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Journal of Al-Nahrain University-Science

Interpretation of the q-Deformed 1-D Quantum Harmonic Oscillator

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

Published Sep 1, 2016• 1 citation
Publisher

Abstract

The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.

DOI
10.22401/jnus.19.3.08

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The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.
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.
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.
In this paper, by using the Wess–Zumino formalism of noncommutative differential calculus, we show that the concept of nonlinear coherent states originates from noncommutative geometry. For this purpose, we first formulate the differential calculus on a GL p, q(2) quantum plane. By using the commutation relations between coordinates and their interior derivatives, we then construct the two-parameter (p, q)-deformed quantum phase space together with the associated deformed Heisenberg commutation relations. Finally, by applying the obtained results for the quantum harmonic oscillator we construct the associated coherent states, which can be identified as nonlinear coherent states. Furthermore, we show that some of the well-known deformed (nonlinear) coherent states, such as two-parameter (p, q)-deformed coherent states, Maths-type q-deformed coherent states, Phys-type q-deformed coherent states and Quesne deformed coherent states, can be easily obtained from our treatment.
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.