Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Scientific Journal ESJ

CANONICAL QUANTIZATION OF DISSIPATIVE SYSTEMS

Ola A. Jarab’ah, Khaled I. Nawafleh, H. B. Ghassib

Published Dec 31, 2013• 5 citations
OpenAlex

Abstract

The canonical method is invoked to quantize dissipative systems using the WKB approximation. The wave function is constructed such that its phase factor is simply Hamilton’s principal function. The energy eigenvalue is found to be in exact agreement with the classical case. To demonstrate our approach, the three examples considered in our previous work (ESJ 9(30), 70-81, 2013) are quantized in detail: the damped harmonic oscillator, a system with a variable mass, and a charged particle in a magnetic field.

Related papers

Quantization of the damped harmonic oscillator

Mohammed Serhan, M. Abusini, Ahmed Al-Jamel +2 more

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.

Journal of Mathematical Physics•Aug 1, 2018•18 citations

QUANTIZATION OF DISSIPATIVE SYSTEMS WITH SECOND-ORDER LAGRANGIAN

Ola A. Jarab’ah, Eyad Hasan Hasan, Khaled I. Nawafleh

This paper examined dissipative systems with second order Lagrangian in the framework of the Hamilton-Jacobi method. The principal function is determined using the method of separation of variables in the same manner as for regular time-independent Lagrangian. The equations of motions are obtained from the action function. By finding the action function, the wave function can be constructed and the quantization has been carried out using the WKB approximation. One example is given to illustrate our formalism.

European Scientific Journal ESJ•Mar 31, 2014•2 citations

PATH INTEGRAL QUANTIZATION OF DISSIPATIVE SYSTEMS

Eyad Hasan Hasan, Ola A. Jarab’ah, Khaled I. Nawafleh

This paper investigated the basic formalism for treating the dissipative Hamiltonian systems within the framework of the canonical method using the path integral quantization. The Hamiltonian treatment of the dissipative systems leads to obtain the equations of motion as a total differential equation in many variables which require the investigation of integrability conditions on the action and the equations of motion. In this formalism, the action integral for dissipative systems is obtained from equations of motion and written in phase space. Besides, the quantization of these systems is investigated using the action to construct the wave function in terms of the canonical phase space coordinates. Two examples are examined, the free particle and the simple harmonic oscillator.

European Scientific Journal ESJ•May 30, 2014•1 citation

Quantization of Dissipative Anisotropic Harmonic Oscillators

Bashar M. Al-khamiseh, Yazen M. Alawaideh, Hayat Issaadi +4 more

In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find the action function by including boundary conditions in the study of a dissipative anisotropic harmonic oscillator. To quantize the system, we employed three distinct methods: the WKB approximation, canonical quantization, and the creation and annihilation operator technique. The quantization of the Heisenberg equations was achieved by reformulating them using Poisson bracket quantum variables. Through the application of creation and annihilation operators, we derived the energy levels and quantum states of the system. The results demonstrated complete consistency across the different methods. This model exhibits strong agreement among the solutions derived from various approaches, highlighting its importance in advancing the understanding of dissipative quantum systems.

European Journal of Pure and Applied Mathematics•May 1, 2025•1 citation

Quantization of Time Independent Damping Systems Using WKB Approximation

Ola A. Jarab’ah

In this work time independent damping systems are studied using Lagrangian and Hamiltonian for time independent damping, which are present through the factor eλq. The Hamilton Jacobi equation is formulated to find the Hamilton Jacobi function S using separation of variables technique. We can form this function in compact form of two parts the first part as a function of coordinate q, and the second part as a function of time t. Finally, we find the ability of these systems to quantize through an illustrative example.

