(i) What are the physical observables associated with the principal quantum numbers $n$, $l$ and $m$ which are used to describe the electronic states of hydrogen?
(ii) State the inequalities which limit the possible combinations of $n$, $l$ and $m$.
(iii) Describe the angular dependence of the hydrogen states [$n = 2$, $l = 0$, $m = 0$], [$n = 2$, $l = 1$, $m = 0$], and [$n = 2$, $l = 1$, $m = -1$].
(iv) (a) How does the spin-orbit correction applied to hydrogen depend on orbital and spin angular momentum?
(b) What is the physical mechanism which gives rise to the spin-orbit correction?
(c) What effect does the spin-orbit correction have on the quantum numbers used to label the states of hydrogen?
(v) The sum of the relativistic fine structure corrections for hydrogen is given by
$\Delta E = -\frac{\alpha^2}{n^2} \left[ \frac{3}{4} - \frac{n}{j + \frac{1}{2}} \right] E_n^0$.
How many distinct spectral lines due to this fine structure are seen for optical transitions between the $n = 3$ and $n = 2$ levels? Give your reasoning.
