Contraction of the Finite One-Dimensional Oscillator

Abstract

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.

Group Theory in Finite Hamiltonian Systems

“…In one space dimension we can have the spin SU(2) group (β = 1) [1,2], the Lorentz group (β = −1) SU(1, 1) [3], the group of Euclidean motions ISO(2) (β = 0) [4], or any three-parameter algebra, or q-algebra [5,6,7,8], which contracts to HW. This postulate thus entails the correspondence principle: when the number and density of points increase without limit in a discrete Hamiltonian system, the limit should be some well-known continuous Hamiltonian system [9,10].…”

Rotation and gyration of finite two-dimensional modes

“…uniformly to the HG n ͑x͒ mode with x R [7]. In two dimensions, N ϫ N pixellated images are linear combinations of the simultaneous Kronecker eigenbases of Q x and Q y , and correspondingly, the two-dimensional finite harmonic oscillator (Kravchuk) functions are the overlaps between the Kronecker eigenbasis of positions and the number eigenbasis of N x and N y .…”

“…Correspondingly, in the su͑2͒ finite model (8), the domestic number operator N o generates the fractional Fourier-Kravchuk trans- form K͑␤͒ = exp͑−i␤N o ͒ [10], which is represented by unitary matrices of dimension N =2j + 1 forming a group with ␤ modulo 2. When N → ϱ, the fractional FourierKravchuk transform becomes (weakly) the fractional Fourier integral transform [7].…”

“…There, the 4-parameter Fourier group is represented by N 2 ϫ N 2 matrices that are unitary and converge (weakly) to the corresponding integral kernels in the N → ϱ limit [7]. The finite analogs of the Hermite-Gauss (HG) and LaguerreGauss (LG) modes will be presented in this paper and subjected to fractional gyration.…”

Finite systems, fractional Fourier transforms and their finite phase spaces

“…• In the limit N → ∞, the su(2) algebra contracts to the Heisenberg-Weyl algebra plus the usual harmonic oscillator Hamiltonian; the fractional Fourier-Kravchuk transforms limit to the Fourier integral transforms for all powers α [19].…”