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

TL;DR

This paper demonstrates that highly-excited hydrogen atom eigenfunctions can be approximated by a superposition of classical elliptic orbits, illustrating the quantum-classical correspondence in the semi-classical limit.

Contribution

It establishes a direct link between quantum eigenfunctions and classical elliptic orbits for the hydrogen atom, showing eigenstates correspond to orbit ensembles.

Findings

Quantum probability distribution matches classical orbit ensemble distribution.

Eigenfunctions approximate superpositions of classical elliptic orbits.

Supports the principle that quantum states reduce to orbit collections semi-classically.

Abstract

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 $L=\sqrt{l(l+1)}\hbar$, and $z$ component of angular momentum $L_z=m\hbar$. This correspondence is established by comparing the quantum probability distribution $|\psi_{nlm}(\vec{r})|^2$ 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.