PRAMANA

c Indian Academy of Sciences Vol. 61, No. 1

— journal of July 2003
physics pp. 7–20

Thermal state of the general time-dependent harmonic

oscillator
JEONG-RYEOL CHOI
Department of Electronic Physics, College of Natural Science, Hankuk University of Foreign
Studies, Yongin 449-791, Korea
Email: choiardor@hanmail.net

MS received 25 July 2002; revised 15 January 2003; accepted 21 February 2003

Abstract. Taking advantage of dynamical invariant operator, we derived quantum mechanical solu-
tion of general time-dependent harmonic oscillator. The uncertainty relation of the system is always
larger than ~=2 not only in number but also in the thermal state as expected. We used the diagonal
elements of density operator satisfying Leouville–von Neumann equation to calculate various ex-
pectation values in the thermal state. We applied our theory to a special case which is the forced
Caldirola–Kanai oscillator.

Keywords. Time-dependent harmonic oscillator; thermal state; density operator.

PACS Nos 03.65.-w; 05.30.-d; 05.70.-a

1. Introduction

Vibration may be one of the most dominant physical aspect that we come upon in everyday
life [1]. For small oscillation, it can be approximated to the motion of harmonic oscillator.
Harmonic oscillator that has time-dependent mass or frequency may be a good example of
time-dependent Hamiltonian systems. Although a large number of dynamical systems have
been investigated using approximation and perturbation method in the literature [2,3], we
confine our concern to the exact quantum solution of the time-dependent system. There
are about three kinds of methods to solve the quantum solution of time-dependent har-
monic oscillator. These are propagator method [4–6], unitary transformation method [7–9]
and invariant operator method [10–16]. We will use invariant operator method and uni-
tary transformation method together to evolve the quantum theory and investigate thermal
state of the general time-dependent harmonic oscillator. The time-dependent harmonic
oscillator has several applications such as electrical behavior of LC-circuit that has time-
dependent parameter [17] and/or RLC-circuit [18], path-integral formulation of real-time
finite-temperature field theory [19–21], dissipative quantum tunnelling effect in macro-
scopic system [22–25] and quantum motion of an ion in a Paul trap [7,26,27].
When a system interacts with environment, its coupling parameters may explicitly de-
pend on time. Even if the system is closed so that it is conserved as a whole, its subsystem

7
 Jeong-Ryeol Choi

may implicitly depend on time through interaction with the remnant of the system. The
main purpose of this paper is to evolve the thermal state of the general time-dependent
harmonic oscillator. The Liouville–von Neumann equation [28,29] for non-equilibrium
dynamics can be applicable to both time-dependent harmonic and unharmonic oscillator.
The density operator of the system can be obtained using the wave function satisfying
Schrödinger equation and can be used to derive various expectation values of variables in
the thermal state.
In x2, we investigate quantum mechanical solution of the general time-dependent har-
monic oscillator. The thermal state of the system is discussed in x3 on the basis of
Liouville–von Neumann approach. In x4, we will apply our theory for a special case which
is the forced Caldirola–Kanai oscillator. Finally, x5 summarizes this paper and concludes
about some physical results of the system.

2. Hamiltonian, invariant operator and wave function

The Hamiltonian of general time-dependent harmonic oscillator can be written as

Ĥ (x̂; p̂; t ) = A(t ) p̂2 + B(t )(x̂ p̂ + p̂x̂) + C(t ) p̂ + D 2 (t )x̂2 + D1 (t )x̂ + D0 (t ); (1)

where A(t ) C(t ) and Di (t ) (i = 0; 1; 2) are time-dependent coefficients. These coefficients

are real and differentiable with respect to t and note that A(t ) 6= 0. The corresponding
equation of motion can be derived from Hamilton’s equation of motion as
 
Ȧ ˙ 2ȦB ȦC
x̂¨ x̂ + 2Ḃ 4B2 + 4AD2 x̂ + Ċ 2BC + 2AD1 = 0: (2)
A A A

The introduction of invariant operator may save the labor of finding quantum mechanical
solution of the system. We let the trial invariant operator as the form

