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Physical Review A

Coordinate and momentum representations of theq-deformed oscillator and their interpretation

A. K. Rajagopal

Published May 1, 1993• 4 citations
Publisher

Abstract

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(-${\mathit{s}}^{2}$), with the respective ``coordinate'' and ``momentum'' operators given by X${\mathrm{^}}_{\mathit{s}}$=y-${\mathit{A}}_{\mathit{s}}$(y), P${\mathrm{^}}_{\mathit{s}}$=-id/dy-${\mathit{iA}}_{\mathit{s}}$(y), where ${\mathit{A}}_{\mathit{s}}$(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-${\mathit{dA}}_{\mathit{s}}$ /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.

DOI
10.1103/physreva.47.r3465

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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\ensuremathσs{\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}\ensuremathσs​(y)/dy, \ensuremathσs{\mathrm{\ensuremath{\sigma}}}_{\mathit{s}}\ensuremathσs​ 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\ensuremath\Elzxhs{\mathrm{\ensuremath{\Elzxh}}}_{\mathit{s}}\ensuremath\Elzxhs​(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.
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.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }
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.