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

Abstract

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 ( − γ 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 = m x ̇ y ̇ − U ( x + 1 2 y ) + U ( x − 1 2 y ) + γ 2 ( x y ̇ − y x ̇ ) − x K ( t ) + y K ( 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 ) 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.