Exercise 6.20
The wave function of a hydrogen-like atom at time $t = 0$ is
$$ \Psi(\vec{r}, 0) = \frac{1}{\sqrt{11}}\left[\sqrt{3}\psi_{2,1,-1}(\vec{r}) - \psi_{2,1,0}(\vec{r}) + \sqrt{5}\psi_{2,1,1}(\vec{r}) + \sqrt{2}\psi_{3,1,1}(\vec{r})\right], $$
where $\psi_{nlm}(\vec{r})$ is a normalized eigenfunction (i.e., $\psi_{nlm}(\vec{r}) = R_{nl}(r)Y_{lm}(\theta, \varphi)$).
(a) What is the time-dependent wave function?
(b) If a measurement of energy is made, what values could be found and with what probabilities?
(c) What is the probability for a measurement of $\hat{L}_z$ which yields $-1\hbar$?
