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Philippine Journal of Science
133 (2): 145-160, December 2004
ISSN 0031 - 7683
Coherent, Squeezed, and Thermal State of
Harmonic Oscillator with Eponentially
Decreasing Frequency
Jeong-Ryeol Choi
Department of New Material Science, Division of Natural Sciences
Sunmoon University, Asan 336-708, Republic of Korea
The solutions of Schrödinger equation for a damped harmonic oscillator with
exponentially decreasing frequency are derived using invariant operator method. We
investigated coherent, squeezed, and thermal states of the system as well as number
state. The uncertainty product, quantum-mechanical energy expectation value, and density
operator of the system are evaluated. The uncertainty product in coherent state are same
as the minimum uncertainty product in number state. We confirmed that the uncertainty
product is always larger than 1/2 in all kind of states we considered. The mechanical energy
expectation value in thermal state decreases in the same way as that in number state.
Keywords: Invariant operator, Coherent state, Squeezed state, Thermal state
Introduction
The harmonic oscillator played an important role in the development of both theoretical and experimental
physics (Dekker 1981). Recently, the investigation of the quantum-mechanical solution for the time-dependent
harmonic oscillator (Um et al. 1997), time-dependent forced harmonic oscillator (Um et al. 1996), and asymmetrical
quantum sextic anharmonic oscillator (Lee et al. 1998) have been attracted considerable interest as well as their
classical analysis. The relations between various results of quantum analysis for the linearly coupled damped
harmonic oscillator and their classical property are studied in detail by Dekker (Dekker 1981).
With the jump on these trends, we are motivated to study the many-sided quantum properties in coherent state,
in squeezed state, and in thermal state as well as in number state for the time-dependent harmonic oscillator. To
do this, we consider damped harmonic oscillator with exponentially decreasing frequency as a model example.
We investigate diverse quantum-mechanical properties of the system such as wave function, uncertainty product,
and quantum-mechanical energy eigenvalues.
One of the powerful tools to solve the quantum-mechanical solution of the time-dependent harmonic oscillator
is the invariant operator method. The invariant operator was first introduced by Lewis in 1967 (Lewis 1967). The
key idea that using invariant operator in order to solve Schrödinger equation for the time-dependent Hamiltonian
system is that the solutions (wave functions) of the Schrödinger equation are the same as the eigenstates of
the invariant operator, except for some time-dependent phase factor (Lewis and Riesenfeld 1969). Although,
the construction of harmonic oscillator that has time-dependent frequency may be not easy substantially, the
investigation of these systems may offer deep intuition for a full understanding of the quantum time-dependent
Hamiltonian systems.
*Corresponding author: [email protected]
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Jeong-Ryeol Choi
The coherent state can be an appropriate basis not confined to optical field. This state describes quantummechanical state that resembled the classical description of the system. In squeezed state, it is possible to reduce
the variances in one quadrature component below coherent state limit at the expense of increased variances in
the canonically conjugate quadrature, such that uncertainty principle hold. By introducing density operator that
satisfies Leouville-von Neumann equation (Ji and Kim 1996), we also investigate thermal state of the system.
This paper is organized in the following order. In Sec. II, we introduce the Hamiltonian of the harmonic oscillator
with exponentially decaying frequency and construct corresponding invariant operator. The eigenstates of invariant
operator are derived using unitary transformation method. Exact wave functions satisfying the Schrödinger equation,
uncertainty product, and quantum-mechanical energy expectation values in number state are evaluated in Sec.
III. We derived coherent states and squeezed states of the system in Sec. IV and V, respectively. After Glauber’s
investigations (Glauber 1963), the coherent states of the harmonic oscillator have attracted considerable attention
in literature and have been frequently used to describe the radiation field. In Sec. VI, we calculated the values
of the diverse quantity in thermal state. Finally, we conclude this paper in Sec. VII with brief a discussion of the
results of previous sections.