Iˆ(t ) = α1 (t )[ p̂ p p(t )]2 + α2 (t )f[x̂ x p (t )][ p̂ p p(t )]

+[ p̂ p p (t )][x̂ x p (t )]g + α3(t )[x̂ x p(t )]2 ; (3)

where α1 (t ) α3 (t ) are time-variable functions which should be determined afterwards

and x p (t ) is a particular solution of the equation of motion in x̂ space, (eq. (2)) and p p (t ) is
the corresponding particular solution of the equation of motion in pˆ space. We can choose
the dimension of Iˆ(t ) the same as that of the Hamiltonian. By virtue of its definition, the
invariant operator must satisfy the following relation:

dIˆ(t ) ∂ Iˆ(t ) 1 ˆ
= + [I (t ); Ĥ ] = 0: (4)
dt ∂t i~
Substituting eqs (1) and (3) in the above equation gives

α1 (t ) = c1 ρ12 (t ) + c2 ρ1 (t )ρ2 (t ) + c3ρ22 (t ); (5)

α2 (t ) =
1
4A
f4[c1ρ12 (t ) + c2ρ1(t )ρ2 (t ) + c3ρ22 (t )]B
[2c1 ρ1 (t )ρ̇1 (t ) + c2 ρ̇1 (t )ρ2 (t ) + c2 ρ̇2 (t )ρ1 (t ) + 2c3 ρ2 (t )ρ̇2 (t )]g; (6)

8 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator

α3 (t ) = f [c ρ̇ 2 (t ) + c2ρ̇1(t )ρ̇2 (t ) + c3ρ̇22 (t )]

1 1
2A2 2 1 1
B[2c1 ρ1 (t )ρ̇1 (t ) + c2ρ̇1 (t )ρ2 (t ) + c2ρ̇2 (t )ρ1 (t ) + 2c3ρ2 (t )ρ̇2 (t )]
+2B [c1 ρ1 (t ) + c2 ρ1 (t )ρ2 (t ) + c3 ρ2 (t )]
2 2 2
g; (7)

where c1 –c3 are constants and ρ̈1;2 (t ) are two independent solutions of the following dif-
ferential equation
 
Ȧ 2ȦB
ρ̈1;2 (t ) ρ̇1;2 (t ) + 2Ḃ 4B 2
+ 4AD2 ρ1;2 (t ) = 0: (8)
A A

To simplify the invariant operator, we introduce the unitary operator defined as

Ût = Û
00Û 0Û ; (9)
   
i i
Û = exp x p p̂ exp p p x̂ ; (10)
~ ~
 
α2 2
Û 0 = exp i x̂ ; (11)
2α1 ~
 
00
Û = exp
i
(x̂ p̂ + p̂x̂) ln(2α 1 ) : (12)
4~
We can transform the invariant operator using the above operator as

Iˆ0 = Ût IˆÛt† : (13)

Then, Iˆ0 reduces to the following simple form

Iˆ0 = p̂2 + ω 2 x̂2 ;

1 1
(14)
2 2
where
1
ω 2 = 4(α1 α3 α22 ) = (ρ ρ̇ ρ̇1ρ2 )2 (4c1 c3 c22) = constant: (15)
4A2 1 2
For convenience, we only discuss the system for ω 2 > 0. Since eq. (14) is the same as that
of the Hamiltonian of the simple harmonic oscillator with unit mass, we can introduce the
ladder operators defined as
r
ω
x̂ + p
i
b̂ = p̂; (16)
2~ 2ω ~
r
ω
b̂ †
=
2~
x̂ pi p̂: (17)
2ω ~

These satisfy the boson commutation relation [ b̂; b̂† ] = 1. In terms of these operators, eq.
(14) can be simplified to

Pramana – J. Phys., Vol. 61, No. 1, July 2003 9

 Jeong-Ryeol Choi
 
Iˆ0 = ~ω b̂† b̂ +
1
: (18)
2

The eigenvalue equation for Iˆ0 can be written as

Iˆ0 jn0 (t )i = λn jn0 (t )i: (19)

We can easily identify the eigenstate in x̂ space as

 ω 1 r   ω 
ω
hx̂jn0(t )i = p 1n
= 4
Hn x̂ exp x̂2 : (20)
~π 2 n! ~ 2~

The eigenstate of the untransformed invariant operator can be obtained from

hx̂jn(t )i = Ût†hx̂jn0(t )i : (21)

Substituting eqs (9) and (20) into the above equation gives
 1 r   
ω 4
=
ω
hx̂jn(t )i = 2α1 ~π
Hn p 1n 2α1 ~
(x̂ x p ) exp
i
~
p p x̂
2 n!
 
