No abstract

We study confinement effects on the energy eigenvalues, dipole moments and Einstein coefficients of a model harmonic oscillator restricted by two impenetrable walls placed either symmetrically or asymmetrically with respect to the potential minimum. The calculations are made using perturbation theory as a function of the position of the potential minimum with respect to the bounding walls. For small boxes, the energy levels resemble more closely those of a free particle in a box, than those of an unbounded harmonic oscillator. When the size of the box increases, the lowest energy levels become more similar to those of the unbounded harmonic oscillator, but the highest energy levels remain similar to those of the free particle in a box. We also show that the selection rules for the confined harmonic oscillator are not the same as those of the unbounded harmonic oscillator.

We construct the Perelomov number coherent states for any three su(1, 1) Lie algebra generators and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. We show that the most general SU (1, 1) coherence-preserving Hamiltonian has the Perelomov number coherent states as eigenfunctions, and we obtain their time evolution. We apply our results to obtain the non-degenerate parametric amplifier eigenfunctions, which are shown to be the Perelomov number coherent states of the two-dimensional harmonic oscillator.

We apply the Schrödinger factorization method to the radial secondorder equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1, 1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.

We apply the Schrödinger factorization to construct the ladder operators for hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions. By generalizing these operators we show that the dynamical algebra for these problems is the su(1, 1) Lie algebra.