I'm reading about atomic physics and though none of the literature blatantly says it, I'm getting the impression that a photon doesn't need to be absorbed or emitted at the exact resonance frequency. It's rather that there is a high probability that it will be absorbed/emitted at this frequency. It could also be absorbed/emitted at deviations from this frequency, it's just that the probability is lower.
Am I on the right track?
This is intimately linked to the line width of the transition. Let's first consider the emission process:
The excited state will have a certain life time (for example limited by spontaneous emission in empty space), typically some few to tens of nanoseconds in the optical regime. You can imagine that during this finite life time the atom will be typically found in the process of emitting the photon (only when you check with a click detector, you'll find that the photon actually arrives a sharply defined (random) times). This probability distribution of arrival times is linked to the "temporal shape" of the photon. This shape can be really arbitrary (see e.g. Peter B R Nisbet-Jones et al 2011 New J. Phys.13 103036). Coming back to our spontaneous emission case: In classical language, the atom emits a wave-train with a characteristic length of the time it took to emit it (let's say 10 nanoseconds). When you are now interested in the spectrum of photons from this emitter you can compute the Fourier transform of these finite time wave-trains. Assuming the temporal shape of the photon is an exponential decay (spontaneous emission gives you a constant probability of decay per time interval, so results in a exponential decay), the Fourier transform of that is a Lorentzian function. And the width of it is just given by roughly the inverse of the lifetime. This is just a fundamental property of waves, that they obey a frequency-time uncertainty relation.
So that means: The shorter the "burp" from the emission process of the atom, the broader its emission spectrum. So the individual emitted photon ,according to the Copenhagen interpretation, doesn't really have a well defined colour, until you measure it with a your single photon sensitive spectrometer of choice. It has a certain temporal envelope, and according spectrum. But once you ask it for its colour with your spectrometer, it will give you a well defined, certain colour, randomly, according to the probability distribution of its spectrum.
With the absorption it is a similar game, reversed. Let's imagine you have at hand a very precise colour light-source like a narrow-band laser, providing a very long, almost never-ending coherent wave and you shine it onto your atom of choice to probe its transition. You will take measures to scan the laser's frequency (colour) across the transition frequency. Then you can think about the situation in the following simple model:
The atomic transition corresponds to an oscillator (the charge distribution of the electronic shell is swinging around the positive core, such that the electric dipole moment of the atom changes harmonically in time; this explains the coupling to the electric field associated with the light). This oscillator is damped. It is damped because any excitation of the atom is constantly decaying via spontaneous emission. So it's really like an oscillator with a finite quality factor, it has a line width. So the atom will still respond to the excitation laser, even if it shines in at a frequency slightly off from the resonance frequency of the atomic dipole oscillator, because of the damping.
