The shell of the three-dimensional isotropic oscillator holds states at one energy
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws
Problem
The shell N of the three-dimensional isotropic oscillator holds (N+1)(N+2)/2 states at one energy, and its orbital quantum numbers run l=N,N−2,… down to 1 or 0. Take the shell N=4.
((a)) List the values of l in the shell and the dimension of the SO(3) multiplet each one carries. Check that the dimensions add up to the shell's degeneracy.
((b)) Which states in the shell does rotational symmetry alone force to share one energy, and which coincidences does it leave unexplained? Is the l=0 state forced to share its energy with any other state?
((c)) Add a small extra potential energy that depends only on r but spoils the oscillator's special form. Which degeneracies in this shell does rotational symmetry still guarantee?