4. Consider a simple harmonic oscillator. Use the orthogonality and the recurrence relations for the Hermite polynomial to do the following.
(a) Derive the matrix elements
$\langle n'|x|n \rangle = \sqrt{\frac{\hbar}{2m\omega}} \left( \sqrt{n}\delta_{n',n-1} + \sqrt{n+1}\delta_{n',n+1} \right)$,
and
$\langle n'|p_x|n \rangle = i\sqrt{\frac{m\hbar\omega}{2}} \left( -\sqrt{n}\delta_{n',n-1} + \sqrt{n+1}\delta_{n',n+1} \right)$,
(b) Find $\Delta x$ and $\Delta p_x$ in state $|n\rangle$, and show that they satisfy the uncertainty relation, $\Delta x \Delta p_x \ge \hbar/2$, for all $n$.
