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Progress of Theoretical Physics Supplement

The Quantum Group as a Symmetry

Ursula Carow-Watamura, Satoshi Watamura

Published Jan 1, 1995• 2 citations
Publisher

Abstract

With the aim to construct a dynamical model with quantum group symmetry, the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the q-Euclidian space, the q-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the q-analysis. We represent the Schrödinger equation by the q-difference equation and solve it by using q-polynomials and q-exponential functions. The problem of the involution corresponding to the reality condition is discussed.

DOI
10.1143/ptps.118.375

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A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.
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Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.