Residual Energy after Photon Adsorption

[sethoflagos]

QM calculations these days are so precise, at least for the simpler species, that electron orbital transition energy levels can be calculated with great precision.

This in turn defines the (minimum?) frequency of the energising photon with similar precision.

The probability of receiving an energising photon of exactly the required minimum energy seems to me to approximate to zero, and yet the excitation definitely happens.

What happens to the mismatch between orbital energy change and photon energy?

[swansont]

Surely you mean absorption here. Photons can’t be at rest, and so can’t be adsorbed.

57 minutes ago, sethoflagos said:

QM calculations these days are so precise, at least for the simpler species, that electron orbital transition energy levels can be calculated with great precision.

This in turn defines the (minimum?) frequency of the energising photon with similar precision.

The probability of receiving an energising photon of exactly the required minimum energy seems to me to approximate to zero, and yet the excitation definitely happens.

What happens to the mismatch between orbital energy change and photon energy?

Transition energies themselves do not have an arbitrarily small precision. They have a natural linewidth, which can be broadened by various mechanisms. The D2 transition in Rb-87, for example, has a ~6MHz linewidth, owing to the ~25 ns lifetime of the excited state.

The transition probability decreases exponentially as you move off resonance. There is no mismatch in energy, as such. The atom absorbs all of the photon energy.

[sethoflagos]

8 hours ago, swansont said:

Surely you mean absorption here. Photons can’t be at rest, and so can’t be adsorbed.

Of course. Blasted autocorrect.

8 hours ago, swansont said:

Transition energies themselves do not have an arbitrarily small precision. They have a natural linewidth, which can be broadened by various mechanisms. The D2 transition in Rb-87, for example, has a ~6MHz linewidth, owing to the ~25 ns lifetime of the excited state.

Aha! So it's bounded by the uncertainty principle. That makes sense. Thank you!

[exchemist]

8 hours ago, sethoflagos said:

Of course. Blasted autocorrect.

Aha! So it's bounded by the uncertainty principle. That makes sense. Thank you!

I think there is also broadening due to the Doppler effects of random atomic motion, at least in gases.

 

[sethoflagos]

2 minutes ago, exchemist said:

I think there is also broadening due to the Doppler effects of random atomic motion, at least in gases.

Good point, however that doesn't appear to challenge the 1st Law in and of itself.

Simple absorption is a little different. Okay, an atom can absorb an incoming photon of say 1MHz above or below its standard excitation frequency, but it I can't see it 'remembering' that it has done so.

When the excited electron drops down to ground state, I'm assuming that it too could be say +/- 1MHz away from standard with the same probability as the absorption case and for the same reason - Heisenberg. But this leaves us with small accounting errors in both the energy and momentum budgets which even if they average out to zero in the long run introduce the same sort of fuzziness as raised in my 'How Sacrosanct is Conservation of Momentum in QM' thread.

I wondered if the discrepancies could be dissipated thermally, but that runs into a degrees of freedom issue. One discrepancy can only balance one conserved quantity I think. 

[swansont]

3 hours ago, exchemist said:

I think there is also broadening due to the Doppler effects of random atomic motion, at least in gases.

 

Yes, as I mentioned; Doppler broadening and collisional broadening are two of the prominent ones. 

2 hours ago, sethoflagos said:

Good point, however that doesn't appear to challenge the 1st Law in and of itself.

Simple absorption is a little different. Okay, an atom can absorb an incoming photon of say 1MHz above or below its standard excitation frequency, but it I can't see it 'remembering' that it has done so.

When the excited electron drops down to ground state, I'm assuming that it too could be say +/- 1MHz away from standard with the same probability as the absorption case and for the same reason - Heisenberg. But this leaves us with small accounting errors in both the energy and momentum budgets which even if they average out to zero in the long run introduce the same sort of fuzziness as raised in my 'How Sacrosanct is Conservation of Momentum in QM' thread.

I wondered if the discrepancies could be dissipated thermally, but that runs into a degrees of freedom issue. One discrepancy can only balance one conserved quantity I think. 

