How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits?

We show that a highly-excited energy eigenfunction ψ_nlm (r) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy E_n and angular momentum $L=\sqrt{l(l+1)}ħ$, and z component of angular momentum L_z = mħ. This correspondence is established by comparing the quantum probability distribution |ψ_nlm (r)|² and the classical probability distribution p_c (r) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.