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Journal of Mathematical Physics

A q deformation of the Gauss distribution

Hans van Leeuwen, Hans Maassen

Published Sep 1, 1995• 65 citations
Publisher

Abstract

The q deformed commutation relation aa*−qa*a=1 for the harmonic oscillator is considered with q∈[−1,1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a+a* in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian.

DOI
10.1063/1.530917

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The q deformed commutation relation aa*−qaa=1 for the harmonic oscillator is considered with q∈[−1,1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a+a in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
Relations between the first, the second quantized representations and deform algebra are investigated. In the case of harmonic oscillator, the axiom of first quantization (the commutation relation between coordinate and momentum operators) and the axiom of second quantization (the commutation relation between creation and annihilation operators) are equivalent. We shown that in the case of q -deformed harmonic oscillator, a violence of the axiom of second quantization leads to a violence of the axiom of first quantization, and inverse. Using the coordinate representation, we study fine structures of the vacuum state wave function depend in the deformation parameter q . A comparison with fine structures of Cooper pair of superconductivity in the coordinate representation is also performed.
A derivation of the coordinate representation of the q-deformed oscillator is given. The Hilbert space associated with it is defined on a finite coordinate space y: (-\ensuremath{\pi}/2s,\ensuremath{\pi}/2s), where s is related to the deformation q via q=exp(-s2{\mathit{s}}^{2}s2), with the respective ``coordinate'' and ``momentum'' operators given by X${\mathrm{^}}_{\mathit{s}}$=y-As{\mathit{A}}_{\mathit{s}}As​(y), P${\mathrm{^}}_{\mathit{s}}$=-id/dy-iAs{\mathit{iA}}_{\mathit{s}}iAs​(y), where As{\mathit{A}}_{\mathit{s}}As​(y)=1/2d ln${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$(y)/dy, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}$ being a measure in this space. The commutation rule obeyed by them is X${\mathrm{^}}_{\mathit{s}}$P${\mathrm{^}}_{\mathit{s}}$-P${\mathrm{^}}_{\mathit{s}}$X${\mathrm{^}}_{\mathit{s}}$=i${\mathrm{\ensuremath{\Elzxh}}}_{\mathit{s}}$(y)=i{1-dAs{\mathit{dA}}_{\mathit{s}}dAs​ /dy}. After constructing a suitable representation in the momentum space, we also set up the Wigner distribution function in this phase space. A brief discussion of these results and their interpretation are given.
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.