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This question involves two cases: electrons bound to a nucleus and free electrons.

Bound electrons

Let's consider the hydrogen atom for simplicity. As far as I know, to be able to excite the electron, the energy of the photon should be in discrete values corresponding to the difference between energy levels inside the hydrogen atom. By the way in this link, the answer states that it is not the electrons that absorbs the photon but the atom in general, which makes sense to me (please correct or clarify if wrong).

The question is how long does the electron stay in that excited state, i.e. how quickly is the photon emitted back? Is it the same for all energy levels and all conditions like particle density (when many atoms together), temperature, presence of electric field, nucleus structure (neutron count) etc?

Free electrons

Again, according to the same link, free electrons do not absorb photons, which means they only undergo Compton scattering. Is this correct? If not, how long does it take for the photon be emitted back? Is the energy gain permanent?

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the answer states that it is not the electrons that absorbs the photon but the atom in general,

That is true, the system nucleus-electrons absorbs the energy. In the usual approximation of a nucleus at rest (due to its much larger mass a good approximation) one talks of the electron changing orbitals going to a higher energy level.

The question is how long does the electron stay in that excited state, i.e. how quickly is the photon emitted back?

The question is answered by the width of the spectral line by the energy-time uncertainty , although to get the correct number one must study the general broadening that can exist.

which means they only undergo Compton scattering.

This is correct, although I would include all kinds of scatterings, (Compton is high energy ).

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  • $\begingroup$ If it's the system nucleus-electrons that absorbs the energy, then it's this same system that changes energy level, not just an electron. $\endgroup$
    – Ruslan
    Oct 3, 2021 at 8:07
  • $\begingroup$ @Ruslan yes, but i, because the nucleus is so tiny and so much heavier, we talk of electrons in orbitals as if the nucleus is at rest, its orbitals confined in such small radiii $\endgroup$
    – anna v
    Oct 3, 2021 at 14:13
  • $\begingroup$ My point is that in the first part of the sentence you talk about nucleus-electrons system, and in the second you silently assume fixed-nucleus approximation, invalidating the quote being answered and the first part of your sentence. $\endgroup$
    – Ruslan
    Oct 3, 2021 at 14:26
  • $\begingroup$ @Ruslan I have clarified $\endgroup$
    – anna v
    Oct 3, 2021 at 14:44
  • $\begingroup$ @annav In your answer you wrote 'the electron's mass is fixed, and if it were able to absorb a photon - at the electron's center of mass - the mass would have to change, which contradicts observations'. But in this link you provided it writes 'The measurement of the mass energy of an unstable particle a large number of times gives a distribution of energies called a Lorentzian or a Breit-Wigner distribution'. How do we observe distribution if electron mass is constant? $\endgroup$
    – Xfce4
    Oct 10, 2021 at 16:13
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An isolated atom in a' excited state woul remain there forever. However, the atom necessarily interacts with electromagnetic field, other atoms, etc., which would cause it to reemit the photon. Some of these processes, such as spontaneous emission are independent on temperature and other conditions. Others, such as stimulated emission or relaxation due to collisions with other atoms can be temperature dependent.

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  • $\begingroup$ An isolated atom in an excited state would remain there forever. However, the atom necessarily interacts with ... That part is so cool. Looking at the QM interpretation of orbitals this is what I would expect. Because once the WF of the electron fits into an orbital (like particle in a 1-D box), why would it change its state without disruption. If it was mandatory for electrons to drop into lower orbitals without an environmental factor, wouldn't they always act the same and drop at the same duration? But still, are you sure about the correctness of that statement? $\endgroup$
    – Xfce4
    Oct 3, 2021 at 17:22
  • $\begingroup$ Yes. But normally atom is coupled to electromagnetic field, even if it is in vacuum. This is why it eventually spontaneously emits a photon. It is called natural lifetime of an excited state $\endgroup$
    – Roger Vadim
    Oct 3, 2021 at 17:44
  • $\begingroup$ ...normally atom is coupled to electromagnetic field, even if it is in vacuum. Is this statement from quantum field theory about the fields covering the whole universe or do you mean the classical electromagnetic field between the electron and the proton? $\endgroup$
    – Xfce4
    Oct 3, 2021 at 17:58
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    $\begingroup$ It is quantum statement - with classical em field spontaneous emission cannit be derived. $\endgroup$
    – Roger Vadim
    Oct 3, 2021 at 18:45

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