Time dependence of joint entropy of oscillating quantum systems
Time dependence of joint entropy of oscillating quantum systems
Özgür ÖZCAN,1, ∗ Ethem AKTÜRK,2, † and Ramazan SEVER3, ‡
1Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey
2Department of Physics, Hacettepe University, 06800, Ankara,Turkey
3Department of Physics, Middle East Technical University, 06531, Ankara,Turkey
Abstract
The time dependent entropy (or Leipnik’s entropy) of harmonic and damped harmonic oscillators
is extensively investigated by using time dependent wave function obtained by the Feynman path
integral method. Our results for simple harmonic oscillator are in agrement with the literature.
However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time
for certain values of damping factor. According to the results, the envelop of the joint entropy
curve increases with time monotonically. This results is the general properties of the envelop of
the joint entropy curve for quantum systems.
Keywords: Path integral, joint entropy, simple harmonic oscillator, damped harmonic oscillator,
negative joint entropy
PACS numbers: 03.67.-a, 05.30.-d, 31.15.Kb, 03.65.Ta
∗E-mail: ozcano@hacettepe.edu.tr
†E-mail: eakturk@hacettepe.edu.tr
‡E-mail: sever@metu.edu.tr
http://arxiv.org/abs/0704.1370v3
mailto:ozcano@hacettepe.edu.tr
mailto:eakturk@hacettepe.edu.tr
mailto:sever@metu.edu.tr
I. INTRODUCTION
The investigation of time dependent entropy of the quantum mechanical systems attracts
much attention in recent years. For both open and closed quantum systems, the different
information-theoretic entropy measures have been discussed [2, 3, 4]. In contrast, the
joint entropy [5, 6] can also be used to measure the loss of information, related to evolving
pure quantum states [7]. The joint entropy of the physical systems which are named MACS
(maximal classical states) were conjectured by Dunkel and Trigger [8]. According to Ref. [8],
the joint entropy of the quantum mechanical systems increase monotonically with time but
this results are not sufficient for simple harmonic oscillator [9].
The aim of this study is to calculate the complete joint entropy information analytically
for simple harmonic and damped harmonic oscillator systems.
This paper is organized as follows. In section II, we explain fundamental definitions
needed for the calculations. In section III, we deal with calculation and results for harmonic
oscillator systems. Moreover, we obtain the analytical solution of Kernel, wave function in
both coordinate and momentum space and its joint entropy. We also obtain same quantities
for damped harmonic oscillator case. Finally, we present the conclusion in section IV.
II. FUNDAMENTAL DEFINITIONS
We deal with a classical system with d = sN degrees of freedom, where N is the particle
number and s is number of spatial dimensions [8]. We assume that the density function
g(x, p, t) = g(x1, ..., xd, p1, ..., pd, t) which is the non-negative time dependent phase space
density function of the system has been normalized to unity,
dxdpg(x, p, t) = 1. (1)
The Gibbs-Shannon entropy is described by
S(t) = − 1
dxdpg(x, p, t)ln(hdg(x, p, t)), (2)
where h = 2πh̄ is the Planck constant. Schrödinger wave equation with the Born interpre-
tation [10] is given by
= Ĥψ. (3)
The quantum probability densities are defined in position and momentum spaces as |ψ(x, t)|2
and |ψ̃(p, t)|2, where |ψ̃(p, t)|2 is given as
ψ̃(p, t) =
