11. The electronic state of an excited hydrogen atom shall be described by the following non-\normalized wave function according to the form
$\psi(r) = \sqrt{\frac{2}{3}}\psi_{2,1,-1,}(r) + \frac{(2+i)}{3}\psi_{3,2,2,}(r) + \frac{1+2i}{3}\psi_{2,1,0,-\frac{1}{2}}(r)$
where the $\psi_{n,l,m,s}$ are normalized wavefunctions, eigenfunctions of the total energy operator.
a. [4p] Normalize $\psi(r)$, briefly explaining your reasoning. Thereafter, work with the correctly
normalized wave function.
b. [3p] When you do a single measurement, what is the probability to find the electron in a p-
orbital?
c. [3p] What is the expectation value of $\hat{L}_z$, the component of the angular momentum vector in z-
direction?
