Quasi-coherent states for damped and forced harmonic oscillatorMustafa Dernek and Nuri Ünal Citation: J. Math. Phys. 54, 092102 (2013); doi: 10.1063/1.4819261 View online: http://dx.doi.org/10.1063/1.4819261 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i9 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS 54, 092102 (2013)

Quasi-coherent states for damped and forcedharmonic oscillator

Mustafa Dernek and Nuri UnalDepartment of Physics, Akdeniz University, Antalya 07058, Turkey

(Received 30 November 2012; accepted 12 August 2013; published online 4 September 2013)

In this study, first we construct the quasi-coherent states for a damped and forcedharmonic oscillator and discuss the transition of the system from the damped oscil-lations to forced steady state oscillations. Second, we generalize the Caldirola-KanaiHamiltonian into the new systems such as the frequency and the mass parameters aretime dependent and discuss three examples of them. C© 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4819261]

I. INTRODUCTION

The damped harmonic oscillator as the simplest example of the systems exhibiting dissipationof energy has found a lot of applications in different areas of physics.1–4 In the usual approach,the dissipative systems are discussed as open systems.5 In this technique, the density function ofthe system is investigated and the transition amplitudes of the physical quantities are evaluated byusing it. However, there is always interest in representing the damped harmonic oscillator by a timedependent phenomenological Hamiltonian in a phase space.6–8 The motivation of these interests isto derive a complementary approach to the same problem and to obtain the same physical resultswith the open system procedure.

In phenomenological approach, the classical equations of motion for the damped harmonicoscillator were derived from a Lagrangian by considering an integration factor, ekt, or from aHamiltonian with time varying a mass parameter (Caldirola-Kanai Hamiltonian), M0ekt. In order toderive the classical equations of motion for a damped harmonic oscillator, an alternative technique isto introduce additional degrees of freedom in Hamiltonian, which represents the external system.9–11

There are continuous interest in the quantum harmonic oscillator with time dependent parameters,the mass, and the frequency, since it is an exactly soluble model.

Furthermore, the forced harmonic oscillator is an open system and was first considered byFeynman,12, 13 and later, by different authors14–16 in order for formulation of interaction for theharmonic oscillator with a classical source. The harmonic oscillator with time dependent parametersmay be considered as a system having quadratic couplings with the external time dependent sources.In contrast to it, the forced harmonic oscillator may be considered as a system which has a linearcoupling with the external time dependent source, F(t).

Lewis and Risenfeld used the invariants of the system to discuss the quantum harmonic oscillatorwith the time dependent parameters17 and the properties of the system have been discussed bydifferent authors using these invariants18–29 and in Refs. 11 and 30, the authors conclude that thequantum state functions are not regular at the limit t → ∞. In a previous study, one of us derived thequasi-coherent states for the Caldirola-Kanai Hamiltonian by using new time dependent holomorphiccoordinates.31 Recently, log-periodic oscillators were proposed32 and their wave functions have beendiscussed.33

The aim of this study is to obtain the quasi-coherent states and the quasi-stationary states for theforced Caldirola-Kanai Hamiltonian.We show that the quasi-coherent states for the Caldirola-KanaiHamiltonian are not singular at the limit t → ∞ and discuss the transition of the system from thedamped oscillations to the steady state oscillations by an effect of external driving force.

0022-2488/2013/54(9)/092102/10/$30.00 C©2013 AIP Publishing LLC54, 092102-1

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092102-2 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

In Sec. II, we find the quasi-coherent states for the forced Caldirola-Kanai Hamiltonian, andderive the expectation values of the position, momentum, current, energy, and their uncertainties. Thediscrete quasi-stationary number states are also derived. When t → ∞, in the configuration space, theprobability distribution becomes a delta function and the center of the probability function followsthe external driving force by a phase difference. In Sec. III, we introduce a conformal time for theharmonic oscillators with time dependent frequency and mass and also discuss the generalizationof the Caldiroli-Kanai Hamiltonians to the new systems. The Schrodinger equation is written inthe conformal time. Three examples of the generalized Caldirola-Kanai Hamiltonian are discussed.Section IV is our conclusions.

II. FORCED CALDIROLA-KANAI HAMILTONIAN

A. Quasi-coherent states

For the forced Caldirola-Kanai system, the Hamiltonian is given as

H0 (t) = p2x

2M0ekt+ 1

2M0ektω2

0x2 − F (t) ekt x,

where M0 is the initial mass of a particle and ω0 is the constant natural oscillation frequency of thesystem, k is an attenuation factor, and F(t) is the external driving force. In dimensionless units, werewrite the Hamiltonian as

H0 (τ ) = P2

2eγ τ+ 1

2eγ τ Q2 − f (τ ) eγ τ Q. (1)

Here, τ = ω0t, γ = kω0

, Q = ( M0ω0�

)1/2x , P = ( M0ω0

�

)−1/2px , and f (τ ) = (

M0ω30�)−1/2

F (t) . Inthe configuration space, (Q), we write the Schrodinger equation as

i∂

∂τ� (Q, τ ) = 1

2

(−e−γ τ ∂2

∂ Q2+ eγ τ Q2 − 2 f (τ )eγ τ Q

)� (Q, τ ) . (2)

We search for a solution of Eq. (2) in the form of a Gaussian wave packet in the following form:

�λ(τ ) (Q, τ ) = N (τ )e− a(τ )2 Q2+λ(τ )Q, (3)

where a(τ ) and λ(τ ) are time dependent parameters. To determine the parameters a(τ ) and λ(τ ),we substitute the wave function,�λ(τ ), into Eq. (2) and find that a(τ ) and λ(τ ) satisfy the followingequations:

−ida(τ )

dτ= −e−γ τ a2(τ ) + eγ τ , (4)

idλ(τ )

dτ= e−γ τ a(τ )λ(τ ) − f (τ )eγ τ . (5)

Here, a(τ ) satisfies the Riccati equation. We define a new time dependent variable, ζ , as

ζ = −ie−γ τ a(τ ).

So, ζ satisfies the following equation:

dζ (τ )

dτ= ζ 2 − γ ζ + 1. (6)

The time independent solutions of Eq. (6) are found as

ζ (τ ) = γ

2± iω1,

where ω1 = (1 − γ 2/4)1/2 for weak damping case. Then, for a regular solution when |Q| → ∞, wechoose a(τ ) as

a(τ ) = eγ τ

(iγ

2+ ω1

). (7)

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092102-3 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

For the strong damping case, γ

2 > 1, ω1 becomes i(γ 2/4 − 1)1/2. In this case, a(τ ) becomes purelyimaginary and �λ(τ )(Q, τ ) will not be square integrable and there will be no oscillatory solutionsfor the classical system.

In order to find λ(τ ), in a weak damping case, we insert a(τ ) in Eq. (7) to Eq. (5) and solveλ(τ ). The result is given as

λ(τ ) = λ (0) e( γ

2 −iω1)τ + ie( γ

2 −iω1)τ∫ τ

f (τ ′)e( γ

2 +iω1)τ ′dτ ′. (8)

We investigate λ(τ ) for two possible cases of driving force, f(τ ):

Case 1. The harmonic force, f (τ ) = ∫dω2π

f0 (ω) cos ωτ.

In this case, we find λ(τ ) as

λ(τ ) = [λ (0) − g (τ )] e( γ

2 −iω1)τ + g (τ ) eγ τ , (9)

where g(τ ) is

g (τ ) =∫

dω

4πf0 (ω)

(eiωτ

ω1 + ω − i γ

2

+ e−iωτ

ω1 − ω − i γ

2

).

In Eq. (9), the second term is dominant at large times, (γ τ > 1), and �λ(τ ) represents the steady statesolutions of the system. The poles of the integral in g(τ ) agree with the spectrum given in Ref. 34.

In the limit f0(ω) = 2πδ(ω), we obtain the constant force case with f0 =1 and then, g(τ ) becomes(i γ

2 + ω1)−1

.

Case 2. δ pulse, f(τ ) = f0δ(τ − τ 0):

λ(τ ) =⎧⎨⎩λ (0) e( γ

2 −iω1)τ , for τ < τ0,(λ (0) + i f0e( γ

2 +iω1)τ0

)e( γ

2 −iω1)τ , for τ > τ0.(10)

Equation (10) shows that if there is a δ function discontinuity in driving force, f(τ ), then, theeigenvalue of a wave packet, λ(τ ), and the center of probability density, <Q >, are discontinues.They change suddenly by the effect of an external δ force. However, the width of the wave packet isalso continuous.

B. Expectation values and uncertainty relations for quasi-coherent states

For the wave packet, �λ(τ )(Q, τ ), the normalized probability density, Pλ(Q, τ ), is given as

Pλ(Q, τ ) =(

(a(τ ) + a∗(τ ))

2π

)1/2

e− a(τ )+a∗(τ )

2

(Q− λ(τ )+λ∗ (τ )

a(τ )+a∗ (τ )

)2

. (11)

So, by using Pλ(Q, τ ), we obtain the following expectation values:

〈Q〉λ = λ(τ ) + λ∗(τ )

a(τ ) + a∗(τ ),

⟨Q2

⟩λ

= 1

a(τ ) + a∗(τ )+ 〈Q〉2

λ . (12)

Thus, the uncertainty in position, (�Q)2λ , is given as

(�Q)2λ = 1

a(τ ) + a∗(τ )= e−γ τ 1

2ω1.

Similarly,

〈P〉λ = −i [−a(τ ) 〈Q〉 + λ(τ )] ,⟨P2

⟩λ

= |a(τ )|2 (�Q)2λ + 〈P〉2

λ , (13)

and

(�P)2λ = |a(τ )|2 (�Q)2

λ = |a(τ )|2a(τ ) + a∗(τ )

= eγ τ 1

2ω1.

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092102-4 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

Then, the product of uncertainties, (�Q�P)λ, becomes

(�Q�P)λ = |a(τ )|a(τ ) + a∗(τ )

= 1

2ω1. (14)

Equation (14) shows that (�Q�P)λ is constant for the quasi-coherent states of Caldirola-KanaiHamiltonian and is always larger than the minimum value of the product of uncertainties for thecoherent states of a harmonic oscillator, 1/2.

The expectation value of Hamiltonian between the quasi-coherent states is given as

〈H0〉λ (15)

= 1

2ω21

(Re λ(τ )

eγ

2 τ

)2

+ 1

2

(Im λ(τ )

eγ

2 τ− γ

2ω1

Re λ(τ )

eγ

2 τ

)2

− 1

ω1

(Re λ(τ )

eγ

2 τ

)eγ τ/2 f (τ ) + 1

2ω1.

Here, the last term comes from the uncertainty relations between Q and P operators, In the absenceof a driving force, the expectation value of the Hamiltonian, 〈H0〉λ, does not decrease but oscillatesin time. If we define the velocity of the particle as

d Q

dτ= P

M(τ ),

then, the expectation value of the current density, 〈J〉λ, is given as

〈J 〉λ = 〈P〉λeγ τ

= e− γ

2 τ

[(Im λ(τ )

eγ

2 τ

)+ γ

2ω1

(Re λ(τ )

eγ

2 τ

)]. (16)

Here, in the absence of a driving force, 〈J〉λ decreases exponentially in time, as expected. 〈J〉λcorresponds to the mechanical velocity,

( d Qdτ

)M

.The following observable quantity is defined as the mechanical energy of a damped harmonic

oscillator, EM, by eliminating the integration factor e− γ τ in 〈H0〉λ:

EM = e−γ τ

[1

2ω21

(Re λ(τ )

eγ

2 τ

)2

+ 1

2

(Im λ(τ )

eγ

2 τ− γ

2ω1

Re λ(τ )

eγ

2 τ

)2

+ 1

2ω1

]− 1

ω1

(Re λ(τ )

eγ τ

)f (τ ) .

For γ τ > 1, the probability density of the quasi-coherent states becomes

Pλ(Q, τ ) → δ

[Q − g(τ ) + g∗(τ )

2ω1

],

as in Ref. 14. We obtain the following expectation values:

〈Q〉λ → g(τ ) + g∗(τ )

2ω1, (�Q)2

λ = e−γ τ 1

2ω1→ 0,

〈J 〉λ = 〈P〉λeγ τ

→[

g(τ ) − g∗(τ )

2+ γ

2ω1

(g(τ ) + g∗(τ )

2

)],

and the value of mechanical energy, EM, is given as

EM

� 1

2ω21

(g(τ ) + g∗(τ )

2

)2

+ 1

2

(g(τ ) − g∗(τ )

2− γ

2ω1

g(τ )+g∗(τ )

2

)2

− 1

ω1

(g(τ )+g∗(τ )

2

)f (τ ) .

C. Quasi-stationary number states

The normalized wave packets are given as

�λ(τ ) (Q, τ ) =(

[a(τ ) + a∗(τ )]

2πe− (λ(τ )+λ∗(τ ))2

a(τ )+a∗(τ )

)1/4

e− a(τ )2 Q2+λ(τ )Q .

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092102-5 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

In the wave packet, �λ, we expand the term, eλ(τ )Q, into the power series of Q as

eλ(τ )Q = e[λ(τ )]2

2[a(τ )+a∗(τ )]

∞∑n=0

Hn

(√a(τ )+a∗(τ )

2 Q

)√

2nn!

(λ(τ )√

a(τ )+a∗(τ )

)n

√n!

.

The time evolution of discrete number states is given as by an evolution factor,(λ(τ )√

(a(τ ) + a∗(τ ))

)n

=(

1√2ω1

λ(τ )

eγ

2 τ

)n

.

In the absence of driving force, the discrete state evolve by the evolution factor

1√n!

(λ(τ )√2ω1e

γ

2 τ

)n

� 1√n!

(λ (0)√

2ω1

)n

e−inω1τ . (17)

Using a different representation of wave functions, it could be shown that the evolution factor givenby Eq. (17) is equal to the result given in Ref. 11. Then, the orthonormal number states �n (Q, τ ) aregiven as

�n (Q, τ ) =(

ω1eγ

2 τ

π

) 14

e−inω1τ e− ( iγ

2 +ω1)2

(e

γ2 τ Q

)2 Hn(√

ω1eγ

2 τ Q)√2nn!

. (18)

For the number state,�n, the probability, Pn(Q, τ ), is given as

Pn(Q, τ ) =(

ω1eγ

2 τ

π

)1/21

2nn!e−ω1

(e

γ2 τ Q

)2 ∣∣∣Hn(√

ω1eγ

2 τ Q)∣∣∣2

. (19)

For the model, the number states are similar to the stationary states of the harmonic oscillator;however, the coordinates of the configuration space is multiplied by an expansion factor, e

γ

2 τ .

Between the discrete number states, �n(Q, τ ), the expectation values of the Hamiltonian, H0, andthe product of uncertainties are given, respectively, as

〈H0〉n =(

n + 1

2

)1

ω1(20)

and

(�Q�P)n =(

n + 1

2

)1

ω1. (21)

For γ τ > 1, �n(Q, τ ) becomes

�n (Q, τ ) =(

ω1eγ

2 τ

π

) 14(

g (τ ) eγ

2 τ e−iω1τ

√2ω1

)n

e− ( iγ

2 +ω1)2

(e

γ2 τ Q

)2 Hn(√

ω1eγ

2 τ Q)√2nn!

, (22)

and the probability Pn(Q, τ ) is given as

Pn(Q, τ ) =(

ω1eγ

2 τ

π

)1/21

2nn!

(|g(τ )| e

γ

2 τ

√2ω1

)2n

e−ω1

(e

γ2 τ Q

)2 ∣∣∣Hn(√

ω1eγ

2 τ Q)∣∣∣2

. (23)

Equations (18) and (22) show that configuration space dependence of �n(Q, τ ) are the same fordamped and forced harmonic oscillators. The difference comes from the time evolutions of thesefunctions.

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092102-6 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

III. GENERALIZATION OF CALDIROLA-KANAI SYSTEM

The Lagrangian of the harmonic oscillator with a time dependent mass, M(t), and frequency,ω(t), is given as

L(x, px , t) = pxdx

dt− 1

2M(t)

(p2

x + [M(t)ω(t)]2 x2).

We write the action functional of the system as

A =∫ [

px dx − dt

2M(t)

(p2

x + [M(t)ω(t)]2 x2)] . (24)

If we introduce the conformal time, φ, by the following relation:

φ(t) =∫ t

ω(t ′)dt ′, (25)

and do the following time dependent coordinate transformation:

q =√

M(t)ω(t)x, and p = px√M(t)ω(t)

, (26)

then, we write the action as

A =∫ {

pdq − dφ

2

[(p2 + q2 + j (φ) (pq + qp)

)]}, (27)

where the attenuation factor, j(φ), is given as

j (φ) = d

2dφln [M(t)ω(t)] . (28)

In terms of time dependent coordinates, q, and the conjugate momentum, p, the Hamiltonian is givenas

H = 1

2

[p2 + q2 + j (φ) (pq + qp)

]. (29)

If j(φ) is a constant, then, H corresponds to the generalizations of Caldirola-Kanai Hamiltonian.In these cases, Eq. (28) becomes

d

dtln [M(t)ω(t)] = γω (t) . (30)

The integration of Eq. (30) gives

M(t)ω(t) = M(t0)ω(t0) exp

[γ

∫ t

t0

ω(t ′) dt ′

], (31)

In these cases, the system satisfies the following Schrodinger equation in the conformal time, φ:

i∂

∂φ� (q, φ) = 1

2

[− ∂2

∂q2+ q2 − iγ

2

(q

∂

∂q+ ∂

∂qq

)]� (q, φ) . (32)

Here, we use the units � = 1 by replacing q by q/√

�. For these generalizations of the Caldirola-Kanai Hamiltonian, one could construct the quasi-coherent states by using the method presented inSec. II.

A. Applications

Five examples for a harmonic oscillator with the time dependent parameters, M(t) and ω(t),were proposed in Ref. 32. For three of these examples, called as log-periodic oscillators, the numberstates have been evaluated by using the Lewis-Riesenfeld invariants Ref. 33. We examine them inthis section again and construct the quasi-coherent states for them. For all of them, for t > t0, the

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092102-7 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

angular frequency is given as ω(t) = ω0t0t and then, by using Eq. (25), we find the conformal time

as φ(t) = ω0t0 ln tt0

.

Example A: If mass is given by M (t) = M0( t0

t

)−1, then M(t)ω(t) becomes

M(t)ω(t) = M0ω0 = const. (33)

So, the attenuation factor is calculated from Eq. (28), asγA

2= 0, (34)

and the time independent Hamiltonian in Eq. (29) becomes HA = 12

[p2 + q2

]. Thus, it corresponds

to a harmonic oscillator with an angular frequency, ω1A = 1.

By using Eqs. (12) and (13), we find the expectation value of position and momentum, respec-tively, as

〈q〉λ = |λ (t0)| cos [φ(t) − φ0] , 〈px 〉λ = − |λ (t0)| sin [φ(t) − φ0] . (35)

Here, we choose initial value of λ(t) as

λ (t0) = |λ (t0)| eiφ0 .

From Eq. (14), the uncertainty relation for coherent states, (�q�p)λ, is given as

(�q�p)λ = 1

2. (36)

We find the number states, �n, from Eq. (18) as

�n (q, φ) = e−inφ(t)

(1

π22n (n!)2

) 14

e− 12 q2

Hn(q). (37)

From Eq. (21), the uncertainty relation for number states is given as

(�q�p)n =(

n + 1

2

). (38)

Example B: If mass is given by M(t) = M0, then M(t)ω(t) becomes,

M(t)ω(t) = M0ω0t0t. (39)

We obtain the attenuation factor from Eq. (28) as

γB

2= − 1

2ω0t0. (40)

Therefore, the system has anti-damping. From Eq. (29), we see that these system has the followingtime independent Hamiltonian:

HB = 1

2

[p2 + q2 + γB

2(pq + qp)

]. (41)

In this example, for weak anti-damping case, ω1B is given as the following:

ω1B =(

1 − γ 2B

4

)1/2

. (42)

We obtain, the expectation value of position, by using Eq. (12) as

〈q〉λ = |λ (t0)|ω1B

cos ω1B [φ(t) − φ0] . (43)

Similarly, we find the expectation value of momentum from Eq. (13) as

〈p〉λ = − |λ (t0)| sin ω1B [φ(t) − φ0] − |λ (t0)| γ

2ω1Bcos ω1B [φ(t) − φ0] . (44)

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092102-8 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

From Eq. (14), the product of uncertainties, (�q�p)λ, is obtained as

(�q�p)λ = 1

2ω1B. (45)

We use Eq. (18) and get the number states as

�n (q, φ) =(

ω1B

π22n (n!)2

) 14

e−inω1Bφ(t)e− 12 (ω1B+i γB

2 )q2Hn

(√ω1Bq

). (46)

Finally by using Eq. (21), we find the uncertainty relation for number states as

(�q�p)n =(n + 1

2

)ω1B

. (47)

In this case, the roles of position and momentum are changed in phase space if we compare thedamped case.

Example C: If mass is given by M (t) = M0( t0

t

)−2, then, M(t)ω(t) product becomes

M(t)ω(t) = M0ω0t

t0. (48)

We calculate the attenuation factor from Eq. (28) as

γC

2= 1

2ω0t0, (49)

and the time independent Hamiltonian is found from Eq. (29) as

HC = 1

2

[p2 + q2 + γC

2(pq + qp)

]. (50)

In this example, for the weak damping case, ω1, is given as the following:

ω1C =(

1 − γ 2C

4

) 12

. (51)

We obtain the expectation value of position by using Eq. (12) as

〈q〉λ = |λ (t0)|ω1C

cos ω1C [φ(t) − φ0] . (52)

Similarly, we find the expectation value of momentum from Eq. (13) as

〈p〉λ = − |λ (t0)| sin ω1C [φ(t) − φ0] − |λ (t0)| γ

2ω1Ccos ω1C [φ(t) − φ0] . (53)

The product of uncertainties, (�q�p)λ, is obtained from Eq. (14) as

(�q�p)λ = 1

2ω1C. (54)

We use Eq. (18) and get the number states as

�n (q, φ) =(

ω1C

π22n (n!)2

) 14

einω1Cφ(t)e− 12 (ω1C+i γC

2 )q2Hn

(√ω1Cq

). (55)

Finally, by using Eq. (21), we find the uncertainty relation for number states as

(�q�p)n =(n + 1

2

)ω1C

. (56)

For these tree examples, the classical solutions in Refs. 32 and 33 are found by choosing

|λ (t0)| = 1 and tan ω1iφ0 = γ

2ω1ii = A,B,C.

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092102-9 M. Dernek and N. Unal J. Math. Phys. 54, 092102 (2013)

Here, the first example, (A) corresponds to a harmonic oscillator in conformal time, but the lasttwo cases, (B) and (C), are the examples of generalized Caldirola-Kanai Hamiltonian with theanti-damping case and the damping case, respectively.

IV. CONCLUSION

In this study, we have constructed the quasi-coherent states for the forced Caldirola-KanaiHamiltonian by searching a Gaussian wave packet with a time dependent width and center parame-ters, a(τ ) and λ(τ ), respectively. The parameter, a(τ ), satisfies a complex Riccati equation. Using thesolutions for a(τ ) and λ(τ ), we obtained �λ(τ ) (Q, τ ) representing square integrable quasi-coherentstates. We evaluated the expectation values for the following quantities: the coordinate, momentum,current, the Caldirola-Kanai Hamiltonian, and the mechanical energy of the system.

In the absence a driving force, the expectation value for Caldirola-Kanai Hamiltonian,〈H0〉λ ,

is oscillating with a frequency of 2ω1. The expectation value of the particle’s position follows theclassical trajectory of the damped harmonic oscillator, x(0)e− γ

2 τ cos (ω1τ + ϕ0) and falls to thecenter. The mechanical energy, EM, also oscillates with the frequency 2ω1 and decays exponentiallyas e− γ τ . After having (�Q)λ and (�P)λ, we also showed that, in contrary to the comments inRef. 11, when τ → ∞, there are no inconsistencies in them. If we consider a mechanical momentumoperator as pM = e−γ τ P, similar to mechanical energy, then, there are inconsistencies in uncertaintyrelations between (�Q)λ and (�pM)λ.

In the presence of driving force, when γ τ > 1, we showed that the probability density goes todelta function and the expectation value of the particle’s position follows the driving force. In thiscase, we also derived the current and the mechanical energy.

In Sec. III, we discussed the possible generalizations of the Caldirola-Kanai system in toharmonic oscillators with time dependent parameters: the mass and angular frequency and showedthat three examples in Refs. 32 and 33 have the Caldirola-Kanai type Hamiltonian in a conformaltime proportional to ln t.

The other problems related to the time dependent harmonic oscillator which cannot be reducedto the Caldirola-Kanai system or could be solved by Lewis-Riesenfeld invariants method may besolved by the technique proposed here. Furthermore, the transition amplitudes which derived hereare in consistent with open system approach.

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