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

@article{Lievens2007HarmonicOC,
  title={Harmonic oscillator chains as Wigner quantum systems: Periodic and fixed wall boundary conditions in gl(1|n) solutions},
  author={Stijn Lievens and N I Stoilova and Joris Van der Jeugt},
  journal={Journal of Mathematical Physics},
  year={2007},
  volume={49},
  pages={073502}
}
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… 

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References

SHOWING 1-10 OF 80 REFERENCES

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

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,

SL(3|N) Wigner quantum oscillators: examples of ferromagnetic-like oscillators with noncommutative, square-commutative geometry

A system of $N$ non-canonical dynamically free 3D harmonic oscillators is studied. The position and the momentum operators (PM-operators) of the system do not satisfy the canonical commutation

Wigner quantum systems. Two particles interacting via a harmonic potential. I. Two‐dimensional space

A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center‐of‐mass position and momentum operators obey the canonical

Representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis and Wigner quantum oscillators

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra in a Gel'fand–Zetlin basis is given. Particular attention is paid to the so-called star type I

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(13), are further investigated. Within each state space W(p), p =

Wigner approach to quantization. Noncanonical quantization of two particles interacting via a harmonic potential

Following the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase‐space variables are quantized

Wigner quantum oscillators. osp(3/2) oscillators

The properties of the three-dimensional non-canonical oscillators osp(3/2) (introduced in J. Phys. A: Math. Gen. 27 (1994) 977) are studied further. The angular momentum M of the oscillators can take

Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom

We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest-neighbour interactions. For a

Classical dynamics of the quantum harmonic chain

The origin of classical predictability is investigated for the one dimensional harmonic chain considered as a closed quantum mechanical system. By comparing the properties of a family of

Many-body Wigner quantum systems

We present examples of many-body Wigner quantum systems. The position and the momentum operators RA and PA, A=1,…,n+1, of the particles are noncanonical and are chosen so that the Heisenberg and the
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