Representations of the Lie superalgebra in a Gel'fand–Zetlin basis and Wigner quantum oscillators

Journal of Mathematical Physics

2006

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. †

“…In particular, one should also consider other classes of unitary gl(1|M ) representations [26] and investigate the corresponding physical properties.…”

Section: Discussionmentioning

confidence: 99%

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

Journal of Mathematical Physics

2006

“…Furthermore, we considered a class of Fock representations [12] of gl(1|n) and analysed the energy spectrum and the position operator spectrum in these Fock representations W (p). These Fock representations are rather restricted, however, and do not illustrate the features of general unitary irreducible representations [13] of gl(1|n) (more precisely, of its compact form u(1|n)).…”

Section: Introductionmentioning

confidence: 98%

“…In the present paper, we consider a more general class of gl(1|n) representations, the so-called ladder representations V (p) [13]. These representations are also easy to describe, but more importantly they show interesting properties of the physical quantities (energy spectrum, position operator spectrum) of the system, far more general than those of the Fock representations W (p).…”

Section: Introductionmentioning

confidence: 99%

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

Journal of Mathematical Physics

2008

We describe a quantum system consisting of a one-dimensional 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 (WQS), 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(1|n). We study a particular class of gl(1|n) representations V (p), the so-called 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. †

“…These unitary representations W = W ([m] n+1 ) of gl(1|n) are well known [16]: they are labeled by some (n + 1)-tuple [m] n+1 subject to certain conditions. Even more: for such representations, a Gel'fand-Zetlin basis has been constructed and the explicit action of the gl(1|n) generators on the basis vectors of the representation is also known [13]. Explicit actions of generators on a Gel'fand-Zetlin basis are usually quite involved, and this is also the case for gl(1|n).…”

Section: Unitary Irreducible Representations Of Gl(1|n)mentioning

confidence: 99%

“…For the finite-dimensional unirreps of gl(1|n), a Gel'fand-Zetlin basis (GZbasis) is known, as well as the explicit action of the generators on the basis vectors of the representation [13]. These actions, especially the non-diagonal ones, on a general GZ-pattern are, however, quite involved and thus some of the tasks at hand (e.g.…”

Section: Introductionmentioning

confidence: 99%

A linear chain of interacting harmonic oscillators: solutions as a Wigner quantum system

2008

J. Phys.: Conf. Ser.

Self Cite

Abstract. We consider a quantum mechanical system consisting of a linear chain of harmonic oscillators coupled by a nearest neighbor interaction. The system configuration can be closed (periodic boundary conditions) or open (non-periodic case). We show that such systems can be considered as Wigner Quantum Systems (WQS), thus yielding extra solutions apart from the canonical solution. In particular, a class of WQS-solutions is given in terms of unitary representations of the Lie superalgebra gl(1|n). In order to determine physical properties of the new solutions, one needs to solve a number of interesting but difficult representation theoretical problems. We present these problems and their solution, and show how the new results yield attractive properties for the quantum system (energy spectrum, position probabilities, spacial properties). IntroductionCoupled systems describing the interaction of oscillating or scattering subsystems and the corresponding operators have been widely used in classical and quantum mechanics [1][2][3][4][5].Recently, we have taken up the study of such systems as Wigner Quantum Systems (WQS) [6][7][8]. The particular system under consideration consists of a string 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, . . . , n) are given byq r andp r respectively; more precisely,q r measures the displacement of the rth mass point with respect to its equilibrium position. The oscillators are coupled by some nearest neighbor coupling, represented by terms of the form (q r −q r+1 ) 2 in the Hamiltonian. The system configuration can either be closed (periodic boundary conditions), i.e.q n+1 =q 1 , or open (fixed wall boundary conditions), i.e.q 0 =q n+1 = 0.When treating a system as a WQS, the canonical commutation relations (CCRs) are not required to hold. Instead, the compatibility of the Hamilton equations and the Heisenberg equations is imposed. Expressing this compatibility leads to the so-called compatibility conditions (CCs), and these need to be solved (subject to some unitarity conditions). In [9, 10], we have shown that the CCs associated with a system consisting of a coupled oscillator chain has solutions in terms of generators of the gl(1|n) Lie superalgebra [11]. We are only going to work with finite-dimensional unitary irreducible representations (unirreps) of gl(1|n), so in fact,