dxe−ipx/h̄
(2πh̄)d/2
ψ(x, t). (4)
Leipnik proposed the product function as [8]
gj(x, p, t) = |ψ(x, t)|2|ψ̃(p, t)|2 ≥ 0. (5)
Substituting Eq. (5) into Eq. (2), we get the joint entropy Sj(t) for the pure state ψ(x, t)
or equivalently it can be written in the following form [8]
Sj(t) = −
dx|ψ(x, t)|2 ln |ψ(x, t)|2 −
dp|ψ̃(p, t)|2 ln |ψ̃(p, t)|2 −
− ln hd. (6)
We find time dependent wave function by means of the Feynman path integral which has
form [11]
K(x′′, t′′; x′, t′) =
∫ x′′=x(t′′)
x′=x(t′)
Dx(t)e
S[x(t)]
∫ x′′
Dx(t)e
L[x,ẋ,t]dt. (7)
The Feynman kernel can be related to the time dependent Schrödinger’s wave function
K(x′′, t′′; x′, t′) =
ψ∗n(x
′, t′)ψn(x
′′, t′′). (8)
The propagator in semiclassical approximation reads
K(x′′, t′′; x′, t′) =
∂x′∂x′′
Scl(x
′′, t′′; x′, t′)
Scl(x
′′,t′′;x′,t′). (9)
The prefactor is often referred to as the Van Vleck-Pauli-Morette determinant [12, 13]. The
F (x′′, t′′; x′, t′) is given by
F (x′′, t′′; x′, t′) =
∂x′∂x′′
Scl(x
′′, t′′; x′, t′)
. (10)
III. CALCULATION AND RESULTS
A. Simple Harmonic Oscillator (SHO)
To get the path integral solution for the SHO, we must calculate its action function. The
Lagrangian of the system is given by
L(x, ẋ, t) =
(ẋ2 − 1
ω2x2) (11)
Following a straightforward calculation, it is given by:
S(xcl(t
′′), xcl(t
′)) =
2 sinωt
[(x′′2cl + x
cl) cosωt− 2x′clx′′cl] (12)
with t = t′′ − t′ and x′cl = x0, x′′cl = x. Substituting Eq. (9) into Eq. (7), we obtain the
Feynman kernel [11]:
K(x, x0; t) = (
2πh̄i sinωt
2 exp{−mω
[(x2 + x0
2) cotωt− 2x0x
sinωt
]}. (13)
By the use of the Mehler-formula
2+y2)/2
)2Hn(x)Hn(y) =
1− z2
4xyz − (x2 + y2)(1 + z2)
2(1− z2)
] (14)
where Hn is Hermite polynomials, we can write the Feynman kernel defining x ≡
mω/h̄x0,
mω/h̄x and z = e−iωT
K(x, x0; t) =
e−itEn/h̄Ψ∗(x0)Ψ(x) (15)
with energy-spectrum and wave-functions:
En = h̄ω(n+
), (16)
Ψn(x) = (
22nπh̄n!2
x) exp(−
x2). (17)
Time dependent wave function of the SHO is defined as
Ψ(x, t) =
K(x, x0; t)Ψ(x0, 0)dx0. (18)
It can be written as
Ψ(x, t) =
− iωt
e−2iωt + αᾱe−iωt
where x̄ or ᾱ is mean of the Gaussian curve. The probability density has
|Ψ(x, t)|2 =
− (α− ᾱ cosωt)2
where α =
x. Thus it can be written as
|Ψ(x, t)|2 =
(x− x̄ cosωt)2
This has been shown in Fig.1. In momentum space, the probability density has the form
|ψ̃(p, t)|2 =
mωπh̄
[ −p2
mωx̄2
cos 2ω(t)− 1
− 2px̄
sinω(t)
. (22)
The joint entropy of harmonic oscillator becomes
Sj(t) = ln
x̄2 sin2 ω(t). (23)
In Fig.2, the joint entropy of this system was plotted by using Mathematica in three dimen-
sion. As known from fundamental quantum mechanics and classical dynamics, displacement
of simple harmonic oscillator from equilibrium depends on harmonic functions (e.g sine or
cosine function). Therefore, other properties of the SHO systems indicate the same har-
monic behavior. If the frequency of the SHO is sufficiently small, the system shows the
same behavior as the free particle[8]. As seen from Fig.3 and Fig.4, envelop of the sinusoidal
curve is also monotonically increase with omega and constant with time at constant omega,
respectively. When the frequency increases, the joint entropy of this system indicates a fluc-
tuation with increasing amplitude with time. If t goes to zero, it is important that Eq.(20)
is in agreement with following general inequality for the joint entropy:
Sj(t) ≥ ln(
) (24)
originally derived by Leipnik for arbitrary one-dimensional one-particle wave functions.
