How Do Energy Levels Interact When They Are Quantized?

[Thyme]

Pretty basic question: If energy levels (say, of an electron) are quantized, how is an interaction resolved wherein incoming energy (say, a photon) is not of an appropriate amount of energy to result in an appropriate response (say, moving from 1s to 2s in a simple hydrogen atom)?

Suppose the photon had just a little bit more energy than the 1s--> 2s transition. What happens to the excess?

 

[f95toli]

Suppose the photon had just a little bit more energy than the 1s--> 2s transition. What happens to the excess?

In order for the photon to induce the transition 1s--> 2s it needs to have the right energy, if its energy is higher nothing happens; i.e. the atom does not absorb the photon at all. This is why there are discrete lines in the spetrum of an atom.
The simplest models (e.g. Bohr's model) sometimes give the impression that the photon would need to have exactly the right energy for a transition to take place which of course is impossible. In reality the levels are "smeared out" (to be more specific Lorentzian) and have a non-zero width ; meaning if the energy is a little bit higher or lower the photon can still be absorbed and induce a transition.

 

[Thyme]

Ok, is the "smearing" a quantum indeterminacy thing, or can the energy level of the photon and the energy of the electron jump really be different? If they can really be different, what happens to the residual energy?

Even if the energy levels have non-zero widths, it seems energy would have to be conserved somehow. For example, if you have a photon source that's putting out photons with 1.001 +- epsilon energy and a sink that's absorbing at 1.000 +/- epsilon energy, it seems that it shouldn't be the case that the sink can take in all the source's photons without there being some residual.

 

[jtbell]

The energy of the state is fundamentally indeterminate, before it is measured or observed via a decay.

 

[Thyme]

Thanks, can I ask this as a thought experiment?
Suppose I set up a weak laser producing sparse photons at a "known" energy level, say 3.5 +/- units, and beam it at a substrate whose closest orbital jump is 3.0 +/- energy units. I presume from this thread that the laser would not be absorbed and that it would travel through the substrate.
Now suppose I set up a battery of lasers of energy 3.4, 3.3, 3.25, 3.2, etc. approaching 3.0 closer and closer.

I presume that the probability of absorption just rises as we get closer and closer. What is the reason that there is no violation of conservation of energy when we get closer (but not quite at) our "perfect" match of 3.0? Is the reason why there is no violation that, in principle, the energy of the photons produced by the laser have exactly the indeterminacy in energy such that the observed frequency of absorption will be observed?

So then, if we set up enough piles of absorbing substrate, the only way we could set them up to absorb every photon would be to have substrates that could also absorb the upper part of the distribution of "indeterminacy" of energy, matching perfectly energy emitted and energy absorbed?

 

[jtbell]

I presume that the probability of absorption just rises as we get closer and closer. What is the reason that there is no violation of conservation of energy when we get closer (but not quite at) our "perfect" match of 3.0?

The fact that a photon with energy slightly different from 3.0 (say, 2.9998) actually got absorbed, indicates that the atom that absorbed it had an energy level that was in fact slightly different from 3.0. But until that absorption process takes place, there is no way to know the actual value of that energy. It might have been 2.9999, or 3.0001, etc. That's what we mean by "indeterminate."

Of course, we're assuming that we know the photon's energy with sufficient precision to be able to make these distinctions. In practice, there is always some indeterminacy in the energy of both the atom and the incoming photon.

Is the reason why there is no violation that, in principle, the energy of the photons produced by the laser have exactly the indeterminacy in energy such that the observed frequency of absorption will be observed?

If the indeterminacy of photon energy is large enough for the particular situation at hand, then this is indeed a possible reason. It all depends on the relative amounts of indeterminacy of the energies of the atom and the photon.

 

[Marco_84]

I suggest also to remeber guys the indetermination between Energy and time... it is a rough method to calculate many things... as mean life-times.
Plus you get that if you have totally knowledge of the system energy you ou don't know whene the fact happened.

regards marco

 

[Thyme]

Reply to jtbell

The fact that a photon with energy slightly different from 3.0 (say, 2.9998) actually got absorbed, indicates that the atom that absorbed it had an energy level that was in fact slightly different from 3.0. But until that absorption process takes place, there is no way to know the actual value of that energy. It might have been 2.9999, or 3.0001, etc. That's what we mean by "indeterminate."

So the energy levels, when they can be calculated from theory (as in the case of Hydrogen) are quantized, but also "smeared"? Taking the Hydrogen example, if we have an electron in 1s, and were to characterize a standard deviation of the indeterminacy smear, what proportion of the 1s->2s jump would the standard deviation of the smear be?

 

[Thyme]

Still curious about the answer to these:

Suppose I set up a weak laser producing sparse photons at a "known" energy level, say 3.5 +/- units, and beam it at a substrate whose closest orbital jump is 3.0 +/- energy units. I presume from this thread that the laser would not be absorbed and that it would travel through the substrate.
Now suppose I set up a battery of lasers of energy 3.4, 3.3, 3.25, 3.2, etc. approaching 3.0 closer and closer.

I presume that the probability of absorption just rises as we get closer and closer?

And, if we set up enough piles of absorbing substrate, the only way we could set them up to absorb every photon would be to have substrates that could also absorb the upper part of the distribution of "indeterminacy" of energy, matching perfectly energy emitted and energy absorbed?