Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

Abstract

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, 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. †

Harmonic oscillator chains as Wigner quantum systems: Periodic and fixed wall boundary conditions in gl(1|n) solutions

“…In [7], it was shown that one can relax the canonical commutation relations for the operatorŝ q r andp r , leading to a larger class of solutions for the system described by (2.1). This is known as a Wigner Quantum System approach [9,10].…”

“…The system studied in [7] consists of a string or chain of n identical harmonic oscillators, each having the same mass m and frequency ω. The position and momentum operator for the rth oscillator (r = 1, 2, .…”

“…The oscillators are coupled by some nearest neighbor coupling, represented by terms of the form (q r −q r+1 ) 2 in the Hamiltonian. In [7], we considered such a system with periodic boundary conditions, where the last oscillator is again coupled to the first one (i.e.q n+1 =q 1 ). In other words, the oscillators are on a circle rather than on a line.…”

“…The original contribution of [7] consisted of the treatment of the system described byĤ P as a Wigner Quantum System [8][9][10]. This quantization procedure is based upon the compatibility of Hamilton's equations and the Heisenberg equations.…”

“…As a consequence, one obtains different classes of solutions for the system, of which the canonical solution is only one. For a more detailed description of WQSs, see [7] and references therein. The main feature of this treatment is that the CCRs are replaced by these compatibility conditions (CCs).…”