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arXiv:math-ph/0401035 · Journal of Physics A Mathematical and General

Finiteq-oscillator

Natig M. Atakishiyev, Anatoliy U. Klimyk, Kurt Bernardo Wolf

Published May 11, 2004• 14 citations
arXivHTMLPDF

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 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.

DOI
10.1088/0305-4470/37/21/005

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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.
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [− 1, 1] that depends on the deformation parameter q ∊ (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q -Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q → 1 − , one recovers the ordinary quantum 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.
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.
Abstract We develop an algebraic formulation for the discrete quantum harmonic oscillator (DQHO) from the Hamiltonian for two, coupled QHOs and provide a physical picture for the Kravchuk function eigenstates of the oscillator. The familiar su ( 2 ) structure of the coupled QHO Hamiltonian divides its spectrum into sets corresponding to the DQHO at different resolutions. In addition to energy ladder operators, the formulation allows for the introduction of resolution ladder operators connecting all DQHOs with different resolutions, thus enabling the dynamic scaling of the resolution of finite degree-of-freedom quantum simulations. The coherent state of the DQHO is constructed, and its expected position is proven to oscillate as a classical harmonic oscillator. The DQHO coherent state recovers that of the quantum harmonic oscillator at large resolution.
We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10−7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann’s complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier–Kravchuk transform.
A finite two-dimensional oscillator is built as the direct product of two finite one-dimensional oscillators, using the dynamical Lie algebra su(2)x⊕su(2)y. The position space in this model is a square grid of points. While the ordinary `continuous' two-dimensional quantum oscillator has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)x⊕u(1)y, because it lacks rotations in the position (and momentum) plane. We show how to `import' an SO(2) group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid. We thus propose a finite analogue for fractional U(2) Fourier transforms.