How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits?
Abstract
Abstract 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 = 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.
Fields of Study & Subjects
Quantum chaos and dynamical systems
Quantum Mechanics and Non-Hermitian Physics
Quantum and Classical Electrodynamics
Publication details
- Journal / source
- Chinese Physics B
- Publisher
- IOP Publishing
- Volume
- 35
- Issue
- 5
- Pages
- 050301–050301
- Publication date
- 4 January 2026