Not quite. If you want to do this in full rigour, then you cannot just quantize the atom $-$ you also need to quantize the electromagnetic field. In rough lines, if you start off in an excited state $|e\rangle$ that will decay to a ground state $|g\rangle$, the total quantum state is an entangled state between the atom and the field, where the latter starts off in the vacuum $|0\rangle$ and ends up in a one-photon wavepacket $|{1;\chi(t)}\rangle$, which may (depending on the configuration) end up depending on time. Thus, the full quantum state reads something like
$$
|\Psi(t)\rangle = e^{-\gamma t}|e\rangle|0\rangle + |g\rangle|{1;\chi(t)}\rangle.
$$

Relevant further reading:


*

*Is there oscillating charge in a hydrogen atom?

*Atomic natural line width

*Can we find the exponential radioactive decay formula from first principles?
