Abstract
We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with T^*R²=C² as classical phase space. We show that the eigenfunctions ψ_n of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space C²/Z_n⊂C². They describe Z_n-invariant motion of particle along a circle S¹ in lens space S³/Z_n⊂C²/Z_n, where Z_n is the cyclic group of rotation by an angle 2π/n on the circle S¹, n=1,2,.... Thus the general solution of the Schr\"odinger equation carries information about an infinite number of admissible classical states ψ_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.