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International Journal of Modern Physics A

CONTRACTION OF THE FINITE RADIAL OSCILLATOR

Natig M. Atakishiyev, GEORGE S. POGOSYAN, Kurt Bernardo Wolf

Published Jan 16, 2003• 9 citations
Publisher

Abstract

The finite radial oscillator model, introduced in [J. Phys.A34, 9399–9415 (2001)], is based on the precontracted dynamical algebra so4, consisting of two position and two momentum operators, total mode number and angular momentum, all with a finite number of eigenvalues. We examine the contraction of this model to the ordinary radial quantum oscillator as the number and density of points increase. This is done on the level of the dynamical algebra, of the Schrödinger difference equation, and of the wave functions.

DOI
10.1142/s0217751x03013752

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The finite radial oscillator model, introduced in [J. Phys.A34, 9399–9415 (2001)], is based on the precontracted dynamical algebra so4, consisting of two position and two momentum operators, total mode number and angular momentum, all with a finite number of eigenvalues. We examine the contraction of this model to the ordinary radial quantum oscillator as the number and density of points increase. This is done on the level of the dynamical algebra, of the Schrödinger difference equation, and of the wave functions.
The finite oscillator model of 2j + 1 points has the dynamical algebra u(2), consisting of position, momentum and mode number. It is a paradigm of finite quantum mechanics where a sequence of finite unitary models contract to the well-known continuum theory. We examine its contraction as the number and density of points increase. This is done on the level of the dynamical algebra, of the Schrödinger difference equation, the (Kravchuk) wave functions, and the Fourier–Kravchuk transformation between position and momentum representations.
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 2 j + 1 'sensor'-points , 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 Schrödinger 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 = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.
The one-dimensional finite oscillator model satisfies that under commutator brackets, all operators close into the (dynamical) Lie algebra su(2); therefore the eigenvalues of position, momentum and energy operators are discrete and the eigenvalue of a compact u(l) central operator fixes the dimension of the su(2) representation. In this work we review the one-dimensional finite oscillator model with u(2) as the dynamical algebra where the wave functions, which involve Kravchuk polynomials, are discrete functions of angular momentum theory.
A finite oscillator is a system that has a discrete and finite number of spatial positions and energy levels accessible. The su(2)-oscillator model takes the generators of the angular momentum su(2) algebra and, giving to each one a definite physical interpretation, provides a consistent mathematical description of a finite oscillator. This work searches an example of identification of generalized coherent states, within the collection of states and wavefunctions of the finite su(2) oscillator model.
Abstract In Chapter 4 we considered a ‘twodimensional’ problem in which L2 and Lz constitute a c.s.c.o. and we derived shift operators for the angular momentum quantum numbers l and m in the eigenkets |lm> of these observables. In Chapter 5 we discussed a ‘one-dimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (eqn (5.25)). Here we use results from Chapters 4 and 5 to construct shift operators for the eigenkets of the three dimensional problem (5.11)–(5.13).
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.