1 ω 
 exp 2α1 ~ 2
+ iα2 (x̂ x p )2 : (22)

The x̂ space Schrödinger solution hx̂jψ n i of the Hamiltonian, eq. (1), is the same as the
ˆ except for some time-dependent phase factor, ε n (t ) [30]
eigenstate of I,

hx̂jψn (t )i = exp[iεn (t )]hx̂jn(t )i : (23)

Inserting the above equation into Schrödinger equation, we derive the relation
 
∂
~ε̇n (t ) = hn(t )j i~ Ĥ jn(t )i : (24)
∂t

Using eqs (1), (22) and (24), ε n (t ) can be obtained as

 Z t Z
A(t 0 ) 0
H p (x p (t 0 ); p p (t 0 ); t 0 )dt 0 ;
1 1 t
εn (t ) = ω n + dt (25)
2 0 α1 ~ 0

where H p (x p (t ); p p (t ); t ) is defined as

H p (x p (t ); p p (t ); t ) = A(t ) p2p (t ) + C(t ) p p(t ) D2 (t )x2p (t ) + D0 (t ): (26)

Substituting eq. (25) into (23), we can obtain the exact wave function as
  Z
A(t 0 ) 0
hx̂jψn(t )i = hx̂jn(t )i exp 1 t
iω n + dt
2 0 α1
Z 
H p (x p (t 0 ); p p (t 0 ); t 0 )dt 0
i t
: (27)
~ 0

10 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator

The p̂ space wave function is related to the x̂ space by the Fourier transformation
Z∞  
h p̂jψn (t )i = p 1
hx̂jψn (t )i exp i ~ dx̂:
p̂x̂
(28)
2π ~ ∞
Using (28), the above equation can be calculated as
 1  1
2ωα1 4
=
(ω 2iα2 )n 2
=

h p̂jψn(t )i = ( i)n
~π
p 1n (ω + 2iα2 )n+1
2 n!
"s #
2α1 ω
Hn ~(ω 2 + 4α22 )
( p̂ p p)
 
 exp ~ x p( p̂ p p) ~α(1ω( p̂+ 2ipαp) )
2
i

  Z t 0 2
A(t ) 0
 exp iω n + 2 0 α dt 1

Z
1

i t 0 0 0 0
H p (x p (t ); p p (t ); t )dt : (29)
~ 0

To express Iˆ(t ) in a simple form, we introduce another ladder operator as

1 h ω  i
â(t ) = p + iα2 (x̂ x p ) + iα1 ( p̂ p p ) ; (30)
ω ~α1 2
1 h ω  i
↠(t ) = p iα2 (x̂ x p ) iα1 ( p̂ p p ) : (31)
ω ~α1 2

These operators also satisfy [â; â † ] = 1. In terms of eqs (30) and (31), (3) can be expressed
as
 
1
Iˆ(t ) = ~ω â â + †
: (32)
2
From eqs (30) and (31), we can confirm that the coordinate and the momentum can be
expressed as
r
~α1 †
x̂ = (â + â ) + x p ; (33)
ω
s
~ h ω  ω  i
p̂ = i + iα2 ↠iα2 â + p p: (34)
α1 ω 2 2

Using eqs (27), (33) and (34), we can calculate the following expectation values

hψn jx̂jψn i = x p ; (35)

hψn j p̂jψn i = p p ; (36)

hψn jx̂2 jψni = ~ωα1 (2n + 1) + x2p ; (37)

Pramana – J. Phys., Vol. 61, No. 1, July 2003 11

 Jeong-Ryeol Choi
 
~ ω2
hψn j p̂ jψn i = α ω
2
4
+ α2 (2n + 1) + p p;
2 2
(38)
1
2α2 ~
hψn j(x̂ p̂ + p̂x̂)jψn i = ω
(2n + 1) + 2x p p p : (39)

Then, we can easily identify the uncertainty relation as

∆x̂∆ p̂ = [hψn jx̂2 jψn i ( hψnjx̂jψn i)2 ]1 2 [hψn j p̂2jψn i (hψnj p̂jψn i)2 ]1 2
= =


~q 1
= ω 2 + 4α22 n + : (40)
ω 2

This is always larger than ~=2 as expected. The uncertainty relation of q-deformed har-
monic oscillator also differs from the uncertainty relation for the simple harmonic oscillator
[31].
By performing a similar procedure, we obtain the expectation value of Hamiltonian as

hψn jĤ jψn i = ω~ [α1 D2 2α2 B + α3A](2n + 1) + H p(x p (t ); p p (t ); t ): (41)

3. Thermal state

We consider an ensemble of particles that satisfies the given general time-dependent har-
monic oscillator motion. Let us assume that these particles conform to the Bose–Einstein
distribution function.
Density operator of the system may satisfy Liouville–von Neumann equation as

∂ ρ̂ (t ) 1
+ [ρ̂ (t ); Ĥ ] = 0: (42)
∂t i~
Then, we can express the density operator in x̂ space as
  
1 ∞ ~ω
ρ̂ (x̂; x̂0 ; t ) = hx̂jψn (t )i exp hψn (t )jx̂0 i
1
Z (t ) n∑
n+ ; (43)
=0
kT 2

where k is the Boltzmann constant and T the temperature of the system at initial time.
The partition function of the system can be given by
∞
Z (t ) = ∑ hψn (t )je Iˆ(t )=(kT )
jψn (t )i : (44)
n=0

Using eq. (27), the partition function, eq. (44) and the density operator, eq. (43) can be
calculated as
1
Z (t ) = ; (45)
2 sinh[~ω =(2kT )]

12 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator
  1 (
ω ~ω =2
iα2
ρ̂ (x̂; x̂0 ; t ) = tanh exp [(x̂ x p )2 0 x p )2 ]
(x̂
2π ~α1 2kT 2~α1
     )
ω 0 2 ~ω 02 ~ω
(x̂ + x̂ 2x p) tanh + (x̂ x̂ ) coth
8~α1 2kT 2kT
 
 exp i (x̂ x̂0 ) p p :
~
(46)

At high temperature, eq. (46) reduces to

 
ω
ρ̂ (x̂; x̂0 ; t ) ' exp (x̂ x̂0 ) p p
1 i
2 (α1 π kT )1=2 ~
 
 exp 2i~αα2 [x̂2 x̂02 2(x̂ x̂0)x p ] kT
4~2 α1
(x̂ x̂0 )2 : (47)
1

If the difference between x̂ and x̂ 0 are sufficiently small compared to (2~α 1 =α2 )1=2 and
~= p p , and 1=(kT ) approaches zero, the density operator may be simply represented as
~ω
ρ̂ (x̂; x̂0 ; t ) ' δ (x̂ x̂0 ): (48)
kT
On the other hand, at low temperature, it becomes
 1  
ω = 2
ρ̂ (x̂; x̂0 ; t ) ' p p (x̂ x̂0 )
i
exp
2π ~α1 ~
(
iα2 2
 exp [x̂ x̂02 2(x̂ x̂0 )x p ]
2~α1
)
ω 02
2 2
(x̂ + x̂ + 2x p 2x̂x p 2x̂0 x p ) : (49)
4~α1

The diagonal element of the density operator, eq. (46), can be written as
  1    
ω ~ω = 2
ω ~ω 2
f (x̂) = tanh exp tanh (x̂ x p) :
2π ~α1 2kT 2~α1 2kT
(50)

The above equation represents the probability that the mass of the oscillator reside at x.
ˆ As
temperature increases, it becomes
 
ω ω2
f (x̂) '
1 2
exp (x̂ x p) : (51)
2 (πα1 kT )1=2 4kT α1

Equation (50) can be used to calculate the expectation value in coordinate space as
Z∞
hx̂l iT = x̂l f (x̂)dx̂: (52)
∞

Pramana – J. Phys., Vol. 61, No. 1, July 2003 13

 Jeong-Ryeol Choi

For example, we can obtain for l = 1; 2 as

hx̂iT = x p ; (53)
 
hx̂2 iT = ~ωα1 coth ~ω
2kT
2
+ x p: (54)

When considering eq. (44), the expectation value of Iˆ in the thermal state can be derived
from

hIˆiT = kT 2 ∂∂T ln Z (t ): (55)

Making use of eq. (45), the above equation becomes

 
~ω
hIˆiT = 12 ~ω coth 2kT
: (56)

Using the same procedure in the xˆ space, the p̂ space representation of the density operator
can be obtained as
  1  
2α1 ω ~ω =2
ρ ( p̂; p̂0 ; t ) = x p ( p̂ p̂0 )
i
tanh exp
~π (ω + 4α2 )
2 2 2kT ~
( (
α1 ω 2iα2 2
 exp [ p̂ p̂02 2p p( p̂ p̂0 )]
~(ω 2 + 4α22 ) ω
"     #))
1 0 2 ~ω 0 2 ~ω
( p̂ + p̂ 2p p) tanh + ( p̂ p̂ ) coth :
2 2kT 2kT
(57)
At high temperature, the above equation becomes
 1 2  
α1 ω 2
=

ρ ( p̂; p̂0 ; t ) ' x p ( p̂ p̂0)

i
exp
π (ω + 4α22 )kT
2 ~
( (
α1
 exp 2iα2 [ p̂2 p̂02 2p p( p̂ p̂0 )]
~(ω 2 + 4α22 )
))
kT 02
( p̂ p̂ ) : (58)
~
On the other hand, at low temperature it can be expressed as
 1  
2α1 ω 2
=

ρ ( p̂; p̂0 ; t ) ' p̂0 )

i
exp x p ( p̂
~π (ω 2 + 4α22) ~
( (
α1 ω 2iα2 2
 exp [ p̂ p̂02 2p p( p̂ p̂0)]
~(ω 2 + 4α22) ω
))
2
( p̂ + p̂
02 2 p̂p p 2 p̂0 p p + 2p2p) : (59)

14 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator

The probability that the mass of the oscillator resides at p̂ is obtained taking the diagonal
elements of eq. (57) as
  1
2α1 ω ~ω =2
f ( p̂) = tanh
~π (ω + 4α2 )
2 2 2kT
   
2α1 ω ~ω
 exp ~(ω + 4α2 )
2 2
tanh
2kT
( p̂ p p)2 : (60)

Using eq. (60), we can calculate the expectation value in p̂ space as

h p̂iT = p p ; (61)
 
~(ω 2 + 4α22 ) ~ω
h p̂ iT = 4α ω coth 2kT + p2p
2
: (62)
1

We can also write the expectation value of Iˆ in the thermal state as

hIˆiT = α1 (h p̂2 iT p2p) + 2α2(hx̂ p̂iT x p p p ) + α3 (hx̂2 iT x2p): (63)
Substituting eqs (54), (56) and (62) into the above equation gives
   
~ω 1 ~ω
hx̂ p̂iT = 2α 4
1
ω 2
(α2 + α1 α3 ) coth
2
2kT
+ x p p p: (64)
2

Then, the expectation value of the Hamiltonian, eq. (1), can be calculated as
 
hĤ iT = ω~ (α3 A 2α2B + α1 D2 ) coth
~ω
2kT
+ H p (x p (t ); p p (t ); t ) (65)

and the uncertainty relation in thermal state can be calculated as

(∆x̂∆ p̂)T = [ hx̂2 iT hx̂i2T )(h p̂2 iT  h p̂i2T)]1 2 =

~ q ~ω
= ω 2 + 4α22 coth : (66)
2ω 2kT
By comparing the above equation with eq. (40), we can confirm that the uncertainty rela-
tion in thermal state varies as time goes by, with the same fashion in number state.

4. Forced Caldirola–Kanai oscillator

We can apply our theory to various kinds of time-dependent Hamiltonian systems. As an

example, let us see for the forced Caldirola–Kanai oscillator [32,33]. For this system, the
time-dependent coefficients in eq. (1) are given by
1
A(t ) = e βt; (67)
2m
1
D2 (t ) = mω02 eβ t ; (68)
2
D1 (t ) = F (t )eβ t ; (69)
B(t ) = C(t ) = D0 (t ) = 0; (70)

Pramana – J. Phys., Vol. 61, No. 1, July 2003 15

 Jeong-Ryeol Choi

so that we can rewrite the Hamiltonian as

βt p̂2 βt 1
Ĥ = e +e mω02 x̂2 eβ t F (t )x̂; (71)
2m 2
where m is the mass, β the damping constant and F (t ) the arbitrary time-dependent driving
force. Equation (8) becomes
ρ̈1;2 + β ρ̇1;2 + ω02 ρ1;2 = 0: (72)
The two classical solutions of the above equation can be written as
β t =2 iω t
ρ1 (t ) = ρ1 (0)e e ; (73)
β t =2 iω t
ρ2 (t ) = ρ2 (0)e e ; (74)
where ω is given by
r
β2
ω= ω02 : (75)
4
We choose c1 –c3 in eqs (5)–(7) as
1
c2 = ; c1 = c3 = 0: (76)
2mρ1 (0)ρ2 (0)
The particular solutions x p and p p satisfy the following relations:
F (t )
ẍ p + β ẋ p + ω02x p = ; (77)
m
p̈ p β ṗ p + ω02 p p = eβ t Ḟ (t ): (78)
The solutions of the above equations depend on F (t ). If, we choose F (t ) as
F (t ) = F0t ; (79)
the solutions of eqs (77) and (78) will be
F0 β F0
x p (t ) = t ; (80)
mω02 mω04
F
p p (t ) = 02 eβ t : (81)
ω0
We will also investigate the system driven by the exponentially decaying force:
γt
F (t ) = F0 e ; (82)
where γ is an arbitrary real constant. In this case, the particular solutions are given by
F0 =m
x p (t ) = e γt ; (83)
γ2 β γ + ω02
γ F0
p p (t ) = e(β γ )t
: (84)
γ 2 β γ + ω02

16 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator

(a) (b)
Figure 1. Ground state (a) and first excited state (b) probability density in number state
for the forced Caldirola–Kanai oscillator that the driving force is given by eq. (79), as a
function of position x̂ and time t. We used F0 = 1, ω0 = 1, β = 0:4, ρ1 (0) = ρ2 (0) = 1,
m = 1 and ~ = 1.

The system, we finally consider is the one driven by the periodic force

F (t ) = F0 cos(ω1t + φ ); (85)

where ω1 is a real driving force and φ an arbitrary phase. Equation (85) has the classical
particular solutions which are given by
F0 =m
x p (t ) = q cos(ω1t + φ δ ); (86)
(ω02 ω12 )2 + β 2 ω12
F0 ω1
p p (t ) = q eβ t sin(ω1t + φ δ ); (87)
(ω02 ω12 )2 + β 2ω12

where
β ω1
δ = tan
1
: (88)
ω02 ω12
We depicted ground state and first excited state probability densities in figures 1–3 for eqs
(79), (82) and (85). The center of probability densities shifted from zero point along x̂-axis
with time according to the magnitude of the driving force.
We will investigate the system driven by the periodic force, eq. (85), in more detail.
For this case, the expectation value of the Hamiltonian, eq. (65), in thermal state can be
evaluated as

hĤ iT = ~2ωω0 coth 2kT

~ω 1 β t 2
2
1
+ e mẋ p mω02 eβ t x2p ;
2 2
~ω02 ~ω βt F02
= coth +e Θ(t ); (89)
2ω 2kT m[(ω02 ω12)2 + β 2 ω12 ]

where
1 2 2
Θ(t ) = [ω sin (ω1 t + φ δ) ω02 cos2 (ω1 t + φ δ )]: (90)
2 1
We can confirm that eq. (89) oscillate with time.

Pramana – J. Phys., Vol. 61, No. 1, July 2003 17

 Jeong-Ryeol Choi

(a) (b)
Figure 2. Ground state (a) and first excited state (b) probability density in number state
for the forced Caldirola–Kanai oscillator that the driving force is given by eq. (82), as a
function of position x̂ and time t. We used F0 = 2, ω0 = 1, β = 0:4, ρ1 (0) = ρ2 (0) = 1,
m = 1, γ = 1 and ~ = 1.

(a) (b)
Figure 3. Ground state (a) and first excited state (b) probability density in number
state for the forced Caldirola–Kanai oscillator that the driving force is given by eq.
(85), as a function of position x̂ and time t. We used F0 = 1, ω0 = 1, ω1 = 1:5, β = 0:4,
ρ1 (0) = ρ2 (0) = 1, m = 1, φ = 0 and ~ = 1.

In general, the mechanical energy is somewhat different from the Hamiltonian for the
time-dependent system such as Caldirola–Kanai oscillator [34]. For the case driven by
eq. (85), the mechanical energy expectation values in the thermal state can be calculated
as [34]

hÊ iT = e 1 2
2β t
2m
h p̂ iT + 12 mω02hx̂2 iT
β t ~ω0 ~ω p2 1
2
2β t p
+ mω 0 x p :
2 2
=e coth +e
2ω 2kT 2m 2
β t ~ω0 ~ω
2
F02
=e coth +
2ω 2kT 2m[(ω02 ω12)2 + β 2 ω12 ]
[ω12 sin2 (ω1t + φ δ ) + ω02 cos2 (ω1t + φ δ )]: (91)
Thus, we see that eq. (91) also oscillates with time since the system exchanges the energy
with the surroundings. However, at ω 1 = ω0 , the energy does not oscillate and becomes
(in the limit t ! ∞):
F02
hÊ iT = 2m[(ω 2 ω 2:
ω12)2 + β 2 ω12 ] 0
(92)
0

18 Pramana – J. Phys., Vol. 61, No. 1, July 2003

 Thermal state of the time-dependent harmonic oscillator

(a) (b)
Figure 4. Hamiltonian (a) and quantum-mechanical energy (b) expectation values for
the forced Caldirola–Kanai oscillator in the thermal state as a function of time t. The
q
solid line is for ω1 = 0:5, the long dotted-line for ω1 = ω0 = 1 and the short dotted line
for ω1 = ω02 β 2 =2 ' 0:959. We used F0 = 1, ω0 = 1, β = 0:4, ρ1 (0) = ρ2 (0) = 1,
m = 1, φ = 0 k = 1, T = 300 and ~ = 1.

There are two resonant frequencies for this system [35]. One is the velocity resonant
frequency which is same as the natural frequency ω 0 and the other is the amplitude resonant
q
frequency which is given by ω02 β 2 =2. In figure 4, we depicted the Hamiltonian and
the mechanical energy expectation values for these two resonant frequency in the thermal
state. Even if the mechanical energy far from the resonance points in frequency gradually
disappears with time, the mechanical energy near the resonance points remained the same.

5. Summary

Taking advantage of the invariant operator, we obtained the solution of the Schrödinger
equation for the general time-dependent harmonic oscillator. We assumed that an ensemble
of particles that satisfies the general time-dependent harmonic oscillator motion conform
to the Bose–Einstein distribution function at equilibrium temperature. We investigated
uncertainty relation in number and thermal states. Comparing eqs (40) and (66), we can
confirm that the uncertainty relation in the thermal state varies in the same manner as in
the number state.
The uncertainty relation is always larger than ~=2 in both number and thermal states. We
determined density operators satisfying the Leouville–von Neumann equation and used it
to calculate various expectation values of the variables in the thermal state.
We applied our theory to a special case which is the forced Caldirola–Kanai oscillator.
The center of probability densities in number state shifted from zero point along the x-axis
ˆ
with time according to the magnitude of the driving force. Even if the mechanical energy
far from the resonance points in frequency gradually disappears with time, the mechanical
energy near the resonance points remained the same.

References

[1] M Moshinsky and Y F Smirnov, The harmonic oscillator in modern physics (Harwood Aca-
demic Publishers, Australia, 1996)

Pramana – J. Phys., Vol. 61, No. 1, July 2003 19

 Jeong-Ryeol Choi

[2] G E Giacaglia, Perturbation methods in non-linear systems (Springer, Berlin, 1972); Appl.
Math. Sci. Series, vol. 18
[3] W T Van Horssen, SIAM J. Appl. Math. 59, 1444 (1999)
[4] D-Y Song, Phys. Rev. A59, 2616 (1999)
[5] K H Yeon, K K Lee, C I Um, T F George and L N Pandey, Phys. Rev. A48, 2716 (1993)
[6] K H Yeon, H J Kim, C I Um, T F George and L N Pandey, Nuovo Cimento Soc. Ital. Fis. B111,
963 (1996)
[7] L S Brown, Phys. Rev. Lett. 66, 527 (1991)
[8] Fu-li Li, S J Wang, A Weiguny and D L Lin, J. Phys. A27, 985 (1994)
[9] L F Landovitz, A M Levine and W M Schreiber, Phys. Rev. A20, 1162 (1979)
[10] S W Qian, B-W Huang and Z-Y Gu, J. Phys. A34, 5613 (2001)
[11] J G Hartley and J R Ray, Phys. Rev. A25, 2388 (1982)
[12] K H Yeon, D H Kim, C I Um, T F George and L N Pandey, Phys. Rev. A55, 4023 (1997)
[13] C I Um, I H Kim, K H Yeon, T F George and L N Pandey, J. Phys. A30, 2545 (1997)
[14] C I Um, I H Kim, K H Yeon, T F George and L N Pandey, Phys. Rev. A54, 2707 (1996)
[15] C I Um, J R Choi and K H Yeon, J. Korean Phys. Soc. 38, 447 (2001); 38, 452 (2001)
[16] J Y Ji, J K Kim and S P Kim, Phys. Rev. A51, 4268 (1995)
[17] B Baseia and A L De Brito, Physica A197, 364 (1993)
[18] H Lamba, S McKee and R Simpson, J. Phys. A31, 7065 (1998)
[19] R L Kobes and K L Kowalski, Phys. Rev. D34, 513 (1986)
[20] N P Landsman and Ch G van Weert, Phys. Rep. 145, 141 (1987)
[21] E Calzetta and B L Hu, Phys. Rev. D37, 2878 (1988)
[22] A O Caldeira and A J Leggett, Ann. Phys. 149, 374 (1983)
[23] S Baskoutas, A Jannussis and R Mignani, J. Phys. A27, 2189 (1994)
[24] A O Caldeira and A J Leggett, Phys. Rev. Lett. 46, 211 (1981)
[25] A Widom and T D Clark, Phys. Rev. Lett. 48, 63 (1982)
[26] H Dehmelt, Rev. Mod. Phys. 62, 525 (1990)
[27] W Paul, Rev. Mod. Phys. 62, 531 (1990)
[28] H S Robertson, Statistical thermophysics (Prentice Hall, Englewood Cliffs, 1993) p. 450
[29] A Isihara, Statistical physics (Academic Press, New York, 1971) p. 154
[30] H R Lewis, Jr. and W B Riesenfeld, J. Math. Phys. 10, 1458 (1969)
[31] P Raychev, Adv. Quantum Chem. 26, 239 (1995)
[32] P Caldirola, Nuovo Cimento 18, 394 (1941)
[33] E Kanai, Prog. Theor. Phys. 3, 440 (1948)
[34] K H Yeon, C I Um and T F George, Phys. Rev. A36, 5287 (1987)
[35] G R Fowles, Analytical mechanics (Saunders College Publishing, 4th ed., Philadelphia, 1986)
pp. 70–76

20 Pramana – J. Phys., Vol. 61, No. 1, July 2003