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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.

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. It is well known that quantum damped harmonic oscillator is studied within two representations of the model system. One representation is the Bateman-Feshbach-Tikochinsky (BFT) oscillator (often called the Bateman oscillator) as a closed system with two degrees of freedom [

The damped harmonic oscillator (DHO) is described by

where the overdot denotes the derivative with respect to

(a) The over-damping case:

(b) The critical-damping case:

(c) The under-damping case:

The last case is the most interesting case and Celeghini et al. [

The BFT (Bateman) damped oscillator [^{1} of classical mechanics to dissipative systems, one can double the numbers of degrees of freedom. The new degrees of freedom are assumed to represent a reservoir, also 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 [

This closed system includes a primary one expressed by

It should be noted that this Lagrangian does not depend on time explicitly. By Legendre transforming Equation (6), Bateman obtained the Hamiltonian:

where we used

In this paper, we treat the Hamiltonian formulation and quantization of the DHO where the oscillator mass is time-dependent and study the effect of these control para- meters

Let us consider the case where the oscillator’s mass is time-dependent:

Now we differentiate Equation (8) with respect to

When the oscillator with variable mass is subject to the external force

Thus, dynamics of the damped harmonic oscillator with variable mass

By applying Bateman’s dual oscillator formulation, the equations of motion for the dual system of the damped harmonic oscillator (10) may be expressed by the following equations of motion:

If we do not employ an explicit time-dependent dissipative function, the Lagrangian leading to Equation (11) can be expressed by

It is interesting to note that the form of Equation (12) is similar to Equation (6) but the mass

Lagrange equations of motion for the Lagrangian (12) reproduce correctly the dual equations of motion (11): the first equation represents a damped harmonic oscillator with variable mass, while the second one can be considered as its time-reversed image.

Let us define the canonical momenta

It should be noted that these canonical momenta defined in Equation (13) are different from the kinetic momenta defined by Equation (8). In order to obtain the Hamiltonian of this dual system, we apply Legendre transformation to the Lagrangian function (12) in a following way:

Expressing Equation (14) in terms of the canonical momenta

This is the extended Bateman dual-Hamiltonian for which Hamilton’s equations of motion reproduce correctly the doubled system. Since the energy of the total system is constant, the system of damped harmonic oscillator and its time-reversed image is a closed system described by the Hamiltonian function (15). We can write the canonical equations of Hamilton as follows:

The Hamilton equations of motion reproduce correctly the classical doubled damped harmonic oscillator systems.

The Poisson brackets of the dual system are

The Poisson bracket formulation of Hamilton’s equations is given by

It should be noted that the Hamiltonian (15) is a constant of the motion since

Let us consider the quantal case. Canonical quantization for the dual Hamiltonian

where position and momentum operators are denoted respectively by

where

Note that the mass variable

Now we introduce the pairs of the annihilation and creation operators

The creation operator

The Hamiltonian (21) can then be expressed in terns of these creation and annihila- tion operators:

where

The second-quantized Hamiltonian (28) is not a simple form and it is difficult to clarify the physical meaning of each term in the particle picture. We perform the following linear canonical transformation by introducing new operators

and their conjugates, which resort to Equations (23)-(26). These new operators

Thus these operators construct a dual Hilbert (Fock) space

The Hamiltonian (21) in the Schrödinger picture (SP) can be expressed in terms of the new operators

where

In order to see the effect of varying mass, let us define the vacuum states,

Then the vacuum state of the Hamiltonian

since any operators on

The SP evolution operator

where the symbol

By using the Hamiltonain

Recalling

This equation forms the basis for further evaluation of the vacuum state of the system associated with oscillator’s variable mass and other parameters characterizing the system. In the following we consider the effect of variable mass

Let us study the effect of variable mass/dissipative force on the present dissipative system by looking at the vacuum states with the use of Equation (39) since the vacuum state sensitively reflects the stability (dissipation) of the system. Here we consider the following cases: (i)

We first consider the case (i) for a constant mass, i.e.,

product of

Next we consider the case (ii). The vacuum state, Equation (39), is then given by

It is interesting to note that the vacuum state does not change with time when the mass changes linearly with time

Next we consider the case (iii), where the mass decreases with time:

Finally we consider the case (iv), where the mass increases exponentially with time:

varying mass,

In order to study the effects of the dissipative Hamiltonian

These are of dissipative nature, their time evolution being controlled by

It has been shown that the proper way to perform the canonical quantization of the damped harmonic oscillator is to work in the framework of Quantum Field Theory (QFT) [

In QFT we have to consider infinitely many degrees of freedom. Thus, by using the continuous limit relation

provided that

To sum up, we have studied the DHO with a variable mass as a simple model for a dissipative system, following the theoretical scheme of Majima and Suzuki [

Introducing the canonical momenta (13) by using the obtained Lagrangian (12) for the time-dependent mass

In Section 3, we extended the theory developed in Section 2 to the quantum case, where we showed and discussed in detail how to derive the second quantized form of the Hamiltonian in terms of creation and annihilation operators

In order to discuss the stability of the system arising from the change of oscillator mass in time, we focus on the change of the vacuum state of the system due to the change in the mass causing dissipation/stability of the system. We derived the general formula of the vacuum state

In Section 4, we studied the effect of variable mass/dissipative force on the quantum dissipative system by using the formula (40). We considered the following cases: (i)

Noticing that the time-dependent mass could be a control parameter for dissipation/ damping of the DHO with varying (time-dependent) mass, we developed the theory to investigate dissipation (damping) in the quantum theory and quantum dissipated system by employing the DHO as the simple model system. The time-dependent oscil- lator mass

Suzuki, A. and Ma- jima, H. (2016) On the Quantum Mechanical Treatment of the Bateman-Morse-Fesh- bach Damped Oscillator with Variable Mass. Journal of Modern Physics, 7, 2329-2340. http://dx.doi.org/10.4236/jmp.2016.716201