Discrete symmetries in classical and quantum oscillators

• URL: http://arxiv.org/abs/2601.01960v1
• Date: Mon, 05 Jan 2026 10:04:39 GMT
• Title: Discrete symmetries in classical and quantum oscillators
• Authors: Alexander D. Popov,
• Abstract summary: We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
• Score: 51.56484100374058
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions $ψ_n{=}z^n$ 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 $E_n=\hbarωn$, $n=0,1,2,...\,$. 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 $n\in\mathbb N$.

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