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If a photon has energy $E$, we know it has angular frequency $\frac{E}{\hbar}$. If an atom has an energy gap $E$ between its ground and first excited state, we know that if the atom is in a superposition $\alpha|{1}\rangle+\beta|{2}\rangle$, then the wavefunction oscillates with an angular frequency of $\frac{E}{\hbar}$. We also know that if we want to absorb/emit a photon with energy $E$, we need an atom with an energy gap $E$.

Is the fact that these frequencies are equal a coincidence, or is there some reason we should expect that the wavefunction oscillation frequency of an atom to have the same frequency as the photon it emits/absorbs? Keep in mind that when we usually think of emission/absorption, we think of the atom as going from $|1\rangle$ to $|2\rangle$ or vice versa; we don't usually think of the atom as being in a superposition to start with. So it seems like the photon shouldn't "know" about the atom's oscillation frequency.

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  • $\begingroup$ I took the freedom to clarify your wording in the first sentence of the second paragraph. I hope it is more clear now, what you mean with frequency of the wave function. $\endgroup$ – user_na Jun 2 '16 at 19:37
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Assuming that with "oscillation of the wave function" you mean that the state evolves in time with a relative phase $E/\hbar$ as $\alpha\lvert 1 \rangle +\mathrm{e}^{\mathrm{i}E/\hbar t}\lvert 2 \rangle$, that's really an "accident" because given an energy, how else are you going to get a frequency in quantum mechanics out of it if not by dividing by $\hbar$? There might be some numerical factor but it's not evident where it should come from, and in this generic case, there aren't any other quantities around we could use. Also, the physical significance of the frequency $E/\hbar$ of the wavefunction in connection with the light is rather unclear.

For low field strengths and energies, it is usually good to think about the electromagnetic field as a classical background with which the quantum system interacts.

For the case of a two-level atom, the interaction with (coherent) infalling light is a standard case of a driven Rabi oscillation. It is not the case that the resonant Rabi frequency is $E/\hbar$, rather it is $\vec E\cdot \vec d /\hbar$ where $\vec E$ is the electric field and $\vec d$ the transition dipole moment. So the frequency of the atom that actually has something to do with the interaction of the light need not be the same as $E/\hbar$.

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