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arXiv:1502.00464 · Journal of Physics Conference Series

The su(2) Krawtchouk oscillator model under theCPdeformed symmetry

E. I. Jafarov, A. M. Jafarova, Joris Van Der Jeugt

Published Apr 13, 2015• 2 citations
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Abstract

We define a new algebra, which can formally be considered as a CP deformed su(2) Lie algebra. Then, we present a one-dimensional quantum oscillator model, of which the wavefunctions of even and odd states are expressed by Krawtchouk polynomials with fixed p = 1/2, K 2 n ( k ; 1/2, 2 j ) and K 2 n ( k — 1; 1/2, 2 j — 2). The dynamical symmetry of the model is the newly introduced su(2) CP algebra. The model itself gives rise to a finite and discrete spectrum for all physical operators (such as position and momentum). Among the set of finite oscillator models it is unique in the sense that any specific limit reducing it to a known oscillator models does not exist.

DOI
10.1088/1742-6596/597/1/012047

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We define a new algebra, which can formally be considered as a CP deformed su(2) Lie algebra. Then, we present a one-dimensional quantum oscillator model, of which the wavefunctions of even and odd states are expressed by Krawtchouk polynomials with fixed p = 1/2, K 2 n ( k ; 1/2, 2 j ) and K 2 n ( k — 1; 1/2, 2 j — 2). The dynamical symmetry of the model is the newly introduced su(2) CP algebra. The model itself gives rise to a finite and discrete spectrum for all physical operators (such as position and momentum). Among the set of finite oscillator models it is unique in the sense that any specific limit reducing it to a known oscillator models does not exist.
A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU (2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials, those of Rahman and those of Tratnik. These polynomials form bases for SU (2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families is related to each other by an SU (2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points tends to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the N → ∞ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.
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.
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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.
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.
The Wigner distribution function for the su(2)\mathfrak {su}(2)su(2) finite oscillator and Dyck paths
Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the su(2)\mathfrak{su}(2)su(2) finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the well known Dyck paths. This combinatorial expression of the pre-Wigner matrix elements turns out to be particularly simple.