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.