Abstract
The harmonic oscillator with time-dependent (indefinite and variable) mass subject to the force proportional to velocity is studied by extending Bateman’s dual Lagrangian and Hamiltonian formalism. To study the quantum analog of such a dissipative system, the Batemann-Morse-Feshback classical Hamiltonian of the damped harmonic oscillator with varying (time-dependent) mass is canonically quantized. In order to discuss the stability of the quantum dissipative system due to the influence of varying mass and the dissipative force, we derived a formula for the vacuum state of the dissipative system with the help of quantum field theoretical framework. It is shown that the formula based on this simple model could be used to study the influence of dissipation such as the instability due to the dissipative force and/or the variable mass. It is understood that the change in the oscillator mass corresponds to a control parameter in quantum dissipative systems.
Highlights
The quantum damped oscillator has been studied by many researchers to understand dissipation in quantum theory since the damped harmonic oscillator is one of the simplest systems revealing the dissipation of energy
In order to discuss the stability of the quantum dissipative system due to the influence of varying mass and the dissipative force, we derived a formula for the vacuum state of the dissipative system with the help of quantum field theoretical framework
It is interesting to note that the time-dependent mass m = m (t ) plays the same role of the control parameter for damping as the damping factor γ in the damped harmonic oscillator (DHO) does in the dissipative system (see Equation (10), Section 2)
Summary
The quantum damped oscillator has been studied by many researchers to understand dissipation in quantum theory since the damped harmonic oscillator is one of the simplest systems revealing the dissipation of energy. The new degrees of freedom are assumed to represent a reservoir, called heat bath Applying this idea to the damped harmonic oscillator one obtains a pair of damped oscillators, so-called Bateman’s dual or mirror image system [1], represented by. The energy dissipated by the oscillator is completely absorbed at the same time by the mirror image oscillator, and the energy of the total system is conserved These equations can be derived from the Lagrangian:. It is interesting to note that the time-dependent mass m = m (t ) plays the same role of the control parameter for damping as the damping factor γ in the DHO does in the dissipative system (see Equation (10), Section 2). We treat the Hamiltonian formulation and quantization of the DHO where the oscillator mass is time-dependent and study the effect of these control parameters m (t ) and γ on dissipation in quantum dissipative systems by examining the stability of vacuum state by using the simple model represented by the DHO with varying (time-dependent) mass by employing the theoretical scheme of Majima and Suzukii [11] and study dissipation in quantum dissipative systems in order to understand the dissipation in quantum dissipative systems
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