In quantum mechanics Hermitian operators acting on the Hilbert space of a system are observables. From what I understand this means that there is some measurement we can do such that the eigenvalues of this operator are the result.

But suppose that we given a system,

  • Then can the measurement corresponding to any general Hermitian operation be experimentally performed?

  • Are there any practical considerations that restrict what we can measure?

  • If not then how does one design an experiment to measure a given observable (For instance in a two level spin system)?
  • $\begingroup$ All physical measurement, in classical physics as well as in quantum mechanics are approximations. We determine from observations how close our experimental setup came to measuring a theoretical observable by comparing systematic and statistical errors. In practice the errors can range from near 100% to a part in $10^{10}$, or so. If you are looking for a recipe to construct an experiment from an operator, that doesn't exist, I am afraid, at least not on the level of actual experimental implementation. $\endgroup$ – CuriousOne Feb 15 '16 at 4:40
  • $\begingroup$ How does one go about designing even an approximate experiment? In the two state case there are an infinite number of $2 \times 2$ Hermitian operators possible. A one-to-one mapping between the space of Hermitian operators to the set of experiments is not obviously possible. But there must be some way to cover the space of observables with a countable number of experiments. If not, then that would mean that there are 'observables' which cannot be measured at all $\endgroup$ – biryani Feb 15 '16 at 4:48
  • $\begingroup$ One goes to university and becomes an experimental physicist who happens to know a lot about phenomenological physics and who has a lot of engineering skills. It also helps to know what's in the catalogs of the science suppliers. It's a lot less "formal", I am afraid, than you might imagine. :-) Having said that, there may be a few exceptions to your liking, like in the case of spin echos. Those, I am fairly certain, can be formalized. $\endgroup$ – CuriousOne Feb 15 '16 at 6:46
  • $\begingroup$ I guess it is hard to talk about designing experiments in complete generality. I am just confused ( and slightly cross) that the sweeping statement "Hermitian operators are observables" are made in elementary QM classes without telling us how to observe them or what is the complexity involved. I guess I need to read up on experimental techniques to understand the situation a bit better. $\endgroup$ – biryani Feb 15 '16 at 6:53
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    $\begingroup$ For teaching purposes it would be better to say: "Under suitable circumstances experimental observables can be approximated by Hermitian operators." rather than promising the impossible, that every Hermitian operator (or even a mathematically well defined subset of them) can be approximated with an experiment. That's simply not true. It's not even true in classical mechanics that we can measure an arbitrary physical quantity. I think some textbooks and classes are oversimplifying the measurement problem to make QM more approachable, creating a confusing and unphysical reality in the process. $\endgroup$ – CuriousOne Feb 15 '16 at 7:01

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