In the postulates of quantum mechanics, physical observables are described by Hermitian matrices on the state space of a system.
In another of my questions, the measurements of Rydberg-Ritz spectral emission lines do not label the energy eigenstates of an emitting atom, but instead these label differences of frequencies associated to the energy eigenstates. If the algebra of transitions "completely describes" the system, then measurements of the emission lines are not eigenvalues of a hermitian operator in this algebra (these can be derived from the frequencies of energy eigenstates, though).
Does the postulate of QM above still assert that the above observed frequencies label eigenstates of some quantum system? Is there some other "matrix algebra" corresponding to transitions between these eigenstates? Even if the above two paragraphs are fundamentally misguided in some way, I'd still like to know:
In what sense is every physical measurement an eigenvalue of some observable?
I know these questions have been pretty naive, but I'm intentionally trying to investigate things from an extraordinarily minimalist perspective…
Perhaps the above is ridiculous, in the sense that the starting point for a computation to yield a physical quantity always is the eigenvalue of an observable. For example, if I am using a quantity that is the result of some horrid formula, the thing I plug into this formula must be a measured observable. This would save the previous question, since knowing the transition frequencies is as good as knowing the frequencies of the energy eigenstates…provided we can ascertain one of the energy eigenstates. It is the ability to get this "starting" energy eigenstate that bothers me…it should be something we can directly measure.
Best Answer
I think the main issue here is that you're attempting to think about a system in isolation -- the energy states of an electron subject to the electrostatic potential created by the positively charged nucleus -- and trying to understand the measurement based on this system. This is hopeless, as the measurement is not a part of this system.
Your first big red flag should be that energy is not conserved in such a transition. Where does the energy in the difference between the two states go?
You might try thinking about how this measurement is performed in a laboratory. Somewhat tangentially, this was an experiment in my undergraduate physics laboratory, because it encouraged you to think about precisely this type of problem[1].
I should also specify that the Rydberg-Ritz "difference of frequencies associated with the spectral lines" really just means "transitions between two excited states of the atom."
When an electron 'relaxes' from one energy state to another, the difference in energy has to go somewhere (by conservation of energy). As an interesting corollary, such a transition is forbidden if there is nowhere for that energy to go[2]. In this case, the electron's potential energy is released in the form of an excitation of the electromagnetic field, also known as a photon.
On the other end of the experiment, we measure the wavelength of this photon, typically with a grating monochrometer or similar apparatus. Using the energy-wavelength relation for a photon in free space, $E=\frac{hc}{\lambda}$, we calculate the energy of the emitted photons.
But we still have not gotten to the measurement in any true quantum mechanical sense. How do we know which wavelengths (energies) correspond to the transition lines of the atom? We plot the expected number of photons per unit time versus energy and look for spikes that look like a Lorentzian (or really, a Voigt profile). The center of those spikes is the energy we associate with the transition.
So the true measurement we are making is the expected value of the number operator, $\left<\hat{N}\right> = \left< a^\dagger a \right>$ when the monochrometer is set to different wavelength values.
In summary, you are correct that the difference in energy levels does not correspond nicely to a measurement. What does correspond to a measurement is the energy of the emitted photon when an electron traverses that energy difference. By conservation of energy, these must have the same value[3].
As an end note, including the measurement apparatus can be a powerful tool in the analysis of quantum systems. In the field of quantum information, it's typically referred to as the 'ancilla' system and allows you to understand the measurement of POVMs
[1] Technically, we were looking at the effects of the change in nuclear mass between Hydrogen and Deuterium, but that is really a tangent.
[2] This is the basic idea behind suppression of spontaneous emission in the Purcell Effect.
[3] I left out an important bit here -- why the entire energy of the transition must be conveyed to a single photon. This is essentially a consequence of the quantization of energy levels and the linearity of the electromagnetic field, though the ability to explain this simply and accurately lies beyond my skills.