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European Scientific Journal ESJ

PATH INTEGRAL QUANTIZATION OF DISSIPATIVE SYSTEMS

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

Published May 30, 2014• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
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.
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
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.