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Iranian Journal of Physics Research

A damped harmonic oscillator in the GUP framework

Seyed Davood Sadatian

Published Oct 1, 2019• 0 citations
Publisher

Abstract

Harmonic oscillator is significantly important in quantum gravity for its dynamical properties. Dynamics of the harmonic oscillator can be modified by inserting a modified uncertainty relation considered in quantum cosmology models such as string theory and quantum loop gravity theory. In this paper, the equation of damped quantum harmonic oscillator has been solved with respect to the modified uncertainty relation and the effects of this modification have been probed in the case of the damped oscillating system.

DOI
10.47176/ijpr.19.3.17772

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Harmonic oscillator is significantly important in quantum gravity for its dynamical properties. Dynamics of the harmonic oscillator can be modified by inserting a modified uncertainty relation considered in quantum cosmology models such as string theory and quantum loop gravity theory. In this paper, the equation of damped quantum harmonic oscillator has been solved with respect to the modified uncertainty relation and the effects of this modification have been probed in the case of the damped oscillating system.
We investigate the effects of the Generalized Uncertainty Principle (GUP), which intro- duces a fundamental minimal observable length, on the quantum dynamics of a damped harmonic oscillator. By incorporating a modified Heisenberg algebra into the framework, we derive the altered uncertainty relations for damped quantum systems and examine their dependence on the discrete energy spectrum. Our analysis reveals significant devia- tions from the standard Heisenberg uncertainty relation, demonstrating that the presence of a minimal length scale fundamentally modifies the quantum-classical correspondence in dissipative systems. We present explicit expressions for the modified uncertainty relations and discuss their physical implications for quantum measurements in damped oscillatory systems. These results contribute to our understanding of quantum mechanics at length scales approaching the Planck regime and provide testable predictions for experimental verification of GUP effects in dissipative quantum systems.
Abstract The generalized uncertainty principle (GUP) is a natural consequence of many proposed quantum gravity theories which further indicates the modification of the quantum mechanics and classical mechanics. A unified classical-quantum framework also leads to the interpretation of a Winger phase-space distribution of a state as a KvN wave function. In this article, we utilize the classical-quantum unified framework to calculate the expression for the GUP modified Wigner function of a classical harmonic oscillator in the configuration space instead of the conventional approach of utilising the momentum space. Further, we plot the modified Wigner functions and show that they are similar to what has been obtained before in the past literature.
Abstract The generalized uncertainty principle (GUP) is a natural consequence of many proposed quantum gravity theories which further indicates the modification of the quantum mechanics and classical mechanics. A unified classical-quantum framework also leads to the interpretation of a Winger phase-space distribution of a state as a KvN wave function. In this article, we utilize the classical-quantum unified framework to calculate the expression for the GUP modified Wigner function of a classical harmonic oscillator in the configuration space instead of the conventional approach of utilising the momentum space. Further, we plot the modified Wigner functions and show that they are similar to what has been obtained before in the past literature.
Different proposed candidates of quantum gravity, such as string theory, noncommutative geometry, loop quantum gravity, and doubly special relativity, all predict the existence of a minimum observable length and/or a maximum observable momentum. These natural cutoffs modify the standard Heisenberg uncertainty principle leading to the so-called Generalized Uncertainty Principle (GUP). In this paper, we study the effects of minimal length and maximal momentum on the dynamics of harmonic oscillations. Within this framework we also study the status of Ehrenfest's theorem with the GUP.
The generalized uncertainty principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of quantum gravity. It consists of a modified commutator between the position and momentum. In this work, we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent, and squeezed states in quantum mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We comment on potential applications.