Line broadening: What is actually broadened? Is it just the spectrum of emitted photons that is broadened by e.g. finite lifetime or Doppler effects, or is it actually the electronic energy spectrum?
In my head it would only make sense if it is actually the energy spectrum itself.
 A: The two processes you've mentioned, finite lifetimes and Doppler effects, have different effects:

*

*The finite excited state lifetime results in an uncertainty in the transition energy due to the energy-time uncertainty principle - heuristically, $\Delta E \Delta t \ge \frac{\hbar}2$ means that a short lifetime will increase $\Delta E$, so the emitted photons will have a slight 'range' in frequencies, via $E = h\nu$. You might view this as intrinsic to the emission process.


*For Doppler effects, the emitted radiation will be Doppler shifted due to the velocity of the particles - you may view this effect as extrinsic to the actual emission process itself.
Of course, both of these are factors in making the spectroscopic lines not infinitesimally thin.
A: Since $E = h\nu$, broadening the spectral (frequency) width of photons necessarily broadens the energy spectrum.
Doppler broadening means that atoms are moving at different directions and speed from thermal motion. When an atom runs upstream into a photon, the frequency is increased from the Doppler effect. Also the photon "hits harder", even though it moves at the speed of light relative to any atom. See How can a red light photon be different from a blue light photon?
If the time between when photon was emitted and absorbed is short, the uncertainty principle increases the uncertainty of the wavelength, and therefore the energy. $\Delta E \Delta t > \hbar$. See The more general uncertainty principle, beyond quantum, a video from 3blue1brown.
