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[tex]\int_a^b \psi^* \hat Q \psi dx.[/tex]

rather than

[tex]\int_{-\infty}^{\infty} \psi \hat Q \psi dx[/tex]

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?