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.

This operator case is surprisingly not necessarily mathematically trivial to describe formally because you often end up getting into the Dirac delta functions/continuous spectrum: as mentioned at: mathematical formulation of quantum mechanics