On the Quantum Mechanical Treatment of the Bateman-Morse-Feshbach Damped Oscillator with Variable Mass

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 Bateman oscillator) as a closed system with two degrees of freedom [1] [2]. The other representation is the Caldirola- Kanai (CK) oscillator as an open system with one degree of freedom [3] [4] [5].
The damped harmonic oscillator (DHO) is described by ( ) x t subject to the 2ndorder linear differential equation with constant coefficients, where coefficient ( ) 0 γ > in the first derivative term is called a damping coefficient and k is the harmonic coefficient while m is the mass coefficient (constant): 0, mx x kx γ + + =   (1) where the overdot denotes the derivative with respect to t . Depending on the relation between damping and harmonic coefficients we have three different cases and the general solutions of Equation (1) are: (a) The over-damping case: The last case is the most interesting case and Celeghini et al. [6] rigorously studied classical and quantum damped harmonic oscillator with a constant mass. In this paper we will study the case where the oscillator's mass changes with time. When 0 γ = , Equation (1) is reduced to the standard harmonic oscillator equation of motion.
Throughout the paper we consider the 0 γ ≠ case along with a time-dependent mass ( ) m m t = , i.e., we consider dissipation by using the simple model. The harmonic oscillator described by Equation (1) represents a dissipative system of which energy is not conserved although the γ is time-independent. In order to establish the canonical formalism for the dissipative system we have to construct a Lagrangian-Hamiltonian form in any case. Bateman's formulation [1] resolves this problem of dissipation, where the dynamics of the system is described by Equation (1) This closed system includes a primary one expressed by x -variable and its time reversed image by y -variable. According to this, the energy dissipated by the oscillator is completely absorbed at the same time by the mirror image oscillator, and thus the energy of the total system is conserved. Actually these equations can be derived from the Lagrangian: It should be noted that this Lagrangian does not depend on time explicitly. By Legendre transforming Equation (6) where we used 2  Suzukii [11] and study dissipation in quantum dissipative systems in order to understand the dissipation in quantum dissipative systems.

Classical Theory
Let us consider the case where the oscillator's mass is time-dependent: ( ) m m t = [12].
The kinetic momentum of the oscillator is then defined by ( ) .
Now we differentiate Equation (8) with respect to t , we obtain When the oscillator with variable mass is subject to the external force ex Thus, dynamics of the damped harmonic oscillator with variable mass ( ) m t is governed by this equation of motion. We note that the second term in Equation (10) arises due to the oscillator mass being time-dependent. Damping occurs from the two  and thus the varying mass plays the same role as the damping coefficient γ , that could be a control parameter for the damping.
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: m t y m t y y ky γ γ 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)  ( ) 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 ( ) , x y p p for our dual oscillator system by using the Lagrangian (12): 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: x y x y xp yp xp yp m t xy xy yx kxy Expressing Equation (14) in terms of the canonical momenta ( ) x y x y p p t =   of the system can then be expressed by 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 It is thus concluded that the energy dissipated by the original oscillator is completely absorbed by the dual of the system.

Quantum Theory
Let us consider the quantal case. Canonical quantization for the dual Hamiltonian  in Equation (15) Note that the mass variable m depends on time t , so that the common frequency of the two damped oscillators Ω defined by Equation (23) The creation operator † j a is the Hermitian conjugate of the annihilation operator j a .
These operators j a and ( ) † 1,2 j a j = satisfy the following commutation rules: The Hamiltonian (21) can then be expressed in terns of these creation and annihilation operators: is given by Equation (23).
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  , which define the canonical transformations [9]: and their conjugates, which resort to Equations (23)-(26). These new operators ( ) ,   obey the same algebra as in Equation (27), that is, the following canonical commutation rules hold for the new operators ( ) † † , , ,     : . We note that the Hamiltonians 0  and 1  in SP are both time-dependent through Ω and Γ since they depend on the time dependent mass ( ) m t .
In order to see the effect of varying mass, let us define the vacuum states, 0  for the system (A) spanned by the operator  and 0  for the system (B) spanned by the operator  : Then the vacuum state of the Hamiltonian  on the dual Hilbert space can be described by the direct product of 0  and 0  : where the symbol T designates the time-ordering operator. Then we can define a vacuum state at a time t for a dissipative system as 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 ( ) m t on the dissipated system.

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) , and (iv) We first consider the case (i) for a constant mass, i.e., ( ) 0 m t m = . The vacuum state, Equation (39), is then given by that of the damped harmonic oscillator obtained in Ref. [11]: Figure 1 shows the time development of the vacuum state 0 in Here we see that the physical vacuum 0 (solid line) increases with time, reaches its maximum value and then decreases. This asymmetric shape of the 0 vs. t plot can be explained as follows: As seen from Equation (38), the vacuum state 0 is the product of ( ) is symmetric about 0 t = , which converges 0 at t = ±∞ , while the function Γ takes the values between 0.1 and 1 for 1 t = Γ . Accordingly, the monotonically increasing function ( )  converges about 2.5 for 2 t = Γ . From these results, we can say qualitatively that ( ) 0 t , which is the product of ( )  , has a peak at 2 t = Γ , and then decreases monotonically and converges 0 as t goes infinity. This asymmetry of the vacuum state seen in Figure 1 is due essentially to the presence of dissipative (resistive) force. In other words, this collapse of an initial state (instability of vacuum state) is characteristic of the dissipative system considered here.
We may say that in our dissipative model system, the time reversal symmetry of the vacuum state breaks down essentially due to the presence of the time reversed resistive (damping) force in the present model system.
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 t . In such a case, the vacuum state remains in a static state.
Next we consider the case (iii), where the mass decreases with time: ( )    varying mass, . Indeed this type of mass change explains the peculiar quantum mechanical features of the system represented by the Kanai Hamiltonian, when misinterpreted as representing a particle of fixed mass subject to a damping force [11]. In our formulation, Equation (39)  as Equation (43) shows. This is a general expression for the V-to-V transition amplitude in the presence of external force field, from which we can study the effects of the external and the dissipative force fields through 0  and 1  on the dissipative system expressed by the Hamiltonian  [see Equations (31)-(33)]. 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) [13]. In our formulation for many degrees of freedom, ( ) 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 ( ) † , a a for this dual system. The resultant Hamiltonian (28) is not a simple form and it is hard to clarify the physical meaning of each term in the particle picture. We showed that the secondquantized Hamiltonian (28) can be expressed in a simple form by introducing new creation and annihilation operators ( ) investigate dissipation (damping) in the quantum theory and quantum dissipated system by employing the DHO as the simple model system. The time-dependent oscillator mass ( ) m t as well as the damping factor γ plays an important role for damping (dissipation). Controlling these parameter, we can study the effect of dissipation in quantum dissipative systems.