Question

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 $E_n$?

          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 $E_n$?

Consider eigenenergies of bound states in a central potential.
(a) The radial quantum number nr = 0, 1, 2, … labels solutions of the radial equation with a given angular momentum l in order of increasing energies, i.e., E0,l < E1,l < E2,l < …
Show that Enr, l with fixed nr increases with l.
Suggestion: apply the Hellmann-Feynman theorem [recall Problem 4] to the radial equation.
(b) Instead of nr and l, the eigenenergies can be labeled by the principal quantum number n = 1, 2, 3 … such that E1 < E2 < E3 < …
What is the largest possible value of l for a given n?
What is the largest possible degeneracy of the bound state with eigenenergy En?

Added by Marie S.

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