I’ve been planning a round of exposition/research on both the classical and quantum Kepler problems myself, so if you write something I’d like to look at it!

]]>The little gadget on the website that shows a few random past posts served this up to me, and I figured I’d give it a re-read. I’ve also been planning to return to the Kepler problem, or rather, the quantum Coulomb problem. I have a heap of old notes on the topic that I need to do *something* with, even if it’s only to get them into shape to put on the arXiv as a pedagogical piece for students.

Fixed! What made you read this blog article just now, Blake? I’ve been planning to get back to work on the Kepler problem, but you seem to have picked that up telepathically.

]]>Ok, thanks for the extra reference. I have read your slides already, naturally. I’ve also looked at Greg Egan’s page, and the reference he gives by Jonas Karlsson, “The SO(4) symmetry of the hydrogen atom” which I also found very helpful.

]]>I spent about 3 months carefully reading *Variations on a Theme by Kepler*. They discuss the map at the classical level, and that’s probably the best way to get an intuition for the quantum map.

I reached my maximal point of understanding this stuff shortly after giving this talk at Georgia Tech in March 2019:

• Hidden symmetries of the hydrogen atom.

But I haven’t thought about this since April, and right now my brain is full of combinatorial species and Hopf algebras, so it would be quite painful for me to suddenly try to remember what I was thinking about the hydrogen atom well enough to explain it. Someday I want to explain it and wow the world with some new discoveries… but today is not that day. For now I recommend *Variations on a Theme by Kepler* and also

• Bruno Cordani, *The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Pertubation*, BirkhĂ¤user, Boston, 2002.

Taken together they explain this quite well. But it’s a bit confusing because there are several interlocking viewpoints, and at first it’s hard to believe they’re all true.

Also, if you haven’t read my talk slides, you might try, since they’re about as good as anything I’d say off the cuff right now.

]]>There is something I don’t understand. How do we explicitly make the transition from the quantum mechanics Hilbert space to ? Is it stereographic projection? How do we understand this physically?

Also, have you seen the book “Variations on a Theme by Kepler” by Guillemin and Sternberg on this material?

]]>Believe it or not, I’ve been continuing to think about some of the things we discussed. However, my first batch of comments will only discuss the bound states, not *all* the states of the hydrogen atom.

Yes, please – and please announce it here so that I am informed…

]]>Thanks! There’s a lot more I want to say by now; I guess it’s time to write another episode of this thread.

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