1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable?
2-What are the criteria to say whether some quantity can be considered as an observable or not?
3-An observable is represented by an operator via a recipe called quantization if it has an analog in classical mechanics. If not, such as spin since it has no classical analog, do we use data from experiment to guess what this operator could look like? are there any other methods for finding that?
4-Are there other observables, besides spin, which also have no classical analog?
Answer
Questions 1,2:
An observable is an element that is obtained from experiments. You can take this as the definition of an observable. The fact that we make an operator and give it some properties does not change/influence the outcome of an experiment. It just so happens that the theory we have ascribes linear, hermitian operators to explain experiments. With this in mind, it is easy to say that not all linear, hermitian operators we cook up describe observables.
Question 3
Initially, the classical-quantum correspondence was used, but people quickly realized that it was of limited use. The modern view is that nature can be described by Group Theory (especially the Poincare Group) and everything that is observed follows from there. With this in mind, you don't have to guess about the existence of the Spin Operator, it comes up naturally. What is more important though, is the representations of the operator. When you relate theory and experiments, remember that you are dealing with the representations of an operator. An operator cannot be measured and is useless by itself unless you specify the basis.
Question 4
I don't know the answer to this, but I can tell you that we never measure spin by itself, but the interaction of a spin with something else. Why? In my view, that is the definition of a measurement.
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