Two spherically symmetric potentials, the Coulomb potential and the isotropic harmonic oscillator potential, lead to solutions of the Schrodinger equation that have more degeneracies than usual. Historically, people seem to have thought they had no deeper significance, so they were termed "accidental degeneracies". We know know however, that the 1/r potential and the r^2 potential share a special conservation law, that of the Runge-Lenz vector, which explains their special properties including stable classical orbits. So how does the Hamiltonian operator commuting with the Runge-Lenz vector operator lead to accidental degeneracies? And while we're at it, how does the Runge-Lenz vector work quantum mechanically? Since it is a conserved quantity, by Noether's theorem it has a corresponding symmetry, so what is the Lie group corresponding to it, and what is the representation theory of that Lie group? Any help would be greatly appreciated. Thank You in Advance.