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arXiv:1611.03682 · Journal of Al-Nahrain University-Science

The Liouville Dynamics of the q-Deformed 1-D Classical Harmonic Oscillator

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

Published Mar 1, 2016• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22401/jnus.19.1.14

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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.
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.
We consider the dynamics of a quantum joint phase-space probability density in an exactly solvable model. The density is defined to be a true (i.e., positive) probability distribution for approximate (and thus simultaneously measurable) position and momentum variables. The dynamics of the quantum density is governed by a second-order partial-differential equation with non-positive-definite second-order coefficients. The quantum dynamics is contrasted with the dynamics of a similar joint density in a classical description. The non-positive-definite second-order terms in the quantum evolution equation, not present in the classical case, are responsible for quantum recurrences and prevent the appearance of fine-scale-structure ``whorls'' predicted in a classical description. The generation of ``squeezing'' in the model is also discussed.
The authors contrast the classical and quantum dynamics of SU(1,1) coherent states by the use of a positive-definite quasiprobability distribution, a Q-function, defined over these states. For Hamiltonians that are linear in the SU(1,1) generators, therefore coherence preserving, the quantum and classical Liouville equations are identical. This is illustrated for a degenerate parametric amplifier. For an anharmonic oscillator, the quantum Liouville equation contains additional terms containing second-order derivatives with non-positive-definite coefficients. These terms give rise to the quantum recurrences not seen in the classical evolution of this system. It is pointed out that the quantum equation for the Q-function goes over to the classical equation as the Bargmann index, k, becomes large, in agreement with previous semiclassical considerations.
In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.
We present a solution for the dynamics of an anharmonic oscillator coupled to a zero-temperature heat bath. Comparison of observable properties in a classical and quantum description uses true joint phase-space probability densities. The time evolution of the density in the quantum case is rapidly "reduced" to that given in the classical description. The rate of reduction is proportional to the product of the damping rate and the oscillator's initial energy. Quite rapidly, typical quantum recurrence effects are destroyed and the classical "whorl" structure restored. We point out the close similarity with rapid destruction of quantum coherence through dissipation.