What are the methods of experimentally measuring the observables in quantum mechanics? Perhaps due to the limited number of textbooks on quantum mechanics I have consulted, I have seen presented the fundamental principles related to observables, but have never seen a somewhat systematic discussion of how specific measurement of each of those observables can be carried out in practice or in thought experiments. 
For instance, the position, energy, or momentum (separately, of course) of a particle in a potential well (approximately infinitely high, in particular), or of a quantum mechanical harmonic oscillator. How do we measure these (perhaps just in thought experiments)?

[text deleted after the first answer: the energy or angular momentum of the electron of a hydrogen atom]
 A: No, what is in a potential well doesn't touch our apparatuses. What we do is another trick. We entangle the particle in the potential well with another particle that gets out from the well and reaches our apparatuses. For instance scattering experiments can give us information about internal structures.
The procedure goes as follows: let an internal particle be in a superposition of states
$\psi = \frac {1}{\sqrt{2}} (|A \rangle + |B\rangle )$,
where $|A\rangle $ and $|B\rangle $ are two internal states. Then we bring another particle $|p\rangle $ that interacting with the state $|A\rangle $ passes to a state of linear momentum $|\vec a\rangle $, and interacting with the state $|B\rangle $ passes to a state with linear momentum $|\vec b\rangle $,
$|p\rangle |\psi\rangle  \to \frac {1}{\sqrt{2}} (|A\rangle |\vec a\rangle  + |B\rangle |\vec b\rangle )$.
Thus, we can get information of internal states. Structures of molecules and many microscopic data are obtained this way. Of course, the actual experiments are more complicated, in this answer I only described the general idea.