Invariant operator
The classical equation of motion for the damped harmonic oscillator with general time-dependent frequency,
w (t ), is given by
(1)
where b is damping constant. We only consider a special case that the frequency of the system decreases
exponentially. Then, w (t ) in the above equation becomes
(2)
where w0 and e are positive constants. The classical solution of Eq. (1) may be expressed in terms of Bessel
functions (Erdély 1953):
(3)
where C± are integral constants, n = b/e, and z = w( )/e
In view of quantum mechanics, the Hamiltonian which gives the classical equation of motion as Eq. (1) can
be written as
(4)
Note that we replaced the classical variable and with quantum mechanical operator and in the above
equation. This satisfies the commutation relation that [ , ] = iħ. We can easily check that the above Hamiltonian
gives Eq. (1) by direct calculation using Hamilton’s equation of motion.
In order to derive quantum-mechanical solutions of the time-dependent Hamiltonian system, it is convenient
to use the invariant operator. To investigate the system quantum mechanically, we introduce the trial invariant
operator as
(5)
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Harmonic Oscillator with Exponentially Decreasing Frequency
where the time-dependent functions, h1(t) – h3(t) , should be determined afterward. Since the time derivative
of quantum-mechanical invariant operator becomes zero, Eq. (5) must satisfy the following differential equation:
(6)
Substitution of Eqs. (4) and (5) into Eq. (6), we can determine the functions
( )–
( ) as (Ji et al. 1995)
(7)
(8)
(9)
where c1 -
c3 are arbitrary constants.
It seems that the eigenvalue equation of the invariant operator, Eq. (5), can not be easily solved since the
separation of variables are difficult. Thus, to simplify the problem, we transform Eq. (5) with an appropriate unitary
operator Û
δ= ÛÎÛ†.
(10)
Û = Û 2 Û 1,
(11)
If we choose Û as
with
(12)
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Jeong-Ryeol Choi
(13)
equation (10) reduces to
(14)
where
(15)
We can confirm that ω1 is time-constant quantity, since the derivative of Eq. (15) with respect to time is zero.
Then, Eq. (14) is obviously time-independent operator.
The eigenvalue equation for δcan be written as
(16)
Adding 〈 | to both side of the above equation from the left, we can easily obtain the solution as
(17)
(18)
where
is th order Hermite polynomial and is the eigenvalue of the number operator. The relation
between eigenstate, 〈 | ( )〉, of untransformed invariant operator and that of transformed invariant operator is
given by (Choi 2002)
†
〈 | ( )〉 = Û 〈 | ‫〉) (׳‬
(19)
Using Eqs. (11) and (17), the above equation becomes
(20)
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Harmonic Oscillator with Exponentially Decreasing Frequency
Exact quantum state
The solutions of the Schrödinger equation, 〈 |ψn( )〉, are the same as the eigenstates of invariant operator
except for some time-dependent phase factor, eiγn(t)(Lewis and Riesenfeld 1969):
(21)
If we insert the above equation into Schrödinger equation, we obtain that
(22)
By substitution of Eq. (4) into the above equation, γn(t) can be derived to be
(23)
Using Eqs. (20), (21), and (23), we can express the full wave function as
(24)
This is the exact solution of the Schrödinger equation of the system. We depicted the absolute square of Eq.
(24) in Figs. 1 and 2 for n=0. The Fourier transformation relation between x and p space representation of the
wave function is
(25)
x
0
2
1
3
0.5
0
1
2
0.5
x
1
0
0.5
t
1 0
Figure 1. Probability density |〈x|ψ0(t)〉2 as a function of
position x and time t for є= 0.2 and β = 1
x
0
2
1
0.8
0.6
0.4
0.2
3
0
1
2
0.5
x
1
0
0.5
t
1 0
Figure 2. Probability density |〈x|ψ0(t)〉2 as a function of
position x and time t for є= 0.5 and β = 1
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Jeong-Ryeol Choi
Substitution of Eq. (24) into the above equation gives
(26)
The absolute square of p space wave function is depicted in Figs. 3 and 4 for п = 0. By comparing Figs. 1 and
2 with Figs. 3 and 4, we can confirm that the wave functions in х space are converges to the center as time goes
by owing to the damping constant β, while that of р space are diverges.