B. Damped Harmonic Oscillator (DHO)
The DHO is very important physical system in all physical systems defining an interaction
with its environment. The Lagrangian of the DHO is given by
L(x, ẋ, t) = eγt
ẋ2 − m
ω2x2 + j(t)x)
. (25)
Damped free particle kernel is
K(x, t; x0, 0) =
( γmeγt/2
4πih̄ sinh 1
( iγmeγt/2
4h̄ sinh 1
(x− x0)2
. (26)
The DHO kernel has the form [14]
K(x, t; x0, 0) =
( mωeγt/2
2πih̄ sinhωt
Scl(x, x0, t)
, (27)
or explicitly
K(x, t; x0, 0) =
( mωeγt/2
2πih̄ sinωt
(ax2 + 2bx20 + 2xx0c+ 2xd+ 2x0e− f)
. (28)
Where the coefficients a, b, c, d, f are [14]
a = (−γ
+ ω cotωt)eγt, (29)
b = (
+ ω cotωt), (30)
c = (−
sinωt
eγt), (31)
m sinωt
j(t′)eγt
′/2 sinωt′dt′, (32)
m sinωt
j(t′)eγt
′/2 sinω(t− t′)dt′, (33)
j(t′)j(s)eγ(s+t
′/2) sinω(t− t′) sinωsdsdt′. (34)
The wave function ψn(x, 0) and energy eigenvalues become
ψn(x, 0) = N0Hn(α0x) exp
h̄ω0 (36)
where Hn(x) is the Hermite polynomial of order n and the coefficients are
α0 = (
)1/2, N0 =
(2nn!
π)1/2
. (37)
The time dependent wave function is obtained as [14]
ψn(x, t) =
dx0K(x, t; x0, 0)ψ(x, 0)
(2nn!)1/2
cot−1×
+ cotωt+ f
exp[−(Ax2 +
+ 2Bx)]Hn[D(x− E)]. (38)
To simplify the evaluation, we set j(t) = 0. Such that kernel and wave function of the DHO
[15] become
K(x, t; x0, 0) =
( mωeγt/2
2πih̄ sinωt
γ(x20 − eγtx2) +
sinωt
× [(x20 + x2eγt) cosωt− 2eγt/2xx0]
where ω = (ω20 − γ2/4)1/2 and
ψn(x, t) =
(2nn!)1/2
cot−1
+ cotωt
Hn[Dx] exp[−Ax2]. (40)
Where D, A and N are
D(t) =
αeγt/2
η(t) sinωt
, (41)
η2(t) =
cosωt+ csc2 ωt, (42)
A(t) =
η2(t) sin2 ωt
− cotωt+ γ/2ω + cotωt
η2 sin2 ωt
, (43)
N(t) =
)1/4 exp(γt
η(t)(sinωt)1/2
. (44)
The ground state wave function is given by
ψ0(x, t) = N(t) exp
cot−1
+ cotωt
exp[−A(t)x2]. (45)
So the probability distribution in coordinate space becomes
|ψ0(x, t)|2 = N(t)2 exp[−2A′(t)x2] (46)
where A′ is defined by
A′(t) =
η2(t) sin2 ωt
. (47)
The probability density in coordinate space is shown in Fig.5 and Fig.6 for the different
values of γ. The probability density in momentum space can be written easily
|ψ0(p, t)|2 =
N(t)2
2A(t)A(t)†h̄
A′(t)
A(t)A(t)†
. (48)
The time dependent joint entropy can be obtained from Eq. (2) as
Sj(t) = N(t)
2A′(t)
(lnN(t)2 −
2A(t)A(t)†
N(t)2
2A(t)A(t)†
− ln 2π.(49)
The joint entropy depends on damping factor γ. When γ → 0, all the above results
are converged to simple harmonic oscillator. However, when the γ 6= 0, the joint entropy
has remarkably different features of the SHO. As can be seen in Fig.7 and Fig.8, the joint
entropy of the DHO has very interesting properties. One of the most important properties
of the joint entropy is the probability of taking values for small γ values. As we know
from literature the joint entropy must be positive and monotonically increase.However, this
system has different properties from literature because of periodically discontinuity of the
joint entropy. On the other hand, envelop of this curve is also monotonically increase with
time for large γ. As can be shown these results, the envelop of the joint entropy curves
has general properties as monotonically increase for quantum systems. Thus, we have found
that the joint entropy is depend on properties of investigated system.
