Discrete symmetries in classical and quantum oscillators

We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions ψ_n{=}zⁿ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with zinmathbb C are the coordinates of a classical oscillator with energy E_n=hbarωn, n=0,1,2,... . They are defined on conical spaces {mathbb C}/{mathbb Z}_n with cone angles 2π/n, which are embedded as subspaces in the phase space mathbb C of the classical oscillator. Here {mathbb Z}_n is the finite cyclic group of rotations of the space mathbb C by an angle 2π/n. The superposition ψ=sum_n c_nψ_n of the eigenfunctions ψ_n arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups {mathbb Z}_n are not imposed and the general solution takes into account all possible initial data parametrized by the numbers ninmathbb N.