p
0
2
0.4
2
0.2
0
2
1.5
1
1
p
t
0.5
0
1
p
0
2
0.4
2
0.2
0
2
1.5
1
1
p
Figure 3. Probability density |〈p|ψ0(t)〉 as a function of
position x and time t for є= 0.2 and β = 1
0.5
0
2 0
2
t
1
2 0
2
Figure 4. Probability density |〈p|ψ0(t)〉 as a function of
position x and time є= 0.5 and β = 1
In terms of Î (Î′), the lowering and raising operators can defined in the form
(27)
(28)
We can easily check that these operators satisfies boson commutation relation: [â, â†] =1. In terms of the
ladder operators, Eq. (5) can be expressed as
(29)
Equations (27) and (28) may be easily solved for the canonical variables and obtained
(30)
(31)
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Harmonic Oscillator with Exponentially Decreasing Frequency
The ladder operators satisfy the following conventional relations
(32)
(33)
Note that the above two equations can be reexpressed using |ψn(t)〉 instead of |n(t)〉 as
(34)
(35)
The variances of x and p in number state can be easily evaluated using Eqs. (30), (31), (34), and (35) as
(36)
(37)
We depicted ∆x in Fig. 5 and ∆p in Fig. 6. These figures say that the variances fluctuate more or less according
to the values of є and t.
1.5
3
x
2
6
1
0
4
0.2
0.6
1
6
0.5
0
t
2
0.4
p
0
Figure 5. Variance ∆x as a function of є and time t for β =0
4
0.2
t
2
0.4
0.6
0
Figure 6. Variance ∆p as a function of є and time t for β =0
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Jeong-Ryeol Choi
By multiplying the above two equations, the uncertainty product obtained
(38)
This is always larger than 1/2 so that the uncertainty principle holds. The uncertainty product is depicted in
Figs. 7 and deviated from standard form. Another example for uncertainty product that deviated from standard is
that of q-deformed harmonic oscillator (Raychev 1995).
The mechanical energy of the system may be represented as
(39)
By averaging the above equation with respect to |ψn(t)〉 using the same method as in the calculation of Eqs.
(36) and (37), the quantum-mechanical energy expectation value can be evaluated to be
(40)
Now we analyze the system classically by introducing action variable defined as
(41)
By making use of Eq. (39), the above equation can be easily integrated as
(42)
Since the action variable is adiabatic invariant quantity (Goldstein 1980) in general, we can write J(0) = J(t).
The classical energy at time t is therefore
(43)
where E(0) is initial classical energy. The classical energy decreases exponentially with time due to the factors
β and є. To compare quantum and classical mechanical energy, we depicted them together in Fig. 8. Since the
momentum at turning point should be zero, Eq. (39) may be reexpressed as
(44)
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Harmonic Oscillator with Exponentially Decreasing Frequency
where xmax (t) is the maximum amplitude of oscillation at time t. By Eqs. (43) and (44) we see that
(45)
where xmax(0) is maximum amplitude at t = 0.
Coherent state
The coherent states are displaced ground state wave functions. They resemble the classical fields as close
as quantum mechanics permits. An expression of coherent state in terms of |n(t)〉 is
(46)
If we multiply both sides from the left by 〈x| we derive x space representation of coherent state
(47)
The same method may be applied in order to obtain p space coherent state:
(48)
Although, in general, the coherent state does not satisfy the orthogonal property, Eqs. (47) and (48) can be
used suitably to describe the time-dependent Hamiltonian system, since they constitute a complete set and the
system can be represented uniquely in terms of them. From Eqs. (30) and (31) we see that the expectation value
of , ,
, and in coherent state are
(49)
(50)
(51)
(52)
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Jeong-Ryeol Choi
The variances for the canonical variables are therefore
(53)
(54)
The uncertainty product in coherent state can be identified by multiplying the above two equations
(55)
where
(56)
We can represent Eq. (55) in another form as
(57)
Equation (57) is exactly same as that in number state for n=0, and note that it is larger than that of simple
harmonic oscillator, 1/2.
