Discrete quantum model of the harmonic oscillator

Details Discrete quantum model of the harmonic oscillator Discrete quantum model of the harmonic oscillator

2007-11-20

arxiv.org/abs/0711.3089v1

Physics

Mathematical Physics

21 pages

Scientific paper

10.1088/1751-8113/41/8/085201

We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bounded, whereas the spectra of position and momentum are a denumerable non-degenerate set of points in [-1,1] that depends on the deformation parameter q from (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 q to 1, one recovers the ordinary quantum harmonic oscillator.

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