Monday, April 8, 2013

More Notes on Accidental Degeneracy in Two Dimensions as a Model for Three

I'd hoped I was going to be able to make an authoritative swoop through the oft-mentioned, (here anyway), AJP article by Shubaya and Wulfman[1] wherein they explain how the accidental degeneracy in the hydrogen atom energy solutions can be visualized by looking at the accidental degeneracy in the corresponding two dimensional problem of orbits around a Coulomb potential.  Unfortunately, about half-way through, I realized I'm still not quite there.  Here's what I have so far.  I've cleared up exactly what the definition of the accidental degeneracy is.  There's a more complete grasp on the skeleton of Shubaya and Wulfman's argument here, as well as what 'projection to a +1 dimensional space' actually means.

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[2], 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[3], 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
Shubaya and Wulfman confine themselves to the two dimensional Kepler orbit around a Coulomb potential problem.  By doing this, they wind up with a two dimensional momentum equation that can be projected onto a three dimensional sphere in momentum space with a radius that is equal to twice the energy of the orbiting particle.  A diagram of the projection is shown in picture 4[1].  The two dimensional momentum plane runs along the x axis in the picture.  A momentum vector is projected onto the three dimensional momentum sphere by tracing a line from the south pole of the sphere up to the tip of the two dimensional momentum vector and then from there onto the intersecting surface of the sphere.  Describing the momentum wave function problem in terms of quantities projected onto this three dimensional sphere, they quickly determine that in three dimensions, the solutions for the wave function are three dimensional spherical harmonics.  As mentioned above, spherical harmonics depend on two indices, l and m.  Shubaya and Wulfman show that the energy of the momentum wave function depends only on the l index, in other words, there is an 'unexpected' degeneracy in the two dimensional case with respect to the m index.

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)[1].


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.


References:
1.  Shubaya and Wulfman
http://dx.doi.org/10.1119%2F1.1971931
Shibuya T.I. (1965). The Kepler Problem in Two-Dimensional Momentum Space, American Journal of Physics, 33 (7) 570. DOI:

2.  Table of spherical harmonics
http://en.wikipedia.org/wiki/Table_of_spherical_harmonics

3.  A version of Fock's translated article can be found on Copasetic Flows at:
http://copaseticflow.blogspot.com/2012/11/fock-so4-and-azimuthal-quantum-number.html

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