Are energy levels seen in cold-neutron beam in the gravitational potential of the earth a first "proof" of the existence of gravitons?

Question

I answered a question here  ,by finding a remembered experiment of cold neutrons  in the  gravitational field of the earth, where distinct energy levels were found , just in the gravitational potential  of the earth.

In analogy with the hydrogen atom and the coulomb potential, where the energy level differences can be interpreted as photon transitions, are these energy levels detected in the gravitational potential an experimental  "proof" of the existence of gravitons?
 

Answer 1

I don't think it can be taken as a "proof". For the hydrogen atom with the Coulomb potential (only), the discrete eigenstates arise from the solution of the Schrödinger equation with the \(1/r\)-potential. No photons show up in the derivation of the energy levels. Once you couple the system additionally to the quantised radiation field, you can study transitions and you get the photons. The energy of an emitted photon corresponds to the energy difference of the eigenstates involved in the transition, but the discrete values of the energies of the eigenstates do not arise from the coupling to the radiation field. The same relations can be expected if you consider a particle in a gravitational field.

Comments

The spectra of atoms were explained by the Schrodinger equation solutions, and enhanced the hypothesis of quantization of light ( in conjunction with black body radiation and the,photoelectric effect). In that sense "proof" of gravitons, of gravitational energy exchanged between levels. QED came much later.

Answer 2

In case of gravitons, there is no a hypothesis of quantization in conjunction with some sort of a black body radiation and something like a photoelectric effect, so the quantization of gravitational waves stays a hypothesis.

Any quantized energy level gets a width $\gamma$ (or a shift) if some additional interaction is included. For independent interaction mechanisms the corresponding gammas are added:$\gamma=\sum_i \gamma_i$. Emmiting a graviton is highly improbable, so an excited state of a neutron $\psi_n$ in the experiments above decays or gets excited even more due to other, much more probable interaction mechanisms.

P.S. Maybe I was not clear in my answer. Let's recognize that not everything must be quantized. Huge ocean waves must not be quantized - their classical description is sufficient for our purposes. If you still like quantization approach to them, consider the practical case when we are bound to deal with always coherent states of high amplitudes, and when a single quantum of an ocean wave may not be observed - under no circumstances.

Comments

BUT neutrons are quantum mechanical entities, have to be  described by quantum mechanical states, they are not water waves?

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Right, they are described with QM, I agree. I have been speaking of gravitational waves (quantization of gravity => gravitons).

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In the analogy with the hydrogen atom spectra, just the existence of the spectra implies quantized energy transitions for electromagnetic waves. So the existence of the gravitational spectra should imply quantized energy transitions for gravitational waves ( which have recently been seen)

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Absolutely wrong. Discere energy levels may be excited and deactivated in collisions, without any photons. It means other mechanisms of the level widths exist apart from photon emission.

The ultra cold neutron levels are quantized in a reflecting neutron wave guide without any gravity too.

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@anna:

In case of a system with discrete energy levels it is clear that the "amounts of energy" absorbed or emitted by the system ultimately have to correspond to the differences between the discrete levels (leaving aside for the moment details like recoil energy of the emitting system, Doppler shifts due to thermal motion, ...). This does not imply the form these "amounts of energy" can take. As pointed out by Vladimir, there are various mechanisms of transition. However, let us confine here to one mechanism, the emission / absorption of radiation. The crucial point is that the discrete levels in the system do not imply how the "amount of energy" is radiated away. The discrete levels of the system furthermore do not imply that radiation of frequency \(\nu\) generally comes in packets of energy \(h\nu\). And while there are multi-photon emission/absorption-processes, where instead of the single-photon case \(\Delta E=h\nu\) we have \(\Delta E=h\nu_1+...+h\nu_n\) (the probability of such a process is not relevant here, the conceptually important point is its existence), there is no process where e.g. the energy \(h\nu\) corresponding to the energy difference between the levels is radiated away in, say, 10 packets of energy, each of radiation with frequency \(\nu\), but each with energy \((h\nu)/10\). The discrete levels of the system do not explain why the former multi-photon process is possible, while the latter is not.

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@anna: You say "After all one can have an experiment with one photon at the time, from radiating hydrogen  spectra." and "...but now there are experiments with single photons at a time. It will be  easy to design an experiment that uses the hydrogen spectrum lines and demonstrate that the lines are built up of one photon at a time."

Here you are referring to experiments on the emitted radiation. And in principle analogous experiments (you have pointed out the analogies) are conceivable for gravitationally bound systems (disregarding practical difficulties like emission probabilities). Such experiments provide direct evidence on the quantisation of the radiation, they do not rely on the existence of discrete energy levels in the emitter as a logical prerequisite of the conclusion on the quantisation of the radiation.

Also, on the one hand you are asking about (experimental) "proof", on the other hand you speak of "indication" and of enhancing an hypothesis. Even if an experimental "proof" in physics is not as rigorous a proof as in mathematics in terms of reliability, I would feel that the three terms are not synonymous. Of course, by analogy, one may suspect from the cited experimental results on neutrons that a gravitationally bound system with discrete energy levels radiates away transition energies in the form of quantised gravitational radiation. But such a suspicion does not yet qualify as experimental proof.

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@anna v: One of examples when we may not quantize waves is a sound wave in a gas with a big enough average inter-atomic (inter-molecular) distance $l$: $l\gg a_0$.