Energy is conserved, and if you have a polyatomic molecule or your gas has collisions, there's a ready reservoir for both energy and momentum conservation. I don't know if anyone has looked at a single neutral atom absorbing and emitting photons in an isolated situation, trying to see if e.g. a red-detuned photon was absorbed and resulted in a red-detuned emission, because how would you do that?

[exchemist]

1 hour ago, swansont said:

Yes, as I mentioned; Doppler broadening and collisional broadening are two of the prominent ones. 

Energy is conserved, and if you have a polyatomic molecule or your gas has collisions, there's a ready reservoir for both energy and momentum conservation. I don't know if anyone has looked at a single neutral atom absorbing and emitting photons in an isolated situation, trying to see if e.g. a red-detuned photon was absorbed and resulted in a red-detuned emission, because how would you do that?

Yes. My understanding is that collisional broadening, or pressure broadening, is also due to the uncertainty principle as the collisions shorten the lifetime of the excited state.

But it's an interesting question @sethoflagos poses. The expectation values of energy e.g. of an ensemble, are conserved of course, but whether that is strictly so for an individual pair of absorption and re-emission events by the same atom I feel less confident.  

[Genady]

Generally, an observable value can be uncertain AND conserved - there is no contradiction. For a simple example, consider an entangled Bell pair of two electrons with opposite spins. The spins of the electrons are maximally uncertain but they certainly sum to zero, i.e., are certainly opposite to each other.

[sethoflagos]

16 hours ago, swansont said:

Yes, as I mentioned; Doppler broadening and collisional broadening are two of the prominent ones.

You really have to look at these interactions holistically don't you. The photon momentum is relatively tiny, but it still must result in a small but quantifiable shift in the kinetic energy of the absorbing molecule which may be +ve or -ve depending on frame of reference. This in turn also impacts the relative values of initial photon energy and electron orbital shift energy. If this effect can be accounted for in doppler broadening, then at least my degrees of freedom issue may go away.

14 hours ago, Genady said:

Generally, an observable value can be uncertain AND conserved - there is no contradiction. For a simple example, consider an entangled Bell pair of two electrons with opposite spins. The spins of the electrons are maximally uncertain but they certainly sum to zero, i.e., are certainly opposite to each other.

I don't think you can use the word 'generally' when your justification is based on the specific case of a boolean relationship. Applying this reasoning to a continuous variable like photon frequency seems too big a jump for me.

[Genady]

10 minutes ago, sethoflagos said:

I don't think you can use the word 'generally' when your justification is based on the specific case of a boolean relationship. Applying this reasoning to a continuous variable like photon frequency seems too big a jump for me.

The direction of spin is continuous. The spin of the electrons can have any one of continuous 3D directions.

[MigL]

17 hours ago, Genady said:

an entangled Bell pair of two electrons with opposite spins. The spins of the electrons are maximally uncertain but they certainly sum to zero

The direction is uncertain before an interaction and collapse.
The spins are always opposite, before or after the collapse.

Some observables are always uncertain.

Edited January 30 by MigL

[swansont]

4 hours ago, sethoflagos said:

You really have to look at these interactions holistically don't you. The photon momentum is relatively tiny, but it still must result in a small but quantifiable shift in the kinetic energy of the absorbing molecule which may be +ve or -ve depending on frame of reference. This in turn also impacts the relative values of initial photon energy and electron orbital shift energy. If this effect can be accounted for in doppler broadening, then at least my degrees of freedom issue may go away.

Yes; this is how laser cooling works. You have to adjust the laser’s or atom’s frequency if you want the interaction to continue as the atom slows down (by “chirping” the laser frequency for the former, or by exploiting the zeeman shift with an external magnetic field for the latter) Otherwise the change in speed eventually shifts the atom out of resonance.

[sethoflagos]

2 hours ago, swansont said:

Yes; this is how laser cooling works. You have to adjust the laser’s or atom’s frequency if you want the interaction to continue as the atom slows down (by “chirping” the laser frequency for the former, or by exploiting the zeeman shift with an external magnetic field for the latter) Otherwise the change in speed eventually shifts the atom out of resonance.

That's actually quite mindblowing. Thanks!