Position operator

The position operator represents a position measurement. In the position representation in quantum mechanics, its one-dimensional action is multiplication, $(X\psi )(x)=x\psi (x)$. The Born rule associates an interval with the integral of $|\psi (x){|}^2$ over that interval, rather than assigning a finite probability to a single point of a continuous distribution.

The expectation value is $⟨X⟩=\int x|\psi (x){|}^{2}dx$ when that integral exists. Its domain in the Hilbert space of square-integrable wavefunctions requires $x\psi (x)$ to be square integrable; a normalized state need not belong to this domain. Together with the momentum operator, the position operator obeys the canonical commutation relation on a suitable common domain, leading to the Heisenberg uncertainty principle.