Discrete symmetries in classical and quantum oscillators

TL;DR

This paper explores the relationship between classical and quantum harmonic oscillators, showing that quantum eigenfunctions correspond to classical phase space structures with discrete symmetries, revealing geometric and group-theoretic insights.

Contribution

It demonstrates that quantum eigenfunctions of the harmonic oscillator are linked to classical phase space structures with discrete symmetries, providing a geometric interpretation of quantum states.

Findings

Eigenfunctions correspond to conical spaces with cyclic group symmetries

Superpositions arise from incomplete initial data without symmetry constraints

Quantum states relate to classical phase space with discrete symmetries

Abstract

We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions $\psi_n{=}z^n$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with $z\in\mathbb C$ are the coordinates of a classical oscillator with energy $E_n=\hbar\omega n$, $n=0,1,2,...\,$. They are defined on conical spaces ${\mathbb C}/{\mathbb Z}_n$ with cone angles $2\pi/n$, 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 $2\pi/n$. The superposition $\psi =\sum_n c_n\psi_n$ of the eigenfunctions $\psi_n$ arises only with incomplete knowledge of the initial data for solving the Schr\"odinger 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 $n\in\mathbb N$.