Discrete energy states (explanations in QFT?)

[tim_lou]

I've been thinking about one of the postulates about one particle quantum mechanics, it says that whenever we measure an energy value, we get one of those eigenvalues.

Firstly, pretty much 99% of the stuffs I know in nonrelativistic QM applies in the realm of electromagnetism. I just don't think a particle stuck in a classical gravity well is realistic. (I may be wrong).

So in QED what do we mean by measuring energy? I believe in all the experiments, we measure the photons emitted by states transitions and use that as the energies. I feel fundamentally, the reason why these photon energies are discrete have to do with QED. The question is, how then? is it possible to deduce this postulate from first principles in QFT? I know it is difficult to treat bound states in QED (specially in a perturbation sense...when interactions aren't small at all). So perhaps there are some simple intuitive explanations?

 

[0xDEADBEEF]

[...]
So in QED what do we mean by measuring energy? I believe in all the experiments, we measure the photons emitted by states transitions and use that as the energies. I feel fundamentally, the reason why these photon energies are discrete have to do with QED. The question is, how then? is it possible to deduce this postulate from first principles in QFT? I know it is difficult to treat bound states in QED (specially in a perturbation sense...when interactions aren't small at all). So perhaps there are some simple intuitive explanations?

Well I am not sure what you mean by discrete photon energy states. Because of the conservation of energy the photon that is emitted when an electron drops into the ground state needs to carry the energy that the electron had before (or very close to it). Otherwise photon energies are not discrete. Only when you consider cavities, like metal boxes, then all photons inside the box form standing waves, and those have discrete energy levels. There is more ways to measure energy then with emitted photons btw.

 

[tim_lou]

After I think about it again, I realized I just answered my own question. The mechanism of energy measurement is by photon emissions, and the frequency of these emitted photons are constrained by a delta function. Also, the transition amplitude is nonzero only with states with one emitted photon. We get a spectrum of these waves and naturally associate E=hf and obtain the discrete energy states in our experiments. So indeed, fundamentally, this is due to energy (and momentum) conservation and that fact that only one photon is emitted during state transitions. (otherwise we'll get a continuous spectrum of photons and see no peaks in our experiment).

I'm interested to know what other ways one can measure these energy states without resort to photoemissions. Perhaps pair productions. In that case, the same argument applies.