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Finite q-oscillator

Natig M. Atakishiyev·Anatoliy U. Klimyk·Kurt Bernardo Wolf
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Abstract

The finite q-oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q-algebra su_q(2). The spectrum of position in this discrete system, in a fixed representation j, consists of 2j+1 "sensor"-points x_s=(1/2)[2s]_q, s=-j, -j+1,..., j, and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wave functions involve dual q-Kravchuk polynomials, which are solutions to a finite-difference Schrodinger equation. Time evolution (times a phase) defines the fractional Fourier-q-Kravchuk transform. In the classical limit q -> 1 we recover the finite oscillator Lie algebra, the N=2j -> infinity limit returns the Macfarlane-Biedenharn q-oscillator and both limits contract the generators to the standard quantum-mechanical oscillator.

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Natig M. Atakishiyev, Anatoliy U. Klimyk, Kurt Bernardo Wolf. 2004-01-19. Finite q-oscillator. https://doi.org/10.1088/0305-4470%2F37%2F21%2F005

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