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We describe a quantum system consisting of a one-dimensional linear chain ofn identical harmonic oscillators coupled by a nearest neighbor interaction. Twoboundary conditions are taken into account: periodic boundary conditions (wherethe nth oscillator is coupled back to the first oscillator) and fixed wallboundary conditions (where the first oscillator and the $n$th oscillator arecoupled to a fixed wall). The two systems are characterized by theirHamiltonian. For their quantization, we treat these systems as Wigner QuantumSystems (WQS), allowing more solutions than just the canonical quantizationsolution. In this WQS approach, one is led to certain algebraic relations foroperators (which are linear combinations of position and momentum operators)that should satisfy triple relations involving commutators andanti-commutators. These triple relations have a solution in terms of the Liesuperalgebra gl(1|n). We study a particular class of gl(1|n) representationsV(p), the so-called ladder representations. For these representations, wedetermine the spectrum of the Hamiltonian and of the position operators (forboth types of boundary conditions). Furthermore, we compute the eigenvectors ofthe position operators in terms of stationary states. This leads to explicitexpressions for position probabilities of the n oscillators in the chain. Ananalysis of the plots of such position probability distributions gives rise tosome interesting observations. In particular, the physical behavior of thesystem as a WQS is very much in agreement with what one would expect from theclassical case, except that all physical quantities (energy, position andmomentum of each oscillator) have a finite spectrum.

Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions