Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

Journal Article · Thu Dec 15 00:00:00 EST 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.08.002· OSTI ID:21583300

Majima, H; Suzuki, A

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity (-{gamma}x) and a time-dependent external force (K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman's system, which is described by the Lagrangian: L=mxy-U(x+1/2 y)+U(x-1/2 y)+({gamma})/2 (xy-yx)-xK(t)+yK(t), which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting U(x{+-}y/2)=1/2 k(x{+-}y/2){sup 2} specifically for a dual extended damped-amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman's Hamiltonian H. The Heisenberg equations of motion utilizing the quantized Hamiltonian H surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped-amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force. - Highlights: > A method of quantizing dissipative systems is presented. > In order to obtain the method, we apply Bateman's dual system approach. > A formula for a transition amplitude is derived. > We use the formula to study the instability of the dissipative systems.

• OSTI ID: 21583300
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 12; Other Information: DOI: 10.1016/j.aop.2011.08.002; PII: S0003-4916(11)00128-X; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DISTURBANCES
EQUATIONS OF MOTION
HAMILTONIANS
HARMONIC OSCILLATORS
INSTABILITY
LAGRANGIAN FUNCTION
ONE-DIMENSIONAL CALCULATIONS
PERTURBATION THEORY
POTENTIALS
QUANTIZATION
STABILITY
TIME DEPENDENCE
TRANSITION AMPLITUDES
VELOCITY
AMPLITUDES
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
MATHEMATICAL OPERATORS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS