Do electrons actually accept/emit a *range* of energies of photons, or is it a limitation of the detecting instrument itself that makes the lines thick enough to see? Thank you.
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They do indeed have a width, for [a number of reasons](https://en.wikipedia.org/wiki/Spectral_line_shape).
* The energy of an electron is uncertain, as are many properties in quantum mechanics. Thus, so are changes in its energy, and thus, so are the energies of the photons emitted to conserve its energy as it transitions between states.
* The atoms emitting light have random thermal motion, resulting in slight Doppler red- or blue-shift of the emitted light.
* The atoms aren’t perfectly isolated, so their energy levels are slightly disrupted by the energy levels of the atoms around them. (It’s this effect that is why you only see spectral lines in low-density gases for the most part.)
* Fine structure: the energy levels you talk about in chemistry use a simplified model of an atom in which electrons have no spin. But the spin of electrons interacts with their orbitals in a way that slightly changes their energy, splitting each line into two (a spin-up and a spin-down line each).
* Hyperfine structure: the distribution of charge within the atom isn’t perfectly pointlike or symmetrical, and correcting for that splits the lines further.
There are others as well. The physics of electrons at sufficiently high precision is much, *much* more complicated than a Bohr model of energy “shells” would suggest – but that model is good enough for the basics, so it’s usually how these things get introduced.
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