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Modern Physics Letters B

DIRECT QUANTIZATION OF OPEN DISSIPATIVE SYSTEMS

Jose A. Magpantay

Published Dec 20, 2000• 0 citations
Publisher

Abstract

Open dissipative systems subject to a random force are directly quantized. The starting point is the effective action derived using the method of Parisi–Sourlas. Since the effective action is second order, the method of Ostrogradsky was used to quantize the system canonically. In the case of the harmonic oscillator, the relevant Green function can be computed exactly. In the general case, a perturbation expansion, involving time-dependent (memory) terms, can be defined.

DOI
10.1142/s0217984900001270

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Open dissipative systems subject to a random force are directly quantized. The starting point is the effective action derived using the method of Parisi–Sourlas. Since the effective action is second order, the method of Ostrogradsky was used to quantize the system canonically. In the case of the harmonic oscillator, the relevant Green function can be computed exactly. In the general case, a perturbation expansion, involving time-dependent (memory) terms, can be defined.
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 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.
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 many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture are developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.
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.