A few basic questions about the absorption/emission of photons

[Hiero]

1) Photons have momentum. So if an atom emits a photon it ought to recoil to conserve momentum. This recoil will change the kinetic energy of the atom (by an amount dependent on the initial momentum). Does this mean the energy of the emitted photon is the difference in energy levels (of the excited electron) minus the change in kinetic energy? Does this give the spectral lines of a gas a bit of width (which would depend on the velocity distribution and hence temperature)?

2) In order to absorb a photon, does it have to be “precisely” the right frequency? And if it can be any larger frequency, does it then simultaneously absorb it while emitting a photon with the energy difference? (I know in the photoelectric effect higher frequencies just give the ejected electron excess kinetic energy, but when considering a single atom we have to conserve momentum, so that isn’t an option.)

3) Consider an atom at rest which absorbs a photon. We could view this from a different frame of reference where the photon is red shifted below the energy difference of the electron states. So then an atom can absorb a photon which doesn’t have enough energy as long as it’s moving towards it? This makes sense in connection with my first question; the excess energy would come from reduced kinetic energy. (In fact I could check that it makes sense quantitatively but I’d have to first derive the Doppler shift formula; I’ll try it in the morning.)I should probably just read a few QM books, but they make me sleepy with all the talk of Hilbert spaces and hermitian opera...

Thanks and good night.

 

[DrClaude]

1) Photons have momentum. So if an atom emits a photon it ought to recoil to conserve momentum. This recoil will change the kinetic energy of the atom (by an amount dependent on the initial momentum). Does this mean the energy of the emitted photon is the difference in energy levels (of the excited electron) minus the change in kinetic energy? Does this give the spectral lines of a gas a bit of width (which would depend on the velocity distribution and hence temperature)?

Yes. Some of the width of spectral lines is due to the Doppler effect.

2) In order to absorb a photon, does it have to be “precisely” the right frequency? And if it can be any larger frequency, does it then simultaneously absorb it while emitting a photon with the energy difference? (I know in the photoelectric effect higher frequencies just give the ejected electron excess kinetic energy, but when considering a single atom we have to conserve momentum, so that isn’t an option.)

No to both questions. An atom having discrete levels with a well-defined energy is an approximation. In reality, coupling to the electromagnetic field broadens the levels: there is an uncertainty on the exact energy, if you will. The width is related to the lifetime of excited state (which decays by spontaneous emission).

3) Consider an atom at rest which absorbs a photon. We could view this from a different frame of reference where the photon is red shifted below the energy difference of the electron states. So then an atom can absorb a photon which doesn’t have enough energy as long as it’s moving towards it? This makes sense in connection with my first question; the excess energy would come from reduced kinetic energy. (In fact I could check that it makes sense quantitatively but I’d have to first derive the Doppler shift formula; I’ll try it in the morning.)

Yes. This is the phenomenon at work in Doppler cooling.

 

[vanhees71]

I should probably just read a few QM books, but they make me sleepy with all the talk of Hilbert spaces and hermitian opera...

Hm, strange. I usually cannot sleep well when reading QM books. They are just too exciting. If it says "hermitian" instead of "self-adjoint", it makes me even angry in addition!

 

[jtbell]

hermitian opera...

You just conjured in my mind an image of someone singing an aria: "I am the very model of a modern quantum physicist..."

 

[Hiero]

Thank you @DrClaude for your clear answers, and for that clever application of Doppler cooling!

(In fact I could check that it makes sense quantitatively but I’d have to first derive the Doppler shift formula; I’ll try it in the morning.)

We could try to use the Doppler shift (and aberration if we boost at an angle) and the fact that the excited state has a higher mass to show that absorbed energy is frame invariant. However it dawned on me that this would be a waste of time. The Doppler shift (and aberration) are encoded in the Lorentz transform of a photon; the fact that we can express the absorbed energy in terms of invariants (system rest energy minus unexcited-atom rest energy) guarantees that it is also invariant.

You just conjured in my mind an image of someone singing an aria: "I am the very model of a modern quantum physicist..."

I had to look up the reference but once I heard the math lyrics I bursted out laughing. Hilarious hahaha. The setup was accidental Well played