On the Boson Number Operator in the Deformed Boson Scheme

Abstract

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