The Geometry Underlying the Quantum Harmonic Oscillator

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^{\ast }{\mathbb{R}}^2={\mathbb{C}}^2$ 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 ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n\subset {\mathbb{C}}^2$. They describe ${\mathbb{Z}}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3\mathrm{/}{\mathbb{Z}}_n\subset {\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n$, where ${\mathbb{Z}}_n$ is the cyclic group of rotation by an angle $2\pi \mathrm{/}n$ on the circle $S^1$, $n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$. 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.