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ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE)

Quantization of a Free Particle in a Dissipative Medium: Interaction of Charged Particles With Matter

Eqab M. Rabei, Abdul-Wali Ajlouni, H. B. Ghassib

Published Jan 1, 2005• 0 citations
Publisher

Abstract

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system of free particles in a dissipative medium is carried out according to the Dirac method. A suitable Schro¨dinger equation is set up and solved for the Lagrangian representing this system. The wave function is plotted and the damping effect manifests itself very clearly. This formalism is then applied to the problem of energy loss of charged particles when passing through matter. The results are plotted and the relation between the energy loss and the range agrees qualitatively with experimental results.

DOI
10.1115/detc2005-84101

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Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system of free particles in a dissipative medium is carried out according to the Dirac method. A suitable Schro¨dinger equation is set up and solved for the Lagrangian representing this system. The wave function is plotted and the damping effect manifests itself very clearly. This formalism is then applied to the problem of energy loss of charged particles when passing through matter. The results are plotted and the relation between the energy loss and the range agrees qualitatively with experimental results.
We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the Hamiltonian, we quantize canonically the classical particle in a viscous media. We study the free quantum particle in a viscous media and the dissipative quantum tunnelling. It is found that the dissipation influences tunnelling probability by a factor that closely resembles the result of Caldeira and Leggett.
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.
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.
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 use of fractional momentum operators and fractionary kinetic energy used to model linear damping in dissipative systems such as resistive circuits and a spring-mass ensambles was extended to a quantum mechanical formalism. Three important associated 1 dimensional problems were solved: the free particle case, the infinite potential well, and the harmonic potential. The wave equations generated reproduced the same type of 2-order ODE observed in classical dissipative systems, and produced quantized energy levels. In the infinite potential well, a zero-point energy emerges, which can be fitted to the rest energy of the particle described by special relativity, given by relationship Er=mc2E_r=mc^2Er​=mc2. In the harmonic potential, new fractional creation and destruction operators were introduced to solve the problem in the energy basis. The energy eigenvalues found are different to the ones reported by earlier approaches to the quantum damped oscillator problem reported by other authors. In this case, a direct relationship between the relativistic rest energy of the particle and the expected value of the fractionary kinetic energy in the base state was obtained. We conclude that there exists a relationship between fractional kinetic energy and special relativity energies, that remains unclear and needs further exploration, but also conclude that the current form of transforming fractionary momentum operators to the position basis will yield non-observable imaginary momentum quantities, and thus a correction to the way of transforming them needs to be explored further.