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I have heard on various occasions that the Hamiltonian/Lagrangian of the Hydrogen atom or that of a particle moving in $1/r$-potential can be transformed into that of a free particle moving on the surface of a three-dimensional sphere embedded in a four-dimensional space. Is this easy to show?

There seems to be a paper by Fock given in citation [10] Laplace–Runge–Lenz vector (at Wikipedia) but unfortunately it is not in English. Is anyone aware of the content of this paper, Zur Theorie des Wasserstoffatoms?

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    The Hydrogen spectrum is not the same as that of a free particle moving on $S^3$, but I think the actual statement is that the hydrogen atom has a SO(4) symmetry, due to the conservation of the Laplace-Runge-Lenz vector, which is the same as the symmetry of $S^3$ in $R^4$. This is treated in e.g. Sakurai. The wiki page https://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector seems like a good starting point too. – Meng Cheng Feb 12 '22 at 13:57
  • You may be looking for a “famous” paper by Peter Higgs: Higgs, Peter W. "Dynamical symmetries in a spherical geometry. I." Journal of Physics A: Mathematical and General 12.3 (1979): 309. There’s a follow up paper by Leemon the same year in the same journal. – ZeroTheHero Feb 12 '22 at 14:02
  • Related: https://physics.stackexchange.com/q/603575/2451 and links therein. – Qmechanic Feb 12 '22 at 14:06

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