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Deformed Oscillators with Interaction

O. Batsevych

Published Jan 1, 1997• 0 citations
OpenAlex

Abstract

Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,

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Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,
It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.
In the weak deformation (WD) approximation of the Weyl–Heisenberg algebra, the corresponding generalized coherent states and displacement operator are constructed. It is shown that those states, and contrary to the non-deformed Weyl-Heisenberg algebra, are not eigenstates of the annihilation operator. Moreover, and as an alternative to the Chaïchian et al. Q-deformed path integral approach (where Q is the deformation parameter), using the Bargmann Fock representation, we propose in the WD approximation, a general simple formalism. As an application, we calculate the propagator and the wave function of the harmonic oscillator.PACS Nos.: 03.65.Fd, 31.15.Kb
We propose a new kind of two-parameter (p, q)-deformed Heisenberg and parabose algebra, which reduces to the Heisenberg algebra for the p = 1 case and to parabose algebra for q = -1 case. Corresponding to the two-parameter deformed oscillator, we also introduce a new kind of (p, q)-deformed derivative which relates to the ordinary derivative and q-deformed derivative in an explicit manner. We study the structure of Fock-like space of the new (p, q)-deformed oscillators and derive a formal solution for the eigenvalue equation of the Hamiltonian.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.