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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.

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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.

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This is a nice answer, Greg! It's precisely trying to understand POVMs that is spurring all of my questions. –  Jon Bannon Nov 11 '12 at 22:56
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The measurements in the usual postulates are a very idealized kind of ''von Neumann measurement''. They don't cover most realistic measurement situations except those of single spins.

In particular, there is no operator of interest that has as eigenvalues the measurable spectral differences of a Hamiltonian spectrum. (Such an operator could be constructed, but is never used.)

The postulates are mainly there to introduce the formalism together with some sort of initial intuition. Once one understands the formalism, one hardly ever goes back to the postulates. The real meaning is not in the measurement postulates but in the derived meaning of means and uncertainties.

If one has to analyze real measurements, one needs a theory of realistic measurements, and this is provided by POVMs and their properties rather than ''observables'' behaving according to the postulates.

[Edit:] A fairly old (1976) but well-written reference on POVM measurements (the state of the art at that time) is the book by Helstrom, Quantum detection and estimation. For more recent references see the fairly good list given in the Wikipedia article http://en.wikipedia.org/wiki/POVM

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This is very interesting, and I have to think about this. A POVM to me is more general than an observable in that one replaces projection-valued measures by positive-operator valued measures. The "von Neumann measurement" projection postulate is more than I want to assume, as well. I'd be interested in a nice reference for the theory of realistic measurements via POVMs. Do you know of a good such reference? –  Jon Bannon Nov 12 '12 at 21:21
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@JonBannon: I added references to my answer. –  Arnold Neumaier Nov 13 '12 at 7:50
    
Thanks, Arnold. –  Jon Bannon Nov 13 '12 at 13:42
    
You're wrong, Arnold. Every measurement has to correspond to a normal (typically Hermitian) linear operator. In particular, the measurement of energy difference in the atom is nothing else than the measurement of the energy of the emitted photon. If there's no associated linear operator linked to a question, the question is physically meaningless according to the basic and universal laws of quantum mechanics. –  Luboš Motl Nov 14 '12 at 14:37
    
@LubošMotl: Eigenvalue differences are measured from spectral lines without ever measuring the energy of a photon. One measures properties of the emitting system by looking at a refraction or diffraction pattern of the classical electromagnetic mean field, not by measuring the energy of photons. –  Arnold Neumaier Nov 14 '12 at 14:56
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