The expectation value of $x$ in eigenstate

We can compute the expectation value of $x$ simply.

$$\begin{eqnarray*} \langle u_n\vert x\vert u_n\rangle &=&\sqrt{\hbar\over 2m\omeg... ... u_{n-1}\rangle+\sqrt{n+1}\langle u_n\vert u_{n+1}\rangle)=0 \\ \end{eqnarray*}$$

We should have seen that coming. Since each term in the $x$ operator changes the eigenstate, the dot product with the original (orthogonal) state must give zero.