Squeezed state
Squeezed state of the harmonic oscillator offers another, near term, example of the well-lined interplay between
theoretical and experimental physics. In a squeezed state, the quantum variances in one variable are reduced at
the expense of the corresponding enlarged variances in the conjugate variable such that the uncertainty principle
conserved. The minimum uncertainty squeezed state can be derived from (Nieto and Truax 1993)
(58)
By operating 〈x| to both side of Eq. (58) from left, the squeezed state in x space reads easily as
(59)
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Harmonic Oscillator with Exponentially Decreasing Frequency
where δ is an arbitrary phase. If we choose the same ∆p as that in coherent state, Eq. (54), the above equation
becomes
(60)
By replacing
and
of the above equation with Eqs. (49) and (50), respectively, and with the choice
of d = (i/4) (a2 - a*2), the above equation recovers exactly to that coherent state, Eq. (47), and the uncertainty
product becomes same as that in coherent state, Eq. (57). There found effort to lower the uncertainty product in
squeezed state of time-dependent harmonic oscillator below the value in coherent state[or below, in our paper’s
yardstick, given in Eq. (57) by Yeon (Yeon et al. 1987).
Thermal state
Let us suppose that the ensemble of the system whose temperature equilibrate to the surroundings follows the
Bose-Einstein distribution. The Leouville-von Neumann equation for density operator, (t), can be written as
(61)
If the Hamiltonian of the system explicitly depended on time, the partition function of the system would have
the form (Ji and Kim 1996)
(62)
where bb = 1/(kbT) with kb is Boltzmann’s constant and T is temperature of the system at initial time. Substitution
of Eq. (5) into the above equation and after some algebra, we obtain
(63)
The density operator in x space can be defined as
(64)
If we use Eq. (24), we have
(65)
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Jeong-Ryeol Choi
As temperature goes high, this becomes
(66)
If the difference between and
above equation approximated as
are sufficiently small compared to 2h1/h2 , and bb approaches to zero, the
(67)
On the other hand, at low temperature the Eq. (65) approaches to
(68)
The probability that the harmonic oscillator particle is at
(65) (Isihara 1971):
can be obtained by taking diagonal elements of Eq.
(69)
This may be used to calculate various expectation values in thermal state.
For high temperature, the above equation reduces to
(70)
If we let
as an arbitrary function of , the expectation value of
in thermal state is given by
(71)
Using Eq. (69), the above equation for
=
k
(k=1,2) can be easily evaluated as
(72)
(73)
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Harmonic Oscillator with Exponentially Decreasing Frequency
From Eq. (62) we can confirm that
(74)
The above relation is exactly the same as that of the mechanical energy expectation values for ordinary simple
harmonic oscillator in thermal state. Substitution of Eq. (63) into the above equation gives
(75)
The p space representation of density operator is
(76)
If we consider Eq. (26), this may be evaluated to be
(77)
As temperature goes high, the above equation approaches to
(78)
On the other hand, at low temperature it becomes
(79)
Applying the same method as that for x space, the probability function is given by
(80)
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Jeong-Ryeol Choi
The probability of p and p2 in thermal state can also be derived as
(81)
(82)
In addition, the expectation value of
is given by
(83)
Making use of these results, we can easily identify the expectation value of Hamiltonian, Eq. (4), and mechanical
energy, Eq. (39), in thermal state
(84)
(85)
Note that the expectation value of Hamiltonian and mechanical energy operator are not same. This universally
holds for the damped harmonic oscillator (Yeon et al. 1987). From the above two equations we see that the relation
between the two expectation value is
(86)
By comparing Eq. (85) with Eq. (40), we can confirm that the mechanical energy expectation value in thermal
state decreases in the same way as that in number state. We may also evaluate the uncertainty product in thermal
state
(87)
This product varies in a same way as Eqs. (38) and (57) as time goes by.
conclusion
We used invariant operator method in order to derive the quantum-mechanical solution of the damped harmonic
oscillator with exponentially decreasing frequency. The uncertainty product, quantum-mechanical energy expectation
value and density operator of the system are investigated in detail.
158
Harmonic Oscillator with Exponentially Decreasing Frequency
We used the convention that c1 = c2= c3 = 1, w0 = 1 and n = 0 for all figures. From Figs. 1 and 2 we can confirm
that the oscillation amplitude in x space decreases as time goes by. This phenomenon is brought out from the
dissipation of energy due to the damping constant b. By comparing these two figures, we see that the constant
e is responsible for the possible enlargement of the oscillation amplitude. On the other hand, from Figs. 3 and 4,
the oscillation amplitude in p space increases as time goes by.
Since Eq. (38) is always larger than 1/2, the uncertainty principle is conserved. Figures 5, 6 and 7 says that the
uncertainty product as well as the variance of and fluctuate more or less. An outstanding tendency to increase
the uncertainty product with the magnitude of e and t is seen from Fig. 7. The expression of the expectation values
of the quantum-mechanical energy are somewhat different according to the kind of state the system is in. As an
example, we represented the quantum-mechanical energy in number state and in thermal state in Eqs. (40) and (85).
Figure 8 says that both the quantum energy and the classical energy decreases as time goes by and the difference
between them is small. By comparing Eq. (38) and (57), we see that the uncertainty product in coherent state are
exactly same as the minimum uncertainty product in number state. The uncertainty product in thermal state also
varied in the same fashion as that in other states with time.
1.5
x p
1
6
0.5
0
4
0.2
t
2
0.4
0.6
0
Figure 7. Variance ∆x∆p as a function of є and time t for β =0
Figure 8. Comparison of quantum(solid line) and classical(dot
line) energy for β = 0.5 and є =0.8. We used quantum and
classical energy given in Eq. (40) and (43), respectively.
We chose n = 0 for quantum energy and the initial classical
energy as E(0) = 0.58
References
1. Choi JR. 2002. Quantum state of pendulum under exponentially increasing gravitation. Physica A 310:109120.
2. Dekker H. 1981. Classical and quantum mechanics of the damped harmonic oscillator. Physics Reports 80:1112.
3. Erdély A. 1953. Higher Transcendental functions Vol. II, Chapter VII. New York: McGraw-Hill.
4. Glauber RJ. 1963. Photon correlations. Physical Review Letters 10:84-86.
5. Glauber RJ. 1963a. The quantum theory of optical coherence. Physical Review 130:2529-2539.
6. Glauber RJ. 1963b. Coherent and incoherent states of the radiation field. Physical Review 131:2766-2788.
7. Goldstein H. 1980. Classical Mechanics 2nd ed. Chapter 11. New York: Addison Wesley.
8. Isihara A. 1971. Statistical Physics. London: Academic Press 285 p.
9. Ji J-Y, Kim JK & Kim SP. 1995. Heisenberg-picture approach to the exact quantum motion of a time-dependent
harmonic oscillator. Physical Review A 51:4268-4271.
10. Ji JY & Kim JK. 1996. Temperature changes and squeezing properties of the system of time-dependent
harmonic oscillators. Physical Review A 53:703-708.
159
Jeong-Ryeol Choi
11. Lee JY, Liu KL & Lo CF. 1998. Asymmetrical quantum sextic anharmonic oscillator: Eigenstates and thermal
properties. Physical Review A 58:3433-3438.
12. Lewis HR Jr. 1967. Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians.
Physical Review Letters 27:510-512.
13. Lewis HR Jr & Riesenfeld WB. 1969. An exact quantum theory of the time-dependent harmonic oscillator and of a
charged particle in a time-dependent electromagnetic field. Journal of Mathematical Physics 10:1458-1473.
14. Nieto MM, Truax DR. 1993. Squeezed states for general systems. Physical Review Letters 71:2843-2846.
15. Raychev P. 1995. Quantum groups: Application to neclear and molecular spectroscopy. Advances in Quantum
Chemistry 26:239-357.
16. Um CI, Kim IH, Yeon KH, George TF & Pandey LN. 1996. Quantum analysis of the susceptibility for identical
atoms subjected to an external force with a tail. Physical Review A 54:2707-2713.
17. Um CI, Kim IH, Yeon KH, George TF, & Pandey LN. 1997. Wave functions and minimum uncertainty states
of the harmonic oscillator with an exponentially decaying mass. Journal Physics A: Mathematical and General
30:2545-2555.
18. Um CI, Kim IH & Tokuyama M. 1997. Quantum equivalence of the several Hamiltonian types for damped
harmonic oscillator. In: Statistical Physics (Proceedings of the 2nd Tohwa University International Meeting)
Tokuyama M & Oppenheim I, eds. Singapore: World Scientific p. 113.
19. Yeon KH, Um CI & George TF. 1987. Coherent states for the damped harmonic oscillator. Physical Review
A 36:5287-5291.
160