Why compute expectation values over all space rather than a limited interval

[gulsen]

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?

 

[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.

Daniel.

 

[PRB147]

you must integrate over entire space!
For infinite potential, there in no leak for
wavefunction beyond the potential boundary.