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Physical Review C

Nonstandardq-deformed realizations of the harmonic oscillator

Ángel Ballesteros, Osvaldo Civitarese, M. Reboiro

Published Jul 13, 2005• 15 citations
Publisher

Abstract

The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use two different boson expansions, each of them including a deformation parameter, defined in terms of exponential and logarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physical effects resulting from the deformation of the generators of the algebra are studied by comparing known finite-range potentials and the effective potentials obtained for each of the considered Hamiltonians.

DOI
10.1103/physrevc.72.014305

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The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use two different boson expansions, each of them including a deformation parameter, defined in terms of exponential and logarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physical effects resulting from the deformation of the generators of the algebra are studied by comparing known finite-range potentials and the effective potentials obtained for each of the considered Hamiltonians.
Considering a simple generalization of the (p,q)-deformed boson oscillator algebra, which leads to a two-parameter deformed bosonic algebra in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first finite Fock states, we establish a new harmonic oscillator realization of the deformed boson operators based on the Bogoliubov (p,q)-transformations. We obtain exact expressions for the transformation coefficients and show that they depend on arbitrary functions of p and q which can be interpreted as the parameters of the (p,q)-deformed GL(2,C) group. We also examine the existence and structure of the corresponding deformed Fock-space representation for our problem.
Concerning the energy of harmonic oscillator, a prescription is proposed for making the original form unchanged even after q-deformation. Applicability of the prescription is limited, but, it can be applied to various cases which are well known. 1 typeset using PTPTEX.sty <ver.1.0>An idea of q-deformation serves us an interesting viewpoint for the understanding of dynamics of many-body systems described by boson operators. For the q-deformation, the present authors recently gave a possible scheme in relation to the generalized boson coherent state. 1)-3) It is, in some sense, an extension of the idea given by Penson and Solomon. 4) Before entering the central part of this paper, first, we recapitulate the basic idea of Ref. 1). We treat a boson system described by one kind of boson (ĉ, ĉ ∗). As the deformation of (ĉ, ĉ ∗), the operator (ˆγ, ˆγ ∗ ) is introduced in the definition ˆγ = F ( ˆ N)ĉ, ˆγ ∗ = ĉ ∗ F ( ˆ N). (1) Here, F ( ˆ N) denotes a function of the boson number operator, ˆ N, defined by ˆN = ĉ ∗ ĉ. (2) The function F ( ˆ N) obeys a certain condition which is proper to each case under investigation. The present deformation is characterized by the function F ( ˆ N). The commutation relations [ĉ, ĉ ∗ ] and [ˆγ, ˆγ ∗ ] are as follows: [ĉ, ĉ ∗ ] = 1, [ˆγ, ˆγ ∗ ] = ( ˆ N + 1)F ( ˆ N) 2 − ˆ NF ( ˆ N − 1) 2. (3a) (3b) Later, we discuss the case in which the relation (3b) does not hold. If following our deformed boson scheme, any operator which is expressed in terms of a function (ĉ, ĉ ∗), Ô = O(ĉ, ĉ ∗), is deformed by replacing (ĉ, ĉ ∗ ) with (ˆγ, ˆγ ∗ ): Ôq = O(ˆγ, ˆγ ∗ ). (4) However, in the case of the Hamiltonian, a special consideration is necessary for the replacement
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.