What does closed orbits mean in Quantum Mechanics?

Question

In classical mechanics, The inverse potential $$\phi=-\frac{C}{r}$$ leads to closed orbits. The planetary motion is one example. The the conservation of the $$\mathbf{n}=\frac{\mathbf{p}\times\mathbf{l}}{m}-\frac{e^2}{r}\mathbf{r}$$ Runge-Lenz vector $\mathbf{n}$ implies closed orbits.

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A similar case can be considered in quantum mechanics, where we have a Runge-Lenz vector operator that commutes with Hamiltonian of Hydrogen atom.

$$[\mathbf{N},H]=0$$

So this too should imply the closed orbit but what does it mean by closed orbit in quantum mechanics where orbits are not defined. How this leads to degeneracy? What is the relation between closed orbit and degeneracy?

Answer 1

If Runge-Lenz vector in CM is a constant of motion that implied the motion is closed as well as bound, as you rightly stated.

Now, in QM because we are dealing with orbitals, we have no way of knowing whether the orbital is closed. Hence in QM, the terms 'closed' and 'bounded' are interchangeably used, which is a major reason for confusion for everyone when we return to CM. Hence conservation of Runge-Lenz in QM simply means that the orbital is bounded.

And if it commutes with Hamiltonian that leads a new degeneracy in energy levels. This is exactly the reason for the apparent accidental degeneracy in say Hydrogen atom. In a hydrogen atom, given any angular momentum there are usually other $l$s with the same energy.