Finite q-oscillator

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 is an element of {-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 wavefunctions 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 as 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 harmonic oscillator.