Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System

TL;DR

This paper explores a quantum model of a chain of harmonic oscillators using Wigner's approach, revealing discrete spectra and eigenvalues through Lie superalgebra representations, with implications for quantum state properties.

Contribution

It introduces a Wigner quantum system framework for coupled harmonic oscillators and analyzes spectra using gl(1|M) Lie superalgebra representations, providing new insights into their eigenvalues and state distributions.

Findings

Discrete spectra for all physical operators.

Complete eigenvalue analysis of Hamiltonian, position, and momentum operators.

Probability distributions of positions in stationary states.

Abstract

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.