Problem
Consider the expectation values of the position $\hat{x}$ and momentum $\hat{p}$ operators with respect to an arbitrary state $|\Psi\rangle$, i.e. $\langle \Psi | \hat{x} | \Psi \rangle$ and $\langle \Psi | \hat{p} | \Psi \rangle$, and show how they transform under the action of the translation operator $\hat{T}(\Delta \textbf{x})$, where $\Delta \textbf{x}$ is a finite (i.e. not an infinitesimal) distance in space. Provide an explicit calculation using the Dirac bra-ket notation.