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New Suggestion to Describe q-deformed Oscillators

Lin Xiu

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

Time dependent equation of q deformed oscillators are decribes with Two Wave interpretation physics equation the results show that movement of the particle is nonlinear. At the same time, when r→0 the theory reduces to the common oscillators theory.

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New Suggestion to Two-dimensional Homogeneous q-deformed Oscillators

Jiayu Huang

Time-dependent equation of two-dimensional homogeneous q-deformed oscillators are describes with Two-wave interpretation physicsequation the resultsshow thatmovementof particle is nonlinear. At the same time,when q→ 1 the theory reduces to the common two-dimensional homogeneous oscillators theory.

Journal of Fujian Normal University•Jan 1, 2000•0 citations

DOUBLE-WAVE DESCRIPTION FOR TWO-DIMENSIONAL q-DEFORMED OSCILLATOR

Xiu‐Min Lin, 中国科学技术大学量子通讯与量子计算开放研究实验室，合肥 230026；福建师范大学物理系，福州 350007

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Acta Physica Sinica•Jan 1, 2000•1 citation

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

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

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.

arXiv:1611.03682•Journal of Al-Nahrain University-Science•Mar 1, 2016•0 citations

Double wave description of the motion of Q-dimensional charged Q-deformed oscillator in uniform electric field

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

Ahmed Shaker Mahmood, Mohammed A. Z. Habeeb

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.

Journal of Al-Nahrain University-Science•Sep 1, 2016•1 citation

A Nonlinear $q$-Deformed Schrödinger Equation

M. A. Rego-Monteiro, E. M. F. Curado

We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter $q$. This operator recovers Newton derivative when $q \rightarrow 1$. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential $V(\vec{x})$. We analytically solve the nonlinear deformed Schrödinger equation for $V(\vec{x}) = 0$ and $q \simeq1$. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of $q$.

nlin.PSCross-listedarXiv:2602.11312•Feb 13, 2026
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.
A Nonlinear qqq-Deformed Schrödinger Equation
We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter qqq. This operator recovers Newton derivative when q→1q \rightarrow 1q→1. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential V(x⃗)V(\vec{x})V(x). We analytically solve the nonlinear deformed Schrödinger equation for V(x⃗)=0V(\vec{x}) = 0V(x)=0 and q≃1q \simeq1q≃1. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of qqq.