The Geometry Underlying the Quantum Harmonic Oscillator

TL;DR

This paper explores the geometric structure of the quantum harmonic oscillator in complex phase space, revealing a correspondence between quantum eigenfunctions and classical circular motions in lens spaces, with implications for the hydrogen atom.

Contribution

It uncovers a geometric interpretation linking quantum eigenfunctions to classical trajectories in complex reduced phase space, extending to the hydrogen atom.

Findings

Eigenfunctions correspond to complex radial coordinates in reduced phase space.

Quantum states relate to classical circular motions in lens spaces.

Classical-quantum correspondence extends to the hydrogen atom.

Abstract

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^*{\mathbb R}^{2}={\mathbb C}^2$ as classical phase space. We show that the eigenfunctions $\psi_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space ${\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2$. They describe ${\mathbb Z}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n$, where ${\mathbb Z}_n$ is the cyclic group of rotation by an angle $2\pi/n$ on the circle $S^1$, $n=1,2,...\,$. Thus the general solution of the Schr\"odinger equation carries information about an infinite number of admissible classical states $\psi_n$ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.