Consider eigenenergies of bound states in a central potential.
(a) The radial quantum number $n_r = 0, 1, 2, \dots$ labels solutions of the radial equation with a given angular momentum $l$ in order of increasing energies, i.e., $E_{0,l} < E_{1,l} < E_{2,l} < \dots$
Show that $E_{n_r, l}$ with fixed $n_r$ increases with $l$.
Suggestion: apply the Hellmann-Feynman theorem [recall Problem 4] to the radial equation.
(b) Instead of $n_r$ and $l$, the eigenenergies can be labeled by the principal quantum number $n = 1, 2, 3 \dots$ such that $E_1 < E_2 < E_3 < \dots$
What is the largest possible value of $l$ for a given $n$?
What is the largest possible degeneracy of the bound state with eigenenergy $E_n$?
