Harmonic oscillator chains as Wigner quantum systems: Periodic and fixed wall boundary conditions in gl(1∣n) solutions
Abstract
We describe a quantum system consisting of a onedimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the nth oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anticommutators. These triple relations have a solution in terms of the Lie superalgebra gl(1n). We study a particular class of gl(1n) representations V(p ), the socalled ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position, and momentum of each oscillator) have a finite spectrum.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 July 2008
 DOI:
 10.1063/1.2948894
 arXiv:
 arXiv:0709.0180
 Bibcode:
 2008JMP....49g3502L
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 J.Math.Phys.49:073502,2008