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Journal of Mathematical Physics

Transformation theory of the q-oscillator

Robert J. Finkelstein, Eran Marcus

Published Jun 1, 1995• 40 citations
Publisher

Abstract

A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.

DOI
10.1063/1.531057

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A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.
It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.
There exist operator solutions of the q-Coulomb problem in both configuration and momentum space. Since the arguments of the corresponding amplitudes are noncommuting, however, there are problems of physical interpretation. Here we answer the question of physical interpretation by associating with the operator amplitude in momentum space a numerically valued amplitude lying in a Hilbert space defined by the SUq(2) algebra. This new amplitude, now depending on commuting arguments, may be interpreted by the usual rules of quantum mechanics and may also be Fourier transformed to yield the amplitude in configuration space.
Relation between Bopp-Kubo formulation and Weyl-Wigner-Moyal symbol calculus, and non-commutative geometry interpretation of the phase space representation of quantum mechanics are studied. Harmonic oscillator in phase space via creation and annihilation operators, both the usual and qqq-deformed, is investigated. We found that the Bopp-Kubo formulation is just non-commuting coordinates representation of the symbol calculus. The Wigner operator for the qqq-deformed harmonic oscillator is shown to be proportional to the 3-axis spherical angular momentum operator of the algebra suq(2)su_{q}(2)suq​(2). The relation of the Fock space for the harmonic oscillator and double Hilbert space of the Gelfand-Naimark-Segal construction is established. The quantum extension of the classical ergodiicity condition is proposed.
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.