Advanced Quantum Mechanics

Evaluate $\sum_{a'} \psi_{a'}^*(\textbf{x}') \psi_{a'}^{} (\textbf{x}'')$ where $\psi_{a'}^{} (\textbf{x}') = \langle \textbf{x}' | a' \rangle$

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.

Show that $[x_i, p_i^n] = in\hbar p^{n-1}_i$, and $[p_i, x_i^n] = -in\hbar x^{n-1}_i$.