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Journal of Physics A Mathematical and General

Comment on the q-analogues of the harmonic oscillator

Yee Jack Ng

Published Mar 21, 1990• 146 citations
Publisher

Abstract

The Heisenberg relation for the q-analogues of the quantum harmonic oscillator (introduced independently by Macfarlane and Biedenharn (1989) for the quantum group SU(2) q ) is derived by a method analogous to that used by Schwinger (1968) for the SU(2) case. The author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics.

DOI
10.1088/0305-4470/23/6/022

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The Heisenberg relation for the q-analogues of the quantum harmonic oscillator (introduced independently by Macfarlane and Biedenharn (1989) for the quantum group SU(2) q ) is derived by a method analogous to that used by Schwinger (1968) for the SU(2) case. The author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics.
Nonlinearity of electromagnetic field vibrations described by q-oscillators is shown to produce essential dependence of second correlation functions on intensity and deformation of Planck distribution. Experimental tests of such nonlinearity are suggested. 1 on leave from Lebedev Physical Institute, Moscow 1 In the framework of investigations on quantum groups [1] the special case of the Heisenberg-Weyl algebra was first considered by Biedenharn [2] and Macfarlane [3]. They introduced ”q-oscillator operators ” starting from a deformed commutation relation containing a dimensionless parameter q. The physical sense of the quantun q-oscillator up to now was not clarified. In [4] a generalization to several modes has been analyzed. An attempt to connect quantum q-oscillator with relativistic oscillator model has been discussed in [5]. The q-oscillators and quantum group SUq(2) have also been considered
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.
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.