The Accidental Degeneracy
In the hydrogen atom there are two kinds of degeneracy with respect to energy. The first kind is related to the quantum number m and is expected. It has to do with the fact that the solutions to the angular part of the hydrogen atom are symmetric in three dimensional space with respect to the m number. Take a look at picture 1. It shows graphical representations of the spherical harmonics, Y, which carry the dependence on m for the hydrogen atom wave function, (picture 2).
Notice that the Y term is the only place where m plays a part. If you take a look at the definitions of the spherical harmonics, you'll see that m only serves to rotate the solution. The table entry for the l = 2 spherical harmonic is shown below in picture 3.
The unexpected, (or accidental), degeneracy is due to the fact that the energy of the hydrogen atom is not dependent on l, only on the quantum number n. As I've mentioned before, Fock, in 1935, showed that if the hydrogen wave function is solved in momentum space and then projected onto a four dimensional sphere, it becomes clear that l plays essentially the same role in four dimensions that m does in three dimensions and that the degeneracies due to the l number are not accidental at all. That's all very nice, but the math is more than a bit complicated, and it's difficult to intuitively visualize a four dimensional sphere. It would be nice to have an easier way to see all this without having to work through a chapter's worth of math. Enter the Kepler orbit version of the problem described by Shubaya and Wulfman in 1965, thirty years later.
Shubaya and Wulfmans' Process
I hope to have a more satisfying explanation of all of this soon. Shubaya and Wulfman do provide a very nice picture that shines a light onto the origin of the problem. They actually modeled the first three dimensional spherical harmonic and placed it on a plane representing momentum, (picture 5).
The picture demonstrates that if you project the lines of the different momentum solutions back onto the momentum plane for m = 0, (shown in the left hand side), you get a completely different solution in the momentum plane than you would for the m = -1, 1, (shown in the right hand side). However, as you can see, the radius of the sphere representing the energy of the system did not change. The sphere was only rotated in three dimensional space by the change of m values. More later.
1. Shubaya and Wulfman
Shibuya T.I. (1965). The Kepler Problem in Two-Dimensional Momentum Space, American Journal of Physics, 33 (7) 570. DOI: 10.1119/1.1971931
2. Table of spherical harmonics
3. A version of Fock's translated article can be found on Copasetic Flows at: