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Engineering » Chemical Engineering » "Recent Advances in Thermo and Fluid Dynamics", book edited by Mofid Gorji-Bandpy, ISBN 978-953-51-2239-5, Published: December 21, 2015 under CC BY 3.0 license. © The Author(s).

# Nonequilibrium Thermodynamic and Quantum Model of a Damped Oscillator

By Gyula Vincze and Andras Szasz
DOI: 10.5772/61010

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# Nonequilibrium Thermodynamic and Quantum Model of a Damped Oscillator

Gyula Vincze1 and Andras Szasz1

## 1. Introduction

Dissipation is essential for the evolution of a quantum-damped oscillator. It is responsible for the decay of quantum states, the broadening of the spectral line, and the shifting the resonance frequency. This has been a persistent challenge for a long time since dissipation causes difficulties in the quantization of the damped oscillator [1, 2, 3]. This problem has remained under intensive investigation [4, 5]. There are some widely accepted Hamilton-like variation theories about the treatment of a linearly damped classic or quantum-damped oscillator. One of these theories is Bateman’s mirror-image model [1], which consists of two different damped oscillators, where one of them represents the main linearly damped oscillator. The energy dissipated by the main oscillator will be absorbed by the other amplified oscillator, and thus the energy of the total system will be conserved. The fundamental commutation relations of this model are time independent; however, the time-dependent uncertainty products, obtained in this way, vanish as time tends to infinity [6]. The Caldirola–Kanai theory with an explicit time-dependent Hamiltonian is another kind of variation theory [7, 8, 9]. In the quantum version of this theory, both the canonical commutation rules and the uncertainty products tend to zero as time tends to infinity. The system-plus-reservoir model [10, 11] is another damped oscillator model. It is coupled linearly to a fluctuating bath. If the bath is weakly perturbed by the system, then it can be modeled with a continuous bath of the harmonic oscillator. A quantum Langevin equation in the form of a Heisenberg operator differential equation can be deduced in this model. However, this equation in general does not obey Onsager’s regression hypothesis [12], i.e., only in case when 0 [13]. A direct consequence of this fact is that the expected value of the fundamental observable does not satisfy the equation of the classic linearly damped oscillator. Another consequence is that no spontaneous dissipative process exists in this theory. The above models are based on the Heisenberg’s mechanical reinterpretation model [14].

A possible reinterpretation model based on irreversible thermodynamics was recently published [15]. This model started from the Rosen–Chambers restricted variation principle of the nonequilibrium thermodynamics [16, 17, 18] and used a Hamilton-like variation approach to the linearly damped oscillator. The usual formalisms of classical mechanics, such as the Lagrangian, Hamiltonian, and Poisson brackets, were also covered by this variational principle. By means of canonical quantization, the quantum mechanical equations of the linearly damped oscillator are given. The resulting Heisenberg operator differential equations of the damped oscillator are consistent with the classical equations of motion and can be solved by using ladder operators, which are time dependent. By this theory, the exponential decay of quantum states, the natural width of the spectral line, and the shifts in the resonance frequency can be explained. This work describes the quantum theory of a linearly damped oscillator, which could be reinterpreted in terms of a classical model based on Onsager’s nonequilibrium thermodynamic theory, corresponding to the Heisenberg reinterpretation principle. The first chapters are devoted to Onsager’s thermodynamic theory and the quantum theory of a damped oscillator. The dissipative quantum theory given in the Heisenberg picture is deduced from the general evolution equation of a Hermitian observable by means of two system-specific constitutive equations. The first of the constitutive equations belongs to unitary dynamics, while the second belongs to the dissipative dynamics of the observable. The fundamental commutators, which are a consequence of the constitutive equations, are time dependent. The quantum mechanical equations of motion of the oscillator in the Heisenberg picture, the Ehrenfest theorem, and the uncertainty principle of that oscillator are given. A significant part of the work deals with applications such as the expected value of the main operators of the damped oscillator, the probability description of the wave packet motion belonging to the damped oscillator, the calculation of the wave function by matrix calculus, the spectral density of the energy dissipation, and the natural width of the spectral line. Another significant part of this work deals the quantum statistics of the damped oscillator. By a generalization of the Liouville–von Neumann equation, the statistical thermodynamic theory of the ensemble of the damped oscillators in contact with a thermal bath is given. By introducing the quantum entropy of the ensemble, it is shown that the entropy of the ensemble grows in a dissipative process and in thermal equilibrium for the probability distribution of the quantum states, such that Gibbs’ canonical distribution is valid. Finally, a wave equation of the linearly damped oscillator is given.

## 2. Nonequilibrium thermodynamic theory of the linearly damped oscillator

Meixner was the first to propose a nonequilibrium thermodynamic theory for linear dissipative networks [19, 20]; for a general overview on network thermodynamics, see [21]. In this theory, it can be shown that, for example, electrical networks are thermodynamic systems, and it is possible to derive the network equations (Kirchhoff equations) by application of the principles of nonequilibrium thermodynamics. In what follows, we give an Onsagerian thermodynamic theory of the linearly damped harmonic oscillator. A damped oscillator, as a primitive network, is considered under the isotherm condition, which is maintained by removing the irreversible heat as it developed in damping resistance, or in other words, by placing the damping resistance of the oscillator in a temperature bath. In this case, it is possible to speak not only of the entropy of the damped oscillator but also of the free energy, and not of entropy production but of energy dissipation instead. To show these, we give the actual form of the first law of thermodynamics in the case of a damped oscillator. To do this, let us introduce from the energy conservation law of the oscillator + thermal bath system (Figure 1), such that we obtain the following:

 ddt(U+Ubath)=0→dUdt=−dUbathdt (1)

where U and Ubath are the internal energies of the oscillator and the thermal bath. Since the oscillator is isolated by a rigid diathermal wall, the energy exchange between the oscillator and the bath must be heat transfer only. Thus, the first law of thermodynamics of the oscillator and the thermal bath has the following forms

 dUdt=dQdt, −dUbathdt=dQdt (2)

where dQdt is the power of exchanged heat on the rigid diathermal wall.

#### Figure 1.

Mechanical equivalent circuit of a linearly damped oscillator.

Assume that the entropy S of the oscillator exists, and it is a state function of the variables U, q. By the second law of nonequilibrium thermodynamics, the entropy S(U,q) satisfies the balanced equation:

 dSdt=dSrdt+dSidt, dSidt≥0 (3)

where dSrdt is the reversible rate of change of the entropy or entropy flux on the rigid diathermal wall and dSidt is the so-called entropy production. From the second law of the thermodynamics follows the well-known expression for entropy flux:

 dSrdt=1TdQdt (4)

On the other hand, the rate of change in the entropy of the oscillator can be written as

 dSdt=∂S∂UdUdt+∂S∂qdqdt (5)

From this equation and Equations (2) and (4), the following relations can result:

 ∂S∂U=1T, dSidt=∂S∂qdqdt≥0 (6)

Thus, the entropy balance equation has the form

 dSdt=1TdQdt+∂S∂qdqdt (7)

Because the temperature of the oscillator is constant, we could introduce

 F=U−TS (8)

i.e., the free energy of the oscillator, and from Equations (2) and (7), we can obtain the simple balanced equation for free energy

 dFdt=−T∂S∂qdqdt≤0 (9)

Also, entropy production and the so-called rate of energy dissipation

 R=TdSidt=T∂S∂qdqdt≥0 (10)

decrease the free energy of the oscillator. In what follows, we shall give the actual form of the rate of energy dissipation. Next, we see from Equation (10) that the rate of energy dissipation must be some explicit function of dqdt and may depend implicitly on U, q and the following equation

 R=R(dqdt;U,q)≥0 (11)

such that

 R(0;U,q)=0 (12)

is therefore quite general. We now expand the energy dissipation Equation (11) in a Taylor series, i.e.,

 R=A0+A1dqdt+12A2(dqdt)2+..... (13)

The sufficient condition of nonnegativity of entropy production, which will always be satisfied, is the complete exclusion of all odd terms in dqdt in Equation (13), and A0 must be zero to exclude entropy production in an equilibrium state. Thus, one need only neglect the fourth-order term in the Taylor series (Equation (13)) to obtain the Rayleigh dissipation function

 R=c(dqdt)2, c=12A2 (14)

In addition, the so-called damping constant of oscillator c must be positive to satisfy the nonnegativity condition in Equation (13). If we assume that the nondissipative elements of the damped oscillator are linear, then the free energy in Equation (8) can be identified with the energy stored in the mass m and the spring of the oscillator (Figure 1.). Also, the free energy is equal to the Hamiltonian of the oscillator, i.e.,

 F=H=12p2m+12kq2 (15)

where q is the displacement of the mass from its equilibrium position, p is the momentum of the mass of the oscillator, and k is the constant of the spring. A direct consequence of the above results is that R can be constructed as a bilinear form, namely,

 R=−dHdt=dqdt(−∂H∂q)+dpdt(−∂H∂p)≥0 (16)

Here we now interpret, in the usual nonequilibrium thermodynamic fashion, the quantities (dqdt,dpdt) in terms of thermodynamic fluxes and the quantities (Hq,Hp) in terms of thermodynamic forces. Since we now have Equation (14) of the rate of energy dissipation of the damped oscillator, Equation (16) can be written in the form

 dqdt(−∂H∂q)+dpdt(−∂H∂p)=c(dqdt)2≥0 (17)

In Onsagerian thermodynamics, the constitutive (kinetic) equations between fluxes and forces are linear

 [dqdtdpdt]=[LqqLqpLpqLpp][−∂H∂q−∂H∂p] (18)

where the kinetic matrix

 L=[LqqLqpLpqLpp] (19)

can be split into nondissipative (so-called reactive) and dissipative parts. To do this, take these kinetic equations into Equation (17), and using Equation (15), with a simple calculation, we obtain for these kinetic matrices

 [dqdtdpdt]=([0−aa0]+[000c])[−∂H∂q−∂H∂p] (20)

Here, a is an arbitrary constant. Now, we see that the dissipative part of the kinetic matrix satisfies the Onsager symmetry relation and the positivity of the damping constant c trivially. The constant a can be evaluated as follows. In the case of zero for the dissipative part of the kinetic matrix, these equations must be transformed into Hamilton equations of a simple harmonic oscillator. From this fact, it follows that a is a universal constant and a=1. Also, the final form of the Onsagerian constitutive (kinetic) equation the of damped oscillator is

 [dqdtdpdt]=([0−110]+[000c])[−∂H∂q−∂H∂p] (21)

It is easy to show that these kinetic equations are equivalent to the Newtonian equations of motion of the linearly damped oscillator, namely,

 dqdt=∂H∂p=pm, dpdt=−∂H∂q−c∂H∂p=−kq−cpm (22)

From this equivalence, it follows that the velocity of oscillator as a generalized thermodynamic flux has only a reactive part, while the rate of momentum, as another thermodynamic flux, has both reactive and dissipative constituents. The presented thermodynamic deduction of equations of a linearly damped oscillator enables us to build a stochastic force Fs into equations of motion (Equation (22)). This stochastic force and the thermodynamic forces introduced above are statistically independent and take into account the effect of the temperature bath. In this case, the thermodynamic fluxes are fluctuations, which obey another type of equation (Equation (22)). The correct form of these equations follows from the regression hypothesis of Onsager [22], which states that “the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process.” From this, we give the Langevin-type equations for a linearly damped oscillator,

 dqdt=∂H∂p, dpdt=−∂H∂q−c∂H∂p+Fs (23)

A consequence of these equations is that the dissipative kinetic coefficient c can be related to the correlation coefficient p(t),p(t+t') via the fluctuation dissipation theorem [23, 24]. According to Equation (22), we can conclude that the fluctuations of thermodynamic fluxes are similar to an impressed macroscopic deviation, except they appear spontaneously.

### 2.1. Bohlin’s first integral

By means of Equation (21) or (22), we can deduce the time rate of change of any observable defined in the phase space of the damped oscillator. Let O(q,p,t) be an observable, such that its time rate of change can be expressed as

 dOdt=∂O∂t+∂Odqdq∂t+∂O∂pdpdt=∂O∂t+∂Odq∂H∂p−∂O∂p∂H∂q−c∂O∂p∂H∂p (24)

where we take into account the Onsagerian equations (Equation (22)). An observable O(q,p,t) is the first integral of the (Onsagerian equation (Equation (21)) of the damped oscillator if

 dOdt=0 (25)

and if it is a constant of the motion. Now, let us give the constant of the motion of the linearly damped harmonic oscillator. It was Bohlin [25] who first dealt with the problem of the constants of motion for a damped linear oscillator. It is easy to prove that Bolin’s observable, defined as

 B(q,p,t)=m2e2βt(dqdt−γq)(dqdt−γ∗q)==e2βt(p22m+β2qp+β2pq+mω022q2) (26)

where

 β=c2m,ω0=km,γ=−β+iω,  γ∗=−β−iω, ω=ω02−β2 (27)

is the first integral of the damped oscillator. Now, we may see that the Bolin’s observable in the case of the undamped oscillator is equal to the Hamiltonian of the oscillator.

## 3. Quantum theory of linearly damped oscillator

In the standard theory of quantum mechanics, two kinds of evolution processes are introduced, which are qualitatively different from each other. One is the spontaneous process, which is a reactive (unitary) dynamical process and is described by the Heisenberg or Schrödinger equation in an equivalent manner. The other is the measurement process, which is irreversible and described by the von Neumann projection postulate [26], which is the rigorous mathematical form of the reduction of the wave packet principle. The former process is deterministic and is uniquely described, while the latter process is essentially probabilistic and implies the statistical nature of quantum mechanics.

### 3.1. The general evolution equation of the Hermitian operator

Unlike classical quantum mechanics, the spontaneous processes of the damped oscillator are irreversible, so its quantum mechanical description needs changes to some instruments of classical quantum mechanics. To do this, we use the Heisenberg picture of quantum processes. In this picture, the observables are time-dependent linear Hermitian operators, and the state vector of the system is time independent. Using the terminology introduced in the first part, the infinitesimal time transformation of the Hermitian operator could happen in two ways:

• By reactive transformation, when the orthonormal eigenvectors of the Hermitian observable turn in time, keeping the orthonormal system with unchanged eigenvalues. The eigenvectors belonging to the different moments are connected with unitary transformation, as in classic quantum mechanics. Dynamics belonging to this transformation are so-called unitary dynamics.

• By dissipative transformation, when the real eigenvalues of the Hermitian operator change irreversibly in time.

Let us study the general evolution equation of the Hermitian operator, considering both the above time-dependent processes. For simplicity in demonstrating the derivation, we suppose a discrete eigenvalue spectrum of the Hermitian operator, although the spectra of the displacement and momentum operators could be continuous. In this case, the orthonormal eigenvectors |ΨOi(t) and eigenvalues λOi(t) are solutions of the eigenvalue equation

 O(t)|ΨOi(t)〉=λOi(t)|ΨOi(t)〉, (i=1,2,3,...) (28)

and the eigenvectors form a complete orthonormal basis in a Hilbert space, when the eigenvalue spectrum is nondegenerate. Thus, the spectral representation of the operator would be

 O(t)=O(t)δ=∑i≥0λOi(t)|ΨOi(t)〉〈Ψ0i(t)|, δ=∑i≥0|ΨOi(t)〉〈Ψ0i(t)| (29)

where δ is the unity operator and ΨOi(t)| is the dual of |ΨOi(t), so ΨOi(t)|ΨOi(t)=1.

Since the observable is Hermitian, the transformation of the eigenvector |ΨOi(0) at t=0 to the eigenvector |ΨOi(t) at time t will be represented by an unitary operator U(t). Hence,

 |ΨOi(t)〉=U(t)|ΨOi(0)〉, 〈ΨOi(t)|=U+(t)〈ΨOi(0)|, U(t)U+(t)=δ (30)

Let us substitute Equation (30) into Equation (29), then we obtain

 O(t)=∑i≥0λOi(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t) (31)

The time derivative of the operator is

 O•(t)=∑i≥0λ•Oi(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t)++∑i≥0λOi(t)U•(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t)++∑i≥0λOi(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U•+(t) (32)

or by using the instantaneous eigenvectors, |ΨOi(t) can be written as

 O•(t)=∑i≥0λ•Oi(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t)++∑i≥0λOi(t)U•(t)U+(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t)++∑i≥0λOi(t)U(t)|ΨOi(0)〉〈Ψ0i(0)|U+(t)U(t)U•+(t)==∑i≥0λ•Oi(t)|ΨOi(t)〉〈Ψ0i(t)|+U•(t)U+(t)(∑i≥0λOi(t)|ΨOi(t)〉〈Ψ0i(t)|)++(∑i≥0λOi(t)|ΨOi(t)〉〈Ψ0i(t)|)U(t)U•+(t)=∑i≥0λ•Oi(t)|ΨOi(t)〉〈Ψ0i(t)|+U•(t)U+(t)O(t)+O(t)U(t)U•+(t) (33)

Using the identity U(t)U+(t)=U(t)U+(t), which is a consequence of unitarity, the general evolution equation of the Hermitian operator results

 O•(t)=∂O(t)∂t+[U•(t)U+(t),O(t)],∂O(t)∂t:=∑i≥0λ•Oi(t)|ΨOi(t)〉〈Ψ0i(t)| (34)

where [,] is Dirac’s symbol of a quantum mechanical commutator and O(t)t is the local time rate of change of the operator in the coordinate system of the instantaneous eigenvectors. This equation is universal in the meaning of its independence of the constitutive behavior of the quantum system.

### 3.2. The Heisenberg equation of motion of the linearly damped oscillator

The actual form of Heisenberg’s dynamic equation can be constructed when the expression U(t)U+(t) is formed from the unitary operator and the local rate of change of the operator O(t)t can be connected to the constitutive properties of the studied physical system. The first term will be ordered to the unitary/reactive and the second to the dissipative dynamics. To do this, we accept Heisenberg’s reinterpretation principle [14] (for the philosophical details of this principle, see [27]), which states the possibility of constructing a quantum mechanical description of a physical system whose classical description is known. In our case, the physical system is a linearly damped oscillator, for which we know its Onsagerian thermodynamic description. This description uses the Hamiltonian and the rate of energy dissipation of the system represented by the Rayleigh potential. Since the Hamiltonian is not the first integral of the motion, Bohlin’s first integral could be used for the unitary dynamics. In classical quantum theory, the Hamiltonian belongs to the unitary dynamics as a constitutive property, i.e.,

 iℏH=U•(t)U+(t) (35)

where the Hamilton operator H is a first integral of the system. In a closed physical system, the local time derivative of the observables is zero since the system is reactive. On this basis, the constitutive equations of classical quantum mechanics are

 iℏH=U•(t)U+(t), ∂O(t)∂t=0 (36)

With these constitutive equations from Equation (34), the general evolution equation we give to Heisenberg’s equation of the observable is

 O•=iℏ[H,O] (37)

Also, the all constitutive properties of the quantum system are contained in the Hamilton operator only, which could have originated from the Hamilton function of the classical model by means of Heisenberg’s reinterpretation principle. Figure 2 shows the above-presented scheme of the deduction of Heisenberg’s equation of motion.

### Figure 2.

The schema of the deduction of Heisenberg’s equation of motion of a Hermitian operator and the role of Heisenberg’s reinterpretation principle.

The Hamiltonian H is a trivial nullifier of Dirac’s commutator in this approach, so H is a conserved observable of motion, as was requested. To summarize, we get Heisenberg’s classical Equation (37) from the general evolution Equation (34), if the O(t)t local rate of change of the operator in the coordinate system of the instantaneous eigenvectors is zero. In the case of the Heisenberg’s equation (Equation (37)), the entropy of the system is constant in time, as proven by von Neumann [26]. However, the entropy cannot remain constant in dissipative processes. Consequently, in a correct description of the dissipative system, it is possible to take into account the local rate of change of the operator. In the case of a damped oscillator, this means that for the time rate of change q(t)t, p(t)t of two fundamental observables of a linearly damped oscillator must be given constitutive equations. In this way, we assume that the constitutive equations of the linearly damped oscillator are

 ∂q(t)∂t=−βq(t), ∂p(t)∂t=−βp(t), iℏB=U•(t)U+(t) (38)

where the Hermitian operator B belonging to unitary/reactive dynamics is the quantum mechanical equivalent of Bohlin’s constant in Equation (26). Following Equation (26), this could be written as

 B=e2βt(p22m+β2pq+β2qp+mω022q2) (39)

Note, the third constitutive equation of (38) is the direct consequence of Stone’s theorem [28]. If we take into account these constitutive equations in the general evolution Equation (34) of the Hermitian operators, then we obtain Heisenberg’s equations of motion of a quantum-damped oscillator

 q•(t)=−βq(t)+iℏ[B,q(t)], p•(t)=−βp(t)+iℏ[B,p(t)] (40)

According to Heisenberg’s reinterpretation principle, these equations could be interpreted by means of the Onsagerian equations of the oscillator. To do this, split the Bohlin operator (Equation (39)) into two parts. The first part contains the Hamilton operator H of the oscillator and the second part D belongs to the dissipation. We then obtain

 q•(t)=−βq(t)+iℏ[B,q(t)]={∂q(t)∂t+iℏe2βt[D,q(t)]}+iℏe2βt[H,q(t)]p•(t)=−βp(t)+iℏ[B,p(t)]={∂p(t)∂t+iℏe2βt[D,p(t)]}+iℏe2βt[H,p(t)]+H=(p22m+mω022q2), D=β2(qp+pq) (41)

The expression in {} is connected to dissipative thermodynamic current by analogy, while the currents outside the bracket are analogous to reactive currents. This interpretation, analogous to Equation (22), is supported by

 ∂q(t)∂t+iℏe2βt[D,q(t)]=0, ∂p(t)∂t+iℏe2βt[D,p(t)]=2∂p(t)∂t=−2βp(t) (42)

where the two first equations of Equation (38) were used. In detail, we could write

 ∂q∂t+iℏe2βt[D,q]=−βq+β2(iℏe2βt[p,q])u+β2q(iℏe2βt[p,q])=0∂p∂t+iℏe2βt[D,p]=−βp−β2p(iℏe2βt[p,q])+β2(iℏe2βt[p,q])p=−2βp (43)

Now, we could see the desired interpretation analogy could be applied when the commutator relation e2βt[p,q]=iδ is valid. Since every operator commutes with itself, we also have the fundamental brackets of our quantum theory

 {p,q}=ℏiδ, {q,q}=0, {p,p}=0, {,}:=e2βt[] (44)

Consequently, the first fundamental bracket in Equation (44) ensures that the dissipative part β2qp+β2pq of the Bohlinian adds βq(t) term to the first equation and the βq(t) term to the second equation in Equation (41). This allows to us change our attention from Bohlinian to Hamiltonian in Equation (41), obtaining

 q•(t)=iℏe2βt[H(t),q(t)]=iℏ{H(t),q(t)},p•(t)=−2βp(t)+iℏe2βt[H(t),p(t)]=−2βp(t)+iℏ{H(t),p(t)} (45)

which are equivalent with the equations

 q•(t)=iℏ{H(t),q(t)}, p•(t)=iℏ{H(t),p(t)}−ciℏ{H(t),q(t)} (46)

The quantum mechanical equations of a damped oscillator with the fundamental brackets in Equation (44), applying the rules of Lie algebra, are as follows

 q•(t)=iℏ{H(t),q(t)}=p(t)m,p•(t)=iℏ{H(t),p(t)}−ciℏ{H(t),q(t)}=−2βp(t)−mω02q(t) (47)

which are the operator differential equations version of Onsager’s equations in Equation (22).

To use the Lie algebraic method in an evaluation of the above-introduced time-dependent commutators, it is assumed that the scalar time functions must necessarily be considered as ordinary numbers (for details, see [15]). In summary, according to the Onsagerian equations of the damped oscillator by application of Heisenberg’s reinterpretation principle, the quantum mechanical equation of a damped oscillator in the Heisenberg picture can be obtained.

### 3.3. Ehrenfest theorem of a linearly damped oscillator

It is easy to show, similar to classical quantum mechanics, that the following operator analytics relations are valid [29]

 iℏ{H,q}=∂H∂p, iℏ{H,p}=−∂H∂q (48)

As a consequence of Equation (48), Equation (47), the quantum mechanical equations of the oscillator, could be written in the form

 q•(t)=∂H∂p=p(t)m, p•(t)=∂H∂q−c∂H∂p=−2βp(t)−mω02q(t) (49)

where the formal equivalence with Onsager’s equations Equation (22) is obvious.

It is well-known in the Heisenberg picture that the expectation value of an operator is defined as

 〈O〉(t)=〈Ψ|O(t)|Ψ〉 (50)

where |Ψ is the time-independent state vector of the oscillator. Thus, from Equation (49), the expectation values of the time rate of change of displacement and momentum can be evaluated as

 d〈q〉(t)dt:=〈Ψ|q•|Ψ〉=iℏ〈Ψ|{H,q}|Ψ〉=〈Ψ|∂H∂p|Ψ〉=1m〈Ψ|p|Ψ〉=〈p〉(t)m,d〈p〉(t)dt:=〈Ψ|p•|Ψ〉=iℏ〈Ψ|{H,p}|Ψ〉−ciℏ〈Ψ|{H,q}|Ψ〉==−〈Ψ|∂H∂q|Ψ〉−c〈Ψ|∂H∂p|Ψ〉=−mω02〈Ψ|q|Ψ〉−cm〈Ψ|p|Ψ〉==−cm〈p〉(t)−mω02〈q〉(t) (51)

where we take into account Equation (49). Also, the expectation values of displacement and momentum of the linearly damped oscillator obey time evolution equations, which are exactly equivalent to those of Onsager’s equations (Equations (21) and (22)). This result is Ehrenfest’s theorem.

## 4. Evaluation of the equations of the quantum linearly damped oscillator

The solutions of the operator differential equations (Equation (49)) are

 q=ℏ2mω0(Ae−iωt+A+eiωt),p=ℏm2ω0(γAe−iωt+γ*A+eiωt),A=ae−βt (52)

By substituting the above two expressions into the first fundamental commutation relation of Equation (44), the time-dependent and time-independent amplitude operators are used to obtain the following commutation relations

 {A,A+}=e−2βtδ→[a,a+]=δ (53)

To solve the damped oscillator problem, we have to determine the operator A because this should be known for the specification of displacement, momentum, and the energy of the oscillator. In the case of a nondamped oscillator, the amplitude operator can be determined from the Hamilton operator of the oscillator, which is a constant of the motion. This is, however, not true for our case; thus, we will use the Bohlin operator introduced earlier. By substituting Equation (52) into the Bohlinian Equation (39), we get

 B=e2βt(p22m+β2pq+β2qp+mω022q2)=ℏωm2e2βt(AA++A+A)==ℏωme2βt(A+A+12e2βt)=ℏωm(a+a+12), ωm=ω2ω0−1 (54)

where the commutation relations (Equation (44)) were used. Pursuant to the above two relations, it is easy to show that the time-independent amplitude operators fulfill the equations

 [B,a+]=ℏwma+,[a,B]=ℏwma (55)

Now, we see that if we replace the operator B by the Hamiltonian H of a simple oscillator, these equations are identical to the corresponding equations of the simple quantum oscillator [30, 31]. According to this strong analogy, we are able to determine the amplitude matrix and the matrices of the Bohlin operator B, the displacement operator and the momentum operator. The results are as follows:

• The operators a,a+,B and the occupation number operator N:=a+a have the same eigenvectors and different eigenvalues. Since N:=a+a is positive definite, B can possess no negative eigenvalue. The lowest eigenvalue of B belongs to the eigenket |0 of the operator a for which the relation a|0=0 holds. From Equation (54), this so-called vacuum state belongs to the 12ωm zero-point Bohlinian eigenvalue and zero occupation number eigenvalue. The Bohlinian eigenvalue belonging to the eigenket |n (where the occupation number is n) can be calculated in the form of |n=a+nn!|0 is (12+n)ωm, while the occupation number eigenvalue is n.

• For the actions of the eigenket |n of the ladder operators, a and a+ can be written as a|n=(n1)|n1,a+|n=(n+1)|n+1, from which it follows for the occupation number operator that N|n=n|n.

• The matrices of the above-introduced operators are

 a=[010.002.000.....], (56)
 N=aa+=[100.020.003.....]. (57)

on the basis of which is formed the orthonormal eigenkets

 |0〉=[100.],|1〉=[010.],|2〉=[001.],etc., (58)

It is easy to see that the matrices that belong to aa and a+a+ are not diagonal. From the above equations, it follows that the rules for the time-dependent ladder and occupation number operators are

 A+(t)|n〉=e−βtn+1|n〉, A(t)|n〉=e−βtn−1|n−1〉,N(t)|n〉=e−2βtn|n〉 (59)

## 5. Applications

### 5.1. Expected values of the main operators of a linearly damped oscillator

Moreover, the expected value of the occupation number in the nth energy eigenstate at time t is

 〈N〉(t):=〈n|N(t)|n〉=N0e−2βt, (60)

where N0 is the occupation number at t=0. This result agrees well with the corresponding result derived from the system-plus-reservoir model [32]. In the energy representation, since the matrices of the operators a,a+,a2,a+2 have zero diagonal elements, the expected values of the operators A,A+,A2,A+2 are zero in every energy eigenstate, i.e.,

 〈A〉=0, 〈A+〉=0, 〈A2〉=0, 〈A+2〉=0 (61)

According to these equations, the expected values q=:n|q|n,p=:n|p|n of the displacement and the momentum operator

 q=ℏ2mω0(Ae−iωt+A+eiωt), p=ℏm2ω0(γAe−iωt+γ*A+eiωt) (62)

in the nth energy eigenstate are zero. The variance q2=n|q2|n,p2=n|p2|n of the displacement and momentum operator in the nth energy eigenstate can be evaluated as

 〈q2〉=ℏ2mω0〈AA++A+A〉=ℏ2mω0〈δe−2βt+2AA+〉==ℏ2mω0e−2βt(1+2n) (63)

and

 〈p2〉=ℏm2ω0〈γγ*(Aeiωt)(A+e−iωi)+γγ*(A+e−iωt)(Aeiωt)〉==mℏω02〈δe−2βt+2(Aeiωt)(A+e−iωt)〉==mℏω0e−2βt(12+〈aa+〉)=mℏω0e−2βt(12+n) (64)

where we considered the commutation relation (Equation (53)). According to these results, we obtain the expected value of the energy of the damped oscillator

 〈H〉(t)=〈p2〉2m+mω02〈q2〉2=ℏω0〈δe−2βt+2AA+〉=ℏω0e−2βt(n+12) (65)

and the uncertainty relation

 (Δq)(Δp)=ℏ2e−2βt(2n+1),→(Δq)(Δp)≥ℏ2e−2βt(Δq)2:=〈(q−δ〈q〉)2〉=〈q2〉, (Δp)2:=(p−δ〈p〉)2¯=〈p2〉 (66)

where we considered that q=0,p=0. Now, we can see Heisenberg’s uncertainty relation is not fulfilled and, in the case of the simple oscillator and also when c=0, this relation transforms into Heisenberg’s relation.

### 5.2. Probability description of the wave packet motion of the damped oscillator

To learn something about the time dependence of our system in a certain state |, we will calculate q|(t) and |q||2(t), which represent the probability amplitude and probability of finding the damped oscillator at q at time t in that state. In particular, it is useful to study the so-called coherent state |a, which is an eigenstate of the non-Hermitian time-dependent operator A, i.e.,

 Aeiωt|a〉=mω02ℏA0e−βteiωt|a〉 (67)

We shall also calculate the probability amplitude Ψa(q)=q|a of the wave packet |a at q. To do this, we shall start the following fact of bracket calculus

 〈q|Aeiωt|q〉=∫〈q|Aeiωt|q'〉〈q'|q〉dq'=∫〈q|Aeiωt|q'〉Ψd(q')dq'==mω02ℏA0e−βteiωtΨa(q) (68)

Now, we are going to express the operator A by using the displacement and the momentum operator (52), so we get

 Aeiωt=2ℏmω0p−mγ∗qiω (69)

Taking this expression into (68), we obtain

 ∫〈q|p−mγ*qi|q′〉Ψa(q′)dq′=mωA0e−βteiωtΨa(q) (70)

According to the following

 〈q|p|q'〉=e−2βtℏidδ(q−q')dq+f(q), 〈q|q|q'〉=qδ(q−q') (71)

coordinate representation of the operators originating from the commutation relation Equation (44) (for details, see [15]), we get

 (ωq+ℏmddpe−2βt)Ψd(q)=ωA0e−βteiwtΨd(q) (72)

an ordinary differential equation, where cq is chosen for the arbitrary function f(q). The solution of this equation in a normalized form is

 Ψd(q)=g(t)(ℏωπ)14e−ωm2ℏ(q−A0e−βteiωt)2e−2βt (73)

where the function g(t) is evaluated in follows. From which the probability will be

 |Ψa(q)|2=(mω2ℏπ)12eβte−mωℏe2βt(q−A0e−βtcos(ωt))2 (74)

Now, we might see that this is the Gaussian distribution with the

 |Ψa(q)|2=12π(ℏmωe−βt)e−(q−A0e−βtcos(ωt))22(ℏmωe−βt)2 (75)

probability density function. Therefore, the motion of the center of the wave packet |a is a damped oscillation, and its uncertainty width decreases exponentially from the initial value of mω to zero (see Figure 3).

### Figure 3.

The evolution of the wave packet |a. The motion of the center of packet and its uncertainty width Δq are represented.

Now, we see that the initial uncertainty of the packet |a keeps getting smaller with the progression of time and becomes negligible as t. Also, the evolution of the wave packet continually proceeds toward the motion of a classic damped oscillator with the progression of time.

### 5.3. Calculation of wave function by matrix calculus

Resulting from Equation (52) using Equation (56), the matrix of the displacement operator in energy representation has the form

 〈n'|q|n"〉:=〈n'|A+A+|n"〉=ℏ2mω0[0100010200020300030400040          etc.]e−βt (76)

Here, the displacement operator was used in a narrow sense. Next, we are going to solve the

 q|q〉=q|q〉 (77)

eigenvalue problem in terms of the dn=n|q probability amplitudes. To do this, we rewrite the above eigenvalue equation in matrix form

 ∑n′≥0〈n|q|n′〉〈n′|q〉=q〈n|q〉,(n=0,1,2,…) (78)

where we take into account the n'0|n'n'|=δ closure relation. In explicit form, this looks like

 [0100010200020300030400040etc.]e−βt[d0d1d3...]=2mω0ℏq[d0d1d3...] (79)

From this, we get the difference schema

 d1=qo1d0,d2=qo2d1−12d0,…,dn=qondn−1−n−1ndn−2qo=2mω0ℏeβtq (80)

After some algebra, we obtain another form

 d'n=2q∘2d'n−1−2(n−1)d'n−2,  d'n:=2nn!dn (81)

By introducing a new coordinate variable, we have the difference equation

 d'n=2q∧d'n−1−2(n−1)d'n−2,  q∧:=q∘2 (82)

This difference equation is satisfied by the Hermitian polynomials. Thus, we obtain

 〈n|q〉=dn=c0(eβtmω0ℏq)12nn!Hn(eβtmω0ℏq), (83)

where c0(eβtmω0q) is a function to be determined. Starting from the fact that q'|q''=δ(q'q''), we get

 〈q′|q″〉=∑n〈q′|n〉〈n|q″〉=∑nc0(eβtmω0ℏq′)12nn!Hn(eβtmω0ℏq′)××c0*(eβtmω0ℏq″)12nn!Hn(eβtmω0ℏq″)=∑nc0(eβtmω0ℏq′)c0*(eβtmω0ℏq″)12nn!Hn(eβtmω0ℏq′)×Hn(eβtmω0ℏq″)=δ(q′−q″) (84)

By using the

 ∑n12nn!Hn(eβtmω0ℏq')×Hn(eβtmω0ℏq'')==πe(eβtmω0ℏq')2δ(eβtmω0ℏq'−eβtmω0ℏq'')==πmω0ℏe(eβtmω0ℏq')2eβtδ(q''−q') (85)

relationship, the final form of Equation (83) is given by

 Ψn(q):=〈n|q〉=dn=e12βt(mω0ℏπ)14e−12e2βtmω0ℏq212nn!Hn(eβtmω0ℏq) (86)

The physical meaning of the strange variable in the Hermit polynomials is that the distance of the nodes of these functions keeps getting smaller with the progression of time by the exponential law eβt. According to this result, the probability density of the nth energy state with displacement q is

 |Ψn(q)|2=|〈n|q〉|2=12nn!eβt(mω0ℏπ)12e−mω0ℏq2e−2βtHn2(e−βtmω0ℏq)=12π(ℏ2mω0e−βt)e−q22(ℏ2mω0e−βt)2[12nn!Hn2(e−βtmω0ℏq)] (87)

This result is exactly identical to the equation given by Kim and Page [33] on the basis of another theory. Now, we might see that this is the density function of a modulated Gaussian distribution, where the modulating term has finite amplitude which runs over in time, while the Gaussian distribution sharpens toward to a Dirac delta distribution. This means that the particle will get closer and closer to the equilibrium point as t. From last result, we can conclude that in the case of β0 we get back to the well-known wave function of the simple oscillator.

### 5.4. Spectrum of the energy dissipation of the linearly damped oscillator

We are going to give the frequency spectrum of radiation and explain the natural width of the spectrum line. As an atom emits photons, its energy drops and the amplitude of transition decreases over time. Therefore, the emission is not harmonic, and a spectrum occurs. We shall see that the natural width of the spectral line can be connected to the attenuation coefficient of the damped oscillator. Inversely, from the width of the spectral line, we might determine the attenuation coefficient of the oscillator.

#### 5.4.1. Spectral density of the energy dissipation

In the first section, the time rate of energy dissipation for a damped oscillator is introduced by the Rayleigh dissipation potential. The quantum version of this quantity, i.e., the time rate of the energy dissipation operator, can be originated from Equation (14) as

 R:=cm1mp2=2βmp2 (88)

From this, it follows that the expected value of the operator of energy dissipation is

 wdiss=∫0∞〈R〉dt=∫0∞〈n|R|n〉dt=2βm∫0∞〈n|p|2n〉dt (89)

Substituting the expression of the momentum operator (Equation (64)) into this equation, then the above equation has the form

 wdiss=βℏω0∫0∞〈(γγ*(Aeiωt)(A+e−iωt)+γγ*(A+e−iωt)(Ae−iωt))〉dt==βℏω0∫0∞〈(δe−2βt+2(Ae−iωt)(A+e−iωt))〉dt=ℏω02+2βℏω0〈aa+〉∫0∞eγteγ*tdt=ℏω02+2βnℏω0∫0∞eγteγ*tdt, γ=β+iω (90)

where we assume that the occupation number is n, i.e., n=aa+. Evidently, the first term of this expression belongs to the vacuum fluctuation, and the second term belongs to the essential dissipative process of the damped oscillator in which the occupation number could change. We will evaluate the second term of this energy dissipation formula. According to the Parseval theorem of the Fourier transformation theory, this term of energy dissipation may be written as

 Δwdiss(n)=2βnℏω0∫0∞eγteγ*tdt=n2βℏω0∫0∞F[eγt]F[eγt]*dω==n2βℏω0∫0∞dω′(ω−ω′)2+β2 (91)

where F[eγt] is the Fourier transform of eγt. Also, the spectral density of the dissipated energy is

 2βℏω0(ω−ω')2+β2 (92)

i.e., a Lorentz distribution about the shifted circular frequency ω of the damped oscillator. This result agrees well with corresponding result derived from the two-state atom model of Wigner and Weisskopf [34, 35] and the system-plus-reservoir model [10, 32, 36]. It is well known that the half value width Δω of this distribution is

 Δω=β (93)

Now, we can see the transition from the nth occupation number state to the vacuum state, in which the oscillator will emit nω0 energy. Indeed, from Equation (91) it follows that

 Δwdiss(n)=n2βℏω0∫0∞dω′(ω−ω′)2+β2=nℏω0 (94)

#### 5.4.2. Natural width of the spectral line

The natural line width of the spectral line is a significant result of the dissipative quantum process which accompanies the spontaneous emission of an atom. We will treat this emission process in a dissipative two-state model. We consider the two states of the atom as the zeroth and the first occupation number state of a linearly damped oscillator. In this case, the spontaneous emission of a photon is the consequence of the transition from the first occupations number state to the equilibrium state of the damped oscillator. In this model, the spectrum density of the emitted photon follows from Equation (92)

 Δwdiss(ω')=2βℏω0(ω−ω')2+β2 (95)

The width of this frequency spectrum of a spontaneous emission of the atom is a direct consequence of the dissipative self-force on the atom due to the back-reaction of the emitted photon. This back-reaction of the emitted photon can be characterized by two physical quantities, namely, the frequency shift ω0ω and the half value width Δω=β of the spectrum. If we consider Δω2 as the energy uncertainty ΔE of the emitted wave packet and the time constant of the emission process Δt=β1 as the time uncertainty, we obtain an uncertainty relation

 ΔEΔt≥ℏ2ββ=ℏ2 (96)

The quantum mechanical interpretation of the width of the natural spectral line should be based on this relation, in which the physical quantities ΔE and Δt=β1 have a precise meaning. In our model, the natural line width occurs at wavelength λ and can be calculated as

 Δλ=|Δ2πcω|=2πcω2Δω=2πcω2β (97)

where Equation (93) was used and c is the vacuum velocity of light. It is well known that in the classical dipole model of light emission, the natural line width can be calculated as

 Δλ=4πε03mc2 (98)

where ε0 is the vacuum permittivity and re:=ε03mc2=2,8181015m is the so-called classical electron radius. From the above two equations, it follows that

 β=23ω2cre=4π3reλω (99)

## 6. Uncertainty relation of the linearly damped oscillator

The standard derivation of Heisenberg’s uncertainty relation neglects the possibility that two operators A and B, say q and p,which fulfill the commutator relation

 {A,B}=iℏ (100)

could have a compatible component which is the first part of the trivial identity

 AB=AB+BA2+AB−BA2 (101)

This observation has importance when we take into account the irreversibility. Due to irreversibility, the damped oscillator proceeds to thermal equilibrium with the thermal bath. This thermal equilibrium can be characterized in terms of classical statistic theory. However, in classical statistics, random variables have a joint distribution function, which could exist in the case of quantum theory if the operators are compatible. The commutator relation (Equation (100)) is compatible this physical picture, but from Equations (100) and (101), we obtain

 AB=AB+BA2+δiℏ2e−2βt (102)

From this relation, in the case of t, the compatibility of the operators follows, i.e.,

 AB=AB+BA2→AB=BA (103)

In what follows, we will show that the above-mentioned arguments appear in the uncertainty relation. The variance of the Hermitian operators A and B can be calculated by the norm of the following vectors

 f=(A−δ〈A〉)|Ψ(0)〉,g=(B−δ〈B〉)|Ψ(0)〉 (104)

Indeed, we can write

 (ΔA)2=‖f‖2=〈Ψ(0)|(A−δ〈A〉)2|Ψ(0)〉(ΔB)2=‖g‖2=〈Ψ(0)|(B−δ〈B〉)2|Ψ(0)〉 (105)

where Ψ(0) is the state vector of the system, A and B are the expected value of the operators defined as

 〈A〉=〈Ψ(0)|A|Ψ(0)〉,〈B〉=〈Ψ(0)|B|Ψ(0)〉 (106)

Thus, the f2g2|f,g|2 Schwarz inequality implies

 (ΔA)2(ΔB)2≥|〈Ψ(0)|(A−δ〈A〉)(B−δ〈B〉)|Ψ(0)〉|2=|〈AB〉−〈A〉〈B〉|2 (107)

By substituting into this expression the identity (Equation (101)), then we get

 (ΔA)2(ΔB)2≥|〈AB〉+〈BA〉2+〈AB−BA〉2−〈A〉〈B〉|2==|〈AB〉+〈BA〉2+iℏ2e−2βt−〈A〉〈B〉|2=ℏ24e−4βt+(〈AB〉+〈BA〉2−〈A〉〈B〉)2 (108)

where we take into account that the quadrate of the absolute value of a complex number is equally the sum of the quadrate of its real and imaginary parts. From the above expression, in the case of t, it follows that

 ΔAΔB≥〈AB〉+〈BA〉2−〈A〉〈B〉, ΔA:=+(ΔA)2, ΔB:=+(ΔB)2 (109)

On the another hand, in this case, the commutating relation (Equation (103)) is valid; thus, we can conclude that

 (ΔA)(ΔB)≥〈AB〉−〈A〉〈B〉→0≤〈AB〉−〈A〉〈B〉(ΔA)(ΔB)≤1 (110)

which is the most primitive “uncertainty relation” of classical statistic theory in which the random variables have a joint distribution function. It states the simple fact that the regression coefficient is smaller than one if the random variables are not statistically independent.

In summary, we can provide a speculative interpretation of irreversibility in quantum mechanics, namely, in an irreversible quantum process. The incompatible operators proceed to compatible ones, which are submitted to the laws of classical statistic theory.

## 7. Quantum statistics of the linearly damped oscillator

It was von Neumann [26] who first dealt with the problem of the quantum statistical ensemble. The density operator is the statistical operator of a quantum statistical ensemble. In our case, the statistical ensemble is a set of linearly damped oscillators of several quantum states in contact with a heat bath with temperature T. The density operator is an operator whose eigenvalues are the classical statistical probability of the chosen microstates denoted by pi. If the chosen microstates are denoted by |i, which are eigenstates of a Hermitian operator but not necessarily the eigenstate of a Bohlinian or Hamiltonian, the general density operator is written as

 ρ:=ρδ=∑i≥0ρ|i〉〈i|=∑i≥0pi|i〉〈i|, δ=∑i≥0|i〉〈i| (111)

From this definition, it follows that ρ is Hermitian and normalized

 ρ=ρ+, Trρ=1 (112)

In the Heisenberg picture, the density operator is time independent and is written as

 ρH=∑i≥0pi(0)|i(0)〉〈i(0)| (113)

The ensemble average of an operator in the Heisenberg picture AH(t) is defined as

 〈A〉(t)=Tr(ρHAH(t))=∑i≥0pi(0)〈i(0)|AH(t)|i(0)〉 (114)

Ensemble averages of time rate of change of the displacement and the momentum of the linearly damped oscillator can be evaluated from Equation (51) as follows

 d〈q〉(t)dt:=Tr(ρHq•)=iℏTr(ρH{H,q})=Tr(ρH∂H∂p)=〈∂H∂p〉=〈pm〉=〈p〉(t)md〈p〉(t)dt:=Tr(ρHp•)=iℏTr(ρH{H,p})−ciℏTr(ρH{H,q})==−Tr(ρH∂H∂q)−cTr(ρH∂H∂p)=−〈∂H∂q〉−c〈∂H∂p〉=mω02〈q〉−c〈pm〉==−cm〈p〉(t)−mω02〈q〉(t) (115)

Here, Equation (50) was used. Now, we see that these equations are equivalent to those of the macroscopic Onsagerian equations (Equation (21) or (22)). In the Schrödinger picture, the density operator is time dependent, but the observables of the oscillator are time independent. We define this density operator as

 ρS(t)=∑i≥0pi(t)|i(t)〉〈i(t)| (116)

where we allowed a time-dependent probability pi(t) to microstate |i(t). Also in this picture, the occupation of microstates is not conserved. The ensemble average of an operator AS in the Schrödinger picture is defined as

 〈A〉(t)=Tr(ρH(t)AS)=∑i≥0pi(t)〈i(t)|AS|i(t)〉 (117)

Two ensemble averages of an observable A must be equal, i.e.,

 Tr(ρS(t)AS)=Tr(ρHAH(t)) (118)

from which, in the case of pure unitary dynamics, follows the well-known transformation

 ρS(t)=U+(t)ρH(t)U(t) (119)

where the unitary operator U(t) belongs to time evolution [37]. Since, as we have seen in the case of dissipative processes, this cannot be written by unitary dynamics only, we will use this requirement in a weaker form. We require that the basic ensemble averaged equations of the damped oscillator have the same forms in each picture, i.e.,

 d〈q〉(t)dt=Tr(ρHq•)=Tr(ρ•Sq)=〈p〉(t)md〈p〉(t)dt=Tr(ρHp•)=Tr(ρ•Sp)=−cm〈p〉(t)−mω02〈q〉(t) (120)

According to this requirement, we could give the actual form of the equation of motion for the density operator in the Schrödinger picture. We will see that this equation corresponds to the Liouville–von Neumann equation in the case of dissipative processes. From Equations (113) and (119), it follows that the density operator in the Schrödinger picture could be written by a Hermitian operator in the form

 ρS(t)=∑i≥0pi(t)U+(t)|i(0)〉〈i(0)|U(t) (121)

From the general evolution equation (Equation (34)) of the Hermitian operator, the equation of motion of the density operator in Schrödinger picture could be derived as follows:

 ρS•=dρS(t)dt=∂ρS(t)∂t+[U•+(t)U(t),ρS]∂ρS(t)∂t:=∑i≥0dpi(t)dtU+(t)|i(0)〉〈i(0)|U(t) (122)

where in the case of a damped oscillator, the unitary transformation belongs to the Bohlin operator of Equation (38), i.e.,

 U•+(t)U(t)=−iℏB (123)

Thus, the equation of motion of Schrödinger’s density operator is

 ρS•=dρS(t)dt=∂ρS(t)∂t+iℏ[B,ρS]=∂ρS(t)∂t+iℏ{H,ρS}+iℏ{D,ρS} (124)

Here, similar to the Heisenberg equations (Equation (41)), we introduced the Hamiltonian and the dissipation operator, by means of the commutator {,}=e2βt[,]. To construct a constitutive equation for the local change in the density operator, one must take into account the consequences of the commutation relation (Equation (44)) and the facts (Equations (102) and (103)) by which the incompatibility of the operators show strict fading over time. A consequence of this fading property could be increasing uncertainty in the distinction of the quantum states of the oscillators. On the other hand, in a canonical ensemble, the oscillators weakly interact with each other. By this means, the occupation of the states could change in the ensemble. Thus, we can assume that the occupation of states is not conserved in the time evolution of the ensemble. As a consequence, the statistical ensemble of the oscillators could proceed to a final state in which the classical probabilities of the microstates correspond to a classical thermal equilibrium distribution. Denoted by the density operator ρS in the instantaneous and ρSequ in the equilibrium final state, then the suggested linear constitutive equation is

 ∂ρS∂t=−β(ρS−ρSequ), Tr(ρS)=Tr(ρSequ) (125)

Thus, the final form of the Liouville–von Neumann Equation (124) is

 ρS•=dρS(t)dt=−β(ρS−ρSequ)+iℏ[B,ρS]==−β(ρS−ρSequ)+iℏ{H,ρS}+iℏ{D,ρS} (126)

We will show that this evolution equation guarantees that the equivalence relation (Equation (120)) is fulfilled, the density matrix proceeds to an equilibrium state and that the entropy of the ensemble of the damped oscillator proceeds the maximum value over time, which corresponds to thermal equilibrium. Indeed, the proof of the relations in Equation (120) proceeds as follows

 d〈q〉(t)dt=Tr(ρ•Sq)=−βTr((ρS−ρSequ)q)+iℏTr({H,ρS}q)+iℏTr({D,ρS}q)==−β〈q〉(t)−−β〈q〉equ+iℏTr(ρS{H,q})+iℏTr(ρS{D,q})==−β〈q〉(t)−β〈q〉equ+Tr(ρS∂H∂p)+βTr(ρSq)=−β〈q〉equ+〈p〉(t)md〈p〉(t)dt=Tr(ρ•Sp)=−βTr((ρS−ρSequ)p)+iℏTr({H,ρS}p)+iℏTr({D,ρS}p)==−β〈p〉(t)−β〈p〉equ+iℏTr(ρS{H,p})+iℏTr(ρS{D,p})==−β〈p〉(t)−β〈p〉equ−Tr(ρS∂H∂q)−βTr(ρSp)==−β〈p〉equ−cm〈p〉(t)−mω02〈q〉(t) (127)

where we take into account the cyclic invariance of the trace and the facts in Equation (42).

Now we see that if we choose an ensemble of a damped oscillator in which qequ=0,qequ=0, the required equivalence of the ensemble averages is fulfilled. To prove the increase in entropy, first we introduce quantum entropy. The kBρSlnρS operator is an operator whose eigenvalues are the terms of Shannon entropy kBpi(t)lnpi(t). Thus, the Shannon entropy is the minus trace of that operator [26]

 S(t):=−kB∑i≥0pi(t)lnpi(t)=−kBTr(ρSlnρS) (128)

Here, kB is the Boltzmann constant. According to this definition, the excess entropy of an ensemble, as suggested by Bedeaux and Mazur [38], is given by

 S(t)−Sequ=−kBTr(ln(δρSρSequ−1)ρS), δρS=ρS−ρSequ (129)

The time rate of change of the entropy in that approximation is

 dS(t)dt=−12kBTr((δρSρSequ−1+ρSequ−1δρS)dρSdt) (130)

Entropy production results by substituting the Liouville–von Neumann equation into this equation

 dS(t)dt=−12kBTr((δρSρSequ−1+ρSequ−1δρS)dρSdt)==−12kBTr((δρSρSequ−1+ρSequ−1δρS)(−βδρS+iℏ{B,ρS}))==β2kBTr((δρSρSequ−1+ρSequ−1δρS)δρS)−12kBTr((δρSρSequ−1+ρSequ−1δρS)(iℏ{B,ρS}))==β2kBTr((δρSρSequ−1+ρSequ−1δρS)δρS)=βkBTr(ρSequ−1(δρS)2)≥0 (131)

where the cyclic invariance of the trace and the fact that the Bohlin operator and the ρSequ1 commutator were used. The positivity of entropy production follows from the fact that the matrices of both operators ρSequ1 and (δρS)2 have nonnegative elements only. Thus, the increase in entropy is demonstrated. It can be seen from the above deduction of entropy production that pure unitary dynamics (in the case of an undamped oscillator) is isentropic and that the entropy production is a direct consequence of the unconserved property of the occupation of states. In the thermal equilibrium, from Equation (126), it follows that B and ρSequ commute, i.e.,

 dρS(t)dt=0, ρS=ρSequ, [B,ρSequ]=0 (132)

So the equilibrium density operator ρSequ is a function of the Bohlinian B. At equilibrium, the entropy is at maximum. Now, we maximize S=kBn0pnlnpn under the conditions

 Tr(ρS)=∑n≥0pn=1,〈B〉=Tr(ρSB)=∑n≥0pnBn=const,Bn=〈n|B|n〉=ℏωm(12+n) (133)

where we use Equation (54). The necessary condition of that maximum is

 −kB∑n≥0δpn(lnpn+1)=0 (134)

where the variations δpn are restricted by the conditions

 ∑n≥0δpn=0, ∑n≥0δpnBn=0 (135)

Applying the method of Lagrange multipliers, we get

 ∑n≥0δpn[(lnpn+1)+βBn+γ]=0 (136)

So

 pn=e−βBn−γ−1 (137)

From the first equation of the conditions (133), the normalized version of the probability distribution is obtained

 pn=e−βBn∑i≥0e−βBi (138)

Choosing β=1kBT as usual, then we get Gibbs’ canonical distribution for the occupation probabilities

 pn=e−βBn∑i≥0e−βBi (139)

Introducing the partition function by definition

 Z:=Tr(e−βB)=∑i≥0e−βBi (140)

then we get the equilibrium density operator

 ρSequ=e−βBZ (141)

Thus, the equilibrium ensemble average of an operator AS can be written as

 〈A〉=Tr(ρSequA)=Tr(e−βBA)Z=∑n≥0〈n|A|n〉e−βEn∑i≥0e−βBi (142)

In particular, for the ensemble average of the Bohlinian, this is

 〈B〉=∑n≥0Bne−βEn∑n≥0e−βBn=−∂lnZ∂β=ℏωm(12+e−ℏωmkBT1−e−ℏωmkBT)=ℏωm2cothℏωmkBT (143)

Introducing the free energy by definition F=kBTZ, then in terms of free energy, the Gibbs distribution is written as usual

 pn=eF−BnkBT (144)

Substituting this into the definition equation of entropy (128), then we get

 S=〈B〉−FT→F=〈B〉−TS (145)

Thus, the ensemble average of the Bohlinian is the equilibrium internal energy. It is evident that the actual choice of the angular frequency ωm in the Bohlinian is a convention. It depends on the normalization of Bohlin’s constant (Equation (26)). What is the correct angular frequency? It seems from the physical aspect that the correct choice is that the angular frequency is ω0. In this case, Bohlin’s constant of motion corresponds to the maximum free energy of the linearly damped oscillator measured at time t=0, so it is the exergy of the linearly damped oscillator.

## 8. Wave equation of the linearly damped oscillator

From the above-presented theory, we can conclude that an ensemble from a pure state always proceeds to a mixed state a consequence of irreversibility. Thus, it is impossible to describe the evolution of the pure state of a damped oscillator in the Schrödinger picture. Consequently, it is impossible to construct a linear Schrödinger equation in which the position and the momentum operator are time independent.

However, when the operators are time dependent, the model could show similarities to Schrodinger’s interpretation, which we show below.

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrödinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin’s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form

 iℏd|Ψ(t)〉dt=(H−D)|Ψ〉 (146)

where the Hamiltonian H and the dissipative term D has the same mathematical form as in the Bohlinian, i.e.,

 H(p∧,q∧)=p∧22m+mω022q∧2, D(p∧,q∧)=β2(p∧q∧+q∧q∧), (147)

The operators p,q in this picture are time dependent and satisfy the classical fundamental commutators

 [p,∧q∧]=ℏiδ,[p∧,p∧]=[q∧,q∧]=0 (148)

The time derivative in Equation (146) is “material” (in the sense of continuum mechanics) because of the time dependence of the observable q^. To construct a wave equation, first we rewrite this abstract wave equation in the eigenbase |q of the position operator. To do this, consider the following one-dimensional eigenvalue problem

 q∧|q'〉=q|q'〉 (149)

which could constitute a continuous spectrum. Thus, we must write the orthonormality condition and completeness relation for the eigenvectors as follows

 〈q|q'〉=δ(q−q'), ∫|q'〉〈q'|dq'=δ (150)

where δ(qq') is Dirac’s distribution and δ is the unity operator. The wave function is the probability amplitude that a position measurement on the damped oscillator in state |Ψ(t) will yield an eigenvalue q, mathematically

 Ψ(q,t)=〈q|Ψ(t)〉 (151)

Inserting Equation (150) of the unity operator in the abstract wave Equation (146) and projecting from the left with q|, then we obtain

 iℏ〈q|a|Ψ(t)〉dt=d〈q|Ψ(t)〉dt=dΨ(q,t)dt=〈q|H(p∧,q∧)∫|q'〉〈q'|Ψ(t)〉dq'+−〈q|D(p∧,q∧)∫|q'〉〈q'|Ψ(t)〉dq' (152)

We chose the differential operator representation for the time-dependent operators in the eigenbase |q of the position operator in the form

 p∧=ℏi∂∂eβtq, q∧=qeβt (153)

which resulted in the following wave equation

 iℏdΨ(eβtq,t)dt=(H−D)Ψ(eβtq,t),H=−ℏ22md2d(eβtq)2+mω022(eβtq)2, D=−iℏ(β2+β(eβtq)dd(eβtq)) (154)

which is a linear partial differential equation. To construct the eigenvalue problem that belongs to this wave equation, we chose the wave function as

 Ψn=e(Eniℏ+β2)tΦn(eβtq) (155)

With this wave function, the eigenvalue equation

 −ℏ22md2Φnd(eβtq)2+mω022(eβtq)2Φn=EnΦn (156)

is obtained from the wave equation because the equation

 −D(eβ2tΦn)=iℏβ(12+(qeβt)dd(qeβt))(eβ2tΦn)=iℏdeβ2tΦndt==iℏβ2eβ2tΦn+iℏeβ2t∂Φn∂(qeβt)qdeβtdt (157)

is satisfied identically for every eigenfunction Φn. By introducing a new variable into the eigenvalue Equation (156) defined as ξ:=mω0eβtq, then we get

 d2Φndξ2+(2Enℏω0−ξ2)Φn=0 (158)

We chose the eigenfunction Φn in the form

 Φn=e−ξ22ψn (159)

then we obtained the differential equation

 d2ψndξ2−2ξψn+(2Enℏω0−1)ψn=0 (160)

which has a solution in terms of Hermitian polynoms if the

 2Enℏω0−1=2n→En=(n+12)ℏω0, n=1,2,3... (161)

relation is fulfilled. With these, the solution of the wave equation is obtained as follows:

 Ψn(ξ,t)=e(Eniℏ+β2)tΦn(eβtq)∝e−i(n+12)ω0eβ2te−ξ22Hn(ξ), ξ:=mω0ℏeβtq (162)

from which the probability density function of the oscillator has the form

 |Ψn(q)|2=12nn!12π(ℏ2mω0e−βt)e−q22(ℏ2mω0e−βt)2[Hn2(eβtmω0ℏq)] (163)

which is exactly identical to Equation (87) resulting from the Heisenberg picture of the damped oscillator and the equation given by Kim and Page [33] using another theory. Due to this correspondence, the quantum decoherence of linearly damped oscillators could be described in the same way as done in the publication by Kim et al. [40].

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