4. Uncertainty relation and time dependence of expectation values
a. Show that the generalized uncertainty principle for the operators $\hat{x}$ and $\hat{H}$ yields
$\sigma_{x}\sigma_{H} \ge \frac{h}{2m}|\langle p \rangle|$.
What can you deduce about the value of $\langle p \rangle$ in a stationary state?
b. Use
$\frac{d}{dt}\langle \hat{A} \rangle = \frac{i}{\hbar}[\langle \hat{H}, \hat{A} \rangle] + \langle \frac{\partial \hat{A}}{\partial t} \rangle$
for the three cases of $\hat{A} = \hat{I}$ (identity operator), $\hat{A} = \hat{H}$ and $\hat{A} = \hat{p}$ to prove, respectively, the conservation of probability, conservation of energy and Ehrenfest's theorem. Hints: You can assume the potential is constant in time and note that the term $\frac{\partial \hat{A}}{\partial t}$ on the right hand side is a derivative of the operator, and only of the operator, with respect to time.
