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

Abstract

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,

Structural localization in the classical and quantum Fermi–Pasta–Ulam model

“…For the purpose of the present work, from Eqs. ( 33) to (35), we can extract the condition α = β (39)…”

“…(100). Despite this critical simplification, the system is still challenging from an analytical perspective [35]. In the present appendix we derive a general expression for the distribution of a subset of normal modes of a canonical mixture, and we use the result for the calculation of the average energy per normal mode.…”

Interacting universes and the cosmological constant

“…Let us consider a multiverse of N interacting de-Sitter universes represented, in the third quantization formalism, by harmonic oscillators like those described in the preceding section, with scale factor dependent mass and frequency, given respectively by M(a) = a and ω 2 (a) ≈ Λa 4 , where Λ is the value of cosmological constant of the de-Sitter universes. Following references [19][20][21], let us assume in the multiverse some kind of 'nearest interaction' described by a total Hamiltonian given bŷ…”

“…Following Refs. [19][20][21], for a given representation, the energy spectrum splits into a large number of different levels, like in other collective phenomena (crystals, phonons,...). Given some conditions, the ground state of the new spectrum approaches to zero.…”

“…The condition the operators a ± r must satisfy in order to comply with the canonical commutation relations turns out to be equivalent than to require that [19][20][21] [Ĥ, a ± r ] = ± ω r a ± r .…”