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*position*of something on our equipment, we then, supposedly, infer the value of a quantum observable Q. How do we do this?

Measurements of the position observable itself may be straightforward (or not, as far as I know). But what about other observables? Given, say, the observable momentum operator P, how do we conclude that our real physical measurement has anything to do with this specific theoretical quantity P? Textbook quantum mechanics says nothing about this. QM tells us the probability of "observing" the system to be in the state for a particular value of P. But it tells us nothing about how to recognize such a momentum eigenstate. Does it? What does a momentum eigenstate "look like?" It seems we need a second theory about how QM relates to the results of the lab.

Let's not try to rigorously answer this in general. Let's just understand one simple case first. Please provide one. (I myself know nothing of experimental physics.) It does not have to be momentum. Use energy, spin, whatever you like. Describe the procedure and then tell us, if possible, how we conclude that the measurement should be associated with the given quantum operator and not some other quantum operator.

Granted that if we have observed

*something*there exists some quantum observable operator for that something, how do we know that this observable operator is the specific observable we want? That is, if we are interested in the energy E, and we perform a (macro) measurement with our real physical lab equipment, we assume there is

*some*quantum observable Q whose eigenstate we have reduced the quantum state to by our observation... ok, but how do we conclude that Q = E?