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arXiv:math-ph/0012029

A unified representation of the q-oscillator and the q-plane

T. Rador

Published Dec 14, 2000• 0 citations
arXivHTMLPDF

Abstract

Using deformations inspired by relativistic considerations and phase space symmetry, we deform the position and momentum operators in one dimension. The resulting algebra is shown to yield the q-oscillator algebra in one limiting case and the q-plane commutation relation in another.
DOI
10.48550/arxiv.math-ph/0012029

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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.
Starting from generalized position operators, we derive complex and quaternionic angular momentum operators along with their commutation algebra as well. These algebras differ from the standard Hermitian ones, especially in terms of commutation relations involving partial and total angular momentum operators. Despite these differences, the effective quantum expectation values obtained from slightly deformed algebras align with those from the conventional Hermitian algebra. This suggests that even though the wave functions and resulting dynamics differ from standard quantum Hermitian behavior, these deformed algebras can still be effectively understood as valid angular momentum algebras.
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The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.