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arXiv:math/9307206 · Journal of Physics A Mathematical and General

The q-harmonic oscillator and an analogue of the Charlier polynomials

Richard Askey, Sergeĭ K. Suslov

Published Aug 7, 1993• 35 citations
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Abstract

A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz (1965) is discussed. A simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analogue of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.

DOI
10.1088/0305-4470/26/15/014

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A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz (1965) is discussed. A simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analogue of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.
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Overlap integrals over the full real line for a family of q- extensions of the linear harmonic oscillator wave functions in quantum mechanics are evaluated. In particular, an explicit form of the squared norms for these q-wave functions is obtained. The classical Fourier-Gauss transform connects the families with different values 0 < q < 1 and q < 1 of the deformation parameter q. An explicit expansion of the q-Hermite polynomials of Rogers in terms of the ordinary Hermite polynomials emerges as a by-product.
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [− 1, 1] that depends on the deformation parameter q ∊ (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q -Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q → 1 − , one recovers the ordinary quantum harmonic oscillator.
Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,ℜ\Reℜ). The reproducing kernel for the wavefunctions of these models is also found.