# Expectation values

How much sense does it make to compute expectation value of an observable in a limited interval? i.e.

$$\int_a^b \psi^* \hat Q \psi dx.$$
rather than
$$\int_{-\infty}^{\infty} \psi \hat Q \psi dx$$

Apparently, it shouldn't make any sense for it gives weird results when you compute e.v. of momentum for a part of infinite potentital well (say well is [0,a] and you do the e.v. integral from [0,a/3]]). Why do we have to integrate over all the space then?

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dextercioby
In the position representation, $\psi (x)$ is well defined on all real axis, even though the system might be constrained to "move" in a box.