Journal of Applied Mathematics and Physics•Jan 1, 2023•0 citations

Canonical averaging in the second order quantized Hamilton dynamics by extension of the coherent state thermodynamics of the harmonic oscillator

Eric M. Heatwole, Oleg V. Prezhdo

A conceptually simple approximation to quantum mechanics, quantized Hamilton dynamics (QHD) includes zero-point energy, tunneling, dephasing, and other important quantum effects in a classical-like description. The hierarchy of coupled differential equations describing the time evolution of observables in QHD can be mapped in the second order onto a classical system with double the dimensionality of the original system. While QHD excels at dynamics with a single initial condition, the correct method for generating thermal initial conditions in QHD remains an open question. Using the coherent state representation of thermodynamics of the harmonic oscillator (HO) [Schnack, Europhys. Lett. 45, 647 (1999)], we develop canonical averaging for the second order QHD [Prezhdo, J. Chem. Phys. 117, 2995 (2002)]. The methodology is exact for the free particle and HO, and shows good agreement with quantum results for a variety of quartic potentials.

The Journal of Chemical Physics•May 28, 2007•21 citations
The canonical method is invoked to quantize dissipative systems using the WKB approximation. The wave function is constructed such that its phase factor is simply Hamilton’s principal function. The energy eigenvalue is found to be in exact agreement with the classical case. To demonstrate our approach, the three examples considered in our previous work (ESJ 9(30), 70-81, 2013) are quantized in detail: the damped harmonic oscillator, a system with a variable mass, and a charged particle in a magnetic field.
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.
This paper examined dissipative systems with second order Lagrangian in the framework of the Hamilton-Jacobi method. The principal function is determined using the method of separation of variables in the same manner as for regular time-independent Lagrangian. The equations of motions are obtained from the action function. By finding the action function, the wave function can be constructed and the quantization has been carried out using the WKB approximation. One example is given to illustrate our formalism.
This paper investigated the basic formalism for treating the dissipative Hamiltonian systems within the framework of the canonical method using the path integral quantization. The Hamiltonian treatment of the dissipative systems leads to obtain the equations of motion as a total differential equation in many variables which require the investigation of integrability conditions on the action and the equations of motion. In this formalism, the action integral for dissipative systems is obtained from equations of motion and written in phase space. Besides, the quantization of these systems is investigated using the action to construct the wave function in terms of the canonical phase space coordinates. Two examples are examined, the free particle and the simple harmonic oscillator.
In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find the action function by including boundary conditions in the study of a dissipative anisotropic harmonic oscillator. To quantize the system, we employed three distinct methods: the WKB approximation, canonical quantization, and the creation and annihilation operator technique. The quantization of the Heisenberg equations was achieved by reformulating them using Poisson bracket quantum variables. Through the application of creation and annihilation operators, we derived the energy levels and quantum states of the system. The results demonstrated complete consistency across the different methods. This model exhibits strong agreement among the solutions derived from various approaches, highlighting its importance in advancing the understanding of dissipative quantum systems.
In this work time independent damping systems are studied using Lagrangian and Hamiltonian for time independent damping, which are present through the factor eλq. The Hamilton Jacobi equation is formulated to find the Hamilton Jacobi function S using separation of variables technique. We can form this function in compact form of two parts the first part as a function of coordinate q, and the second part as a function of time t. Finally, we find the ability of these systems to quantize through an illustrative example.
A conceptually simple approximation to quantum mechanics, quantized Hamilton dynamics (QHD) includes zero-point energy, tunneling, dephasing, and other important quantum effects in a classical-like description. The hierarchy of coupled differential equations describing the time evolution of observables in QHD can be mapped in the second order onto a classical system with double the dimensionality of the original system. While QHD excels at dynamics with a single initial condition, the correct method for generating thermal initial conditions in QHD remains an open question. Using the coherent state representation of thermodynamics of the harmonic oscillator (HO) [Schnack, Europhys. Lett. 45, 647 (1999)], we develop canonical averaging for the second order QHD [Prezhdo, J. Chem. Phys. 117, 2995 (2002)]. The methodology is exact for the free particle and HO, and shows good agreement with quantum results for a variety of quartic potentials.