Position operator

Summary

The position operator is the observable that corresponds to the position of a particle. On $L^2(\R)$ it multiplies a wave function by the coordinate: $(X\psi)(x) = x\psi(x)$. It is an unbounded self-adjoint operator with spectrum $\R$ and no eigenvalues. Its spectral projections are multiplications by indicator functions, which is why $\abs{\psi}^2$ is the probability density of the position.