Position operator

Remarks

$X$ is an unbounded self-adjoint operator, so it is an observable. It is the unbounded analogue of a multiplication operator. Its spectrum is $\R$, and it has no eigenvalues, since $x\psi(x) = a\psi(x)$ almost everywhere forces $\psi = 0$ almost everywhere. Its projection-valued measure is $P^X(B)\psi = \mathbf{1}_B \psi$, so the probability that the position lies in $B$ is $\int_B \abs{\psi}^2 \, dx$.

Example: $\psi(x) = \pi^{-1/4} e^{-x^2/2}$ lies in $D(X)$, since $\int x^2 \pi^{-1/2} e^{-x^2}\, dx = 1/2 < \infty$. Its expectation value is $\ip{\psi}{X\psi} = \int x \pi^{-1/2} e^{-x^2}\, dx = 0$.

Non-example: $\psi(x) = \pi^{-1/2}(1 + x^2)^{-1/2}$ is a unit vector of $L^2(\R)$, but $\int x^2 \abs{\psi}^2 \, dx = \pi^{-1} \int x^2/(1 + x^2)\, dx = \infty$, so $\psi \notin D(X)$.