I know that the 2s and 2p (as well as 3s, 3p, and 3d) electron subshells are degenerate in the hydrogen atom. I was wondering if they are also degenerate in other one-electron species, such as the $\ce{He+}$ atom.

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    $\begingroup$ He+ is pretty much the same as H. $\endgroup$ – Ivan Neretin Aug 20 '18 at 20:19
  • $\begingroup$ I'd point out that there is only one 2s orbital so it can't be degenerate. The various p, d, and f orbitals can be, and they are degenerate in the $\ce{He+}$ atom in the ground state. However if the electron is in a 2p orbital say, then the other two are degenerate, but the one with the electron has a different energy. $\endgroup$ – MaxW Aug 20 '18 at 20:34
  • $\begingroup$ @MaxW Sorry, I meant subshells, not orbitals. I know that the 2px, 2py, 2px orbitals are degenerate, but I was wondering if the 2s and 2p subshells are degenerate in all one-electron species. $\endgroup$ – user8081591 Aug 20 '18 at 21:18

Yes, because this system is identical to the original if we simply rescale the charge (or, if you'd like, modify the units of our new system). On a more fundamental level, degeneracies are caused by symmetries of the system, and these are affected not by the magnitudes of various constants or parameters, but rather by the form and structure of the relevant interactions.

The degeneracy of the subshells in the hydrogen atom corresponds to the SO(4) symmetry of the hydrogen atom, which leads to the conservation of the quantum-mechanical form of the Laplace-Runge-Lenz vector. This symmetry depends on the inverse-square law nature of the Coulomb force and appears also in celestial mechanics, where the corresponding inverse-square law force is now gravity.

Changing the charge of your hydrogenic atom will not break this symmetry. On the other hand, fine structure corrections to the hydrogen atom will: if you start looking at relativistic and spin-orbit corrections, you change the form of the Hamiltonian, break the symmetry, and hence resolve the degeneracy.

  • $\begingroup$ Thanks for your answer. To clarify, all subshells of given energy level in a one-electron atom (like H, or Li2+) are degenerate? $\endgroup$ – user8081591 Aug 20 '18 at 22:51
  • $\begingroup$ @user8081591, yes. $\endgroup$ – a-cyclohexane-molecule Aug 20 '18 at 23:04

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