ON FINITE QUANTUM OSCILLATORS

We construct and study new models of a finite quantum oscillator satisfying the quantum mechanics relations [H,Q] = -iP and [H,P] = iQ. These models are related to finite dimensional representations of the quantum algebra su(q) (2). The position and momentum operators and the Hamiltonian are elements of the algebra su(q) (2). The models depend on the index j of an irreducible representation of su(q) (2). The spectra of the position and momentum operators consist of a finite set of points (for this reason, these models are called finite). The time evolution for each of the models is evaluated explicitly. The limit j -> 1 leads to a Macfarlane-Biedenharn q-oscillator.