How Are Quantum Eigenfunctions of Hydrogen Atom Related To Its Classical Elliptic Orbits?

We show that a highly-excited energy eigenfunction ${\psi }_{nlm}(\vec{r})$ of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy $E_n$ and angular momentum $L=\sqrt{l(l+1)}\mathrm{\hslash }$, and $z$ component of angular momentum $L_z=m\mathrm{\hslash }$. This correspondence is established by comparing the quantum probability distribution $\mathrm{\mid }{\psi }_{nlm}(\vec{r}){\mathrm{\mid }}^2$ and the classical probability distribution $p_c(\vec{r})$ 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.