Hydrogen spectrum for transitions of $\ell$, $m_{\ell}$

Question

In this question, there is a spectrum exhibiting transitions corresponding to the azimuthal quantum number $\ell$ of a system corresponding to a spinning $\rm{Cs_2}$ molecule.

Regarding hydrogen-like atoms, I can find spectrum (e.g. Lyman series) corresponding to transitions of the principal quantum number $n$, but no equivalent for the azimuthal and magnetic quantum numbers $\ell$, $m_l$.

Are there experimental evidences for the quantification of $\ell$, $m_l$ in the case of an hydrogen-like atom?

A link to experimental results exhibiting such quantification would be appreciated.

Answer 1

In the Coulomb model, which is manifestly invariant under rotations, the $m$ labeled levels are necessarily degenerate. There is also a hidden $SO(4)$ symmetry that leaves the states of same $l$ degenerate. Leaving the energy only dependent on $n$. Note this same double degeneracy leaves the orbital energy of a planet independent of its eccentricity—-a result that often surprises new students of orbital mechanics.

The degeneracy is lifted by the fine structure, which includes several terms, and the hyperfine structure which is due to coupling with nuclear magnetic moments.