On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

Abstract

We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics, based on a q-extension of the classical Hermite polynomials H n (x), recently introduced by us in [1]. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line R with respect to a positive weight function. A detailed description of the corresponding q-system is carried out.

A discrete quantum model of the harmonic oscillator

“…Contrary to other models that use discrete q-Hermite polynomials [17,18,19], our models (the present one and that in Ref. [8]) fulfill the basic Hamilton equations in the form [H, Q] = −iP and [H, P ] = iQ, with standard commutators -and not q-commutators [1,2].…”

On dynamical symmetry algebra for aq-extension of the harmonic oscillator

“…Finally, as in the case of the non-relativistic linear harmonic oscillator, one can construct q-coherent states for this model as eigenfunctions of the lowering operator a(x; q), that is, a(x; q) ~r (x; q) = ~ FC(x; q), (3.20) where ~ is some arbitrary number. The explicit form of these normalized q-coherent states ~i(x; q) are [2] IEq~/2 (--(1--ql/2)~ 2) Eq~/2 (ql/4vIl-ql/2x~) #r = v ~-~172~/2E-~ Eq (ql/2(1-ql/2)(2) (3.21)…”

“…Such a family enables one to build a q-deformed version of the linear harmonic oscillator in quantum mechanics, which is still defined on the whole real line R and enjoys the continuous orthogonality property on R with respect to a positive weight function. This q-model was constructed and studied in detail in [2]. The main goal in this paper is to derive a dynamical symmetry algebra for the above-mentioned q-extension of the linear harmonic oscillator.…”

“…The main goal in this paper is to derive a dynamical symmetry algebra for the above-mentioned q-extension of the linear harmonic oscillator. Section 2 collects those results from [1] and [2], which are needed in Section 3 in order to derive an explicit form of dynamical symmetry algebra for this q-model. **) E-maih mesuma 9 fc .uaem.mx t) E-marl: natig 9 where 0 < q < 1, Eq(x) is the q-exponential function, Eq(x) := (x;q)~, and the normalization constant dn(q) := q-n/a V/~ (@/2; q)1/2 (@/2; ql/2)n (we employ standard notations of q-analysis, see, for example [3] or [4]).…”

Generalized Picard singular integrals

“…Another needed formula is q-extension of Euler integral representation for the gamma function given in [17,18] for…”