Suppose I have an atom at rest which is at energy level $E_i$. Would it be possible to raise it to the next higher level $E_{i+1}$ by shooting a photon of energy $E_{i+1}-E_i$ at it?
I ask because the photon will impart some momentum to the atom, so the atom will end up with some kinetic energy after the absorption, and clearly this kinetic energy must come out of the photon's energy budget. So if the energy levels $E_i$ and $E_{i+1}$ do not already account for this kinetic energy [and I don't see how they could], I would need to use a photon of energy slightly higher than $E_{i+1}-E_i$.
Conversely, if the atom is at rest at energy level $E_{i+1}$, and it emits a photon to fall back to energy level $E_i$, I assume that the emitted photon will have energy slightly lower than $E_{i+1}-E_i$, to account for the kinetic energy of the atom after the emission. Is that correct?
I was led to this question while thinking about the energy balance in laser cooling.
