scholarly journals The Two-dimensional Harmonic Oscillator on a Noncommutative Space with Minimal Uncertainties

10.14311/1799 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Sanjib Dey ◽  
Andreas Fring
Keyword(s):  
Harmonic Oscillator ◽  
Pt Symmetry ◽  
Exceptional Point ◽  
Noncommutative Space ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Eigenvalue Spectrum ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals THREE-PARAMETER (TWO-SIDED) DEFORMATION OF HEISENBERG ALGEBRA

2012 ◽  
Vol 27 (21) ◽  
pp. 1250114 ◽  
Author(s):  
A. M. GAVRILIK ◽  
I. I. KACHURIK
Keyword(s):  
Deformation Parameter ◽  
Particle Number ◽  
Heisenberg Algebra ◽  
Unusual Property ◽  
Oscillator Algebra ◽  
Number Operator ◽  
The Third ◽  
Defining Relation ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

A three-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the L.H.S. of basic defining relation and certain deformation of its R.H.S., is introduced and studied. The third deformation parameter μ appears in an extra term in the R.H.S. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on particle number operator N. Such a novel N-dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

Representations of the deformed oscillator algebra for different choices of generators

2000 ◽  
Vol 100 (2) ◽  
pp. 2061-2076 ◽  
Author(s):  
V. V. Borzov ◽  
E. V. Damaskinskii ◽  
S. B. Yegorov
Keyword(s):  
Oscillator Algebra ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

On a family of photon-added deformed coherent states associated with two-parameter deformed boson oscillator algebra

2004 ◽  
Vol 82 (8) ◽  
pp. 623-646 ◽  
Author(s):  
M H Naderi ◽  
M Soltanolkotabi ◽  
R Roknizadeh
Keyword(s):  
Coherent States ◽  
Photon Number ◽  
Dimensional Subspace ◽  
Oscillator Algebra ◽  
Infinite Dimensional ◽  
Quadrature Squeezing ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Two Parameter ◽  
Infinite Dimensional Subspace

By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w

scholarly journals Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR

1995 ◽  
Vol 10 (33) ◽  
pp. 2543-2551 ◽  
Author(s):  
KAZUO FUJIKAWA ◽  
L.C. KWEK ◽  
C.H. OH
Keyword(s):  
Deformation Parameter ◽  
Matrix Representation ◽  
Index Theorem ◽  
Oscillator Algebra ◽  
Quantum Deformation ◽  
Phase Operator ◽  
Negative Norm ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Index Condition

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).

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