IV. CONCLUSION
We have investigated the joint entropy for explicit time dependent solution of one-
dimensional harmonic oscillators. We have obtained the time dependent wave function by
means of Feynmann Path integral technique. Our results show that in the simple harmonic
oscillator case, the joint entropy fluctuated with time and frequency. This result indicates
that the information periodically transfer between harmonic oscillators.
On the other hand, in the DHO case, the joint entropy shows a remarkable smooth
discontinuities with time. It also depends on choice of initial values of parameter i.e. ω.
These results can be explained as the information exchange between harmonic oscillator
and system which is supplied damping. But the information exchange appears in certain
values of time for damping. If the damping factor increases, the information entropy has
not periodicity anymore. Moreover, for certain values of the damping factor, the transfer of
information between systems is exhausted.
V. ACKNOWLEDGEMENTS
This research was partially supported by the Scientific and Technological Research Coun-
cil of Turkey.
[1] E. Aydiner, C. Orta and R.Sever, E-print:quant-ph/0602203
[2] W.H. Zurek, Phys. Today 44(10), 36 (1991).
[3] R. Omnes, Rev. Mod. Phys. 64, 339 (1992).
[4] C. Anastopoulos, Ann. Phys. 303, 275 (2003).
[5] R. Leipnik, Inf. Control. 2, 64 (1959).
[6] V.V.Dodonov, J.Opt. B: Quantum Semiclassical Opt. 4, S98 (2002).
[7] S. A. Trigger, Bull. Lebedev Phys. Inst. 9, 44 (2004).
[8] J. Dunkel and S. A. Trigger, Phys. Rev.A71, 052102 (2005).
[9] P. Garbaczewski, Phys. Rev. A 72, 056101 (2005).
[10] M. Born, Z. Phys. 40, 167 (1926).
[11] R.P. Feynmann, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, USA
(1965).
[12] D.C. Khandekar, S.V. Lawande, K.V. Bhagwat, Path-Integral Methods and Their Applica-
tions, World Scientific, Singapore (1993).
[13] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial
Markets, World Scientific, 3rd Edition (2004).
[14] C.I. Um, K.H. Yeon and T.F.George,Physics Reports, 362,63-192 (2002).
[15] K.H. Yeon and C.I. Um, Phys. Rev. A 36,11 (1987).
http://arxiv.org/abs/quant-ph/0602203
ÈΨ@x,tDÈ2
FIG. 1: |Ψ(x, t)|2 versus time and coordinate
0 0.05
0.1 0.15
S jHtL
FIG. 2: The 3D graph of joint entropy of simple harmonic oscillator.
0 0.2 0.4 0.6 0.8 1
FIG. 3: The joint entropy of simple harmonic oscillator versus ω .
0 5 10 15 20
FIG. 4: The joint entropy of simple harmonic oscillator versus time.
ÈΨ0@x,tDÈ
FIG. 5: The probability function as a function of time and coordinate at γ = 0.1.
ÈΨ0@x,tDÈ
FIG. 6: The probability function as a function of time and coordinate at γ = 0.5.
FIG. 7: The 3D graph of the joint entropy of damped harmonic oscillator for damping factor(γ)
at ω0 = 2.
FIG. 8: The 3D graph of the joint entropy of damped harmonic oscillator for damping factor(γ)
at ω0 = 1.
	Introduction
	Fundamental Definitions
	CALCULATION AND RESULTS
	Simple Harmonic Oscillator (SHO)
	Damped Harmonic Oscillator (DHO)
	Conclusion
	Acknowledgements
	References
