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Science International

Quantization of Damped Mechanical Systems Based on the Doubling of Degrees of Freedom

Khaled I. Nawafleh

Published May 19, 2025• 0 citations
Publisher

Abstract

Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature.This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system.The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation.Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model.These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.

DOI
10.17311/sciintl.2025.21.26

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Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature.This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system.The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation.Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model.These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.
Today, two of the most prosperous fields of physics are quantum computing and spintronics. In both, the loss of information and dissipation play a crucial role. In the present work, we formulate the quantization of the dissipative oscillator, which aids the understanding of the abovementioned issues, and creates a theoretical frame to overcome these issues in the future. Based on the Lagrangian framework of the damped spring system, the canonically conjugated pairs and the Hamiltonian of the system are obtained; then, the quantization procedure can be started and consistently applied. As a result, the damping quantum wave equation of the dissipative oscillator is deduced, and an exact damping wave solution of this equation is obtained. Consequently, we arrive at an irreversible quantum theory by which the quantum losses can be described.
In this work, a suitable Hamiltonian that describes the damped harmonic oscillator is constructed. Besides, the Hamilton-Jacobi equation for this dissipative system is written and the action function is obtained. Then, the system is quantized using two methods: the WKB approximation and the canonical quantization. In addition, evolution of the probability density is plotted at different times, and it is observed that the width of these densities decreases with time. The average energy of the system is also calculated and found to decrease with time.
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.
Abstract In this work, we present a quantization scheme for the damped harmonic oscillator (QDHO) using a framework known as momentous quantum mechanics. Our method relies on a semiclassical dynamical system derived from an extended classical Hamiltonian, where the phase-space variables are given by expectation values of observables and quantum dispersions. The significance of our study lies in its potential to serve as a foundational basis for the effective description of open quantum systems (OQS), and the description of dissipation in quantum mechanics. By employing the Bateman’s dual model as the initial classical framework, and undergoing quantization, we demonstrate that our description aligns exceptionally well with the well-established Lindblad master equation for the QDHO. Furthermore, the robust formulation of our approach suggests a broad applicability in the context of OQS, extending implementations into more general systems, making it a versatile and powerful tool to study diverse phenomena.
Abstract The Hamilton–Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics, where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton–Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.