Answer
a. $l=4$
b. n = 5 or more.
Work Step by Step
a. $L_z=2\hbar$ and $L=\sqrt{l(l+1)}\hbar$.
We are told:
$cos63.4^{\circ}=\frac{L_z}{L}=\frac{2}{\sqrt{l(l+1)}}$
Square the equation
$cos^2 63.4^{\circ}=\frac{4}{ l(l+1)}$
$ l(l+1)cos^2 63.4^{\circ}=4$
$ l(l+1) \approx 20$
$l=4$
b. For any $n, l$ can have the values $l = 0, 1, 2, … , n – 1$.
Here $l = 4$, so $n \geq 5$. The minimum n is 5.
