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The energy of Hydrogen electron ground state should depend on $n$ and $\ell$, but it only depends on $n$. What is the reason behind this accidental degeneracy; I know that the reason lies in symmetry; but can someone explain me in simple terms. PS: I do not have knowledge of Group Theory!

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marked as duplicate by Emilio Pisanty quantum-mechanics Feb 22 '18 at 16:56

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    $\begingroup$ There are plenty of technical answers to the original question. The question is what transformation transforms circles into ellipses? (and it's not a "squish"). In this case its a hidden SO(4) symmetry that conserves the Laplace-Runge-Lenz vector. The Lagrangian is invariant under these hyper-rotations (which are w.r.t to the orbital plane and in 4th spatial dimension), with the projection of the orbit into $E3$ being ellipses of varying eccentricity (and hence different total angular momentum). Not easy to visualize. $\endgroup$ – JEB Feb 22 '18 at 18:15