PHYSICAL REVIEW A 71, 052102 共2005兲
Time-dependent entropy of simple quantum model systems
Jörn Dunkel* and Sergey A. Trigger
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
共Received 23 November 2004; published 10 May 2005兲
Information-theoretic entropy measures are useful tools for quantifying the spreading of quantum states in
phase space. In the present paper, we compare the time evolution of the joint entropy for three simple quantum
systems: 共i兲 a free Gaussian wave packet, 共ii兲 a wave packet in a monochromatic electromagnetic field, and 共iii兲
a wave packet tunneling through a ␦ barrier. As initial condition maximal classical states are used, which
minimize the Heisenberg uncertainty and the entropy. It is found that, in all three cases, the joint entropy
increases in time.
DOI: 10.1103/PhysRevA.71.052102
PACS number共s兲: 03.65.Ta, 03.67.⫺a, 05.30.⫺d, 03.65.Xp
I. INTRODUCTION
The transition between quantum and classical behavior of
physical systems has been in the center of intensive research
during the past decades 共for reviews, see 关1–4兴兲. New theoretical insights were complemented by modern experiments,
which have impressively confirmed quantum phenomena
such as the superposition of states 关5,6兴 or the existence of
Schrödinger cat states 关7,8兴. Thus, considerable progress has
been achieved in the conceptual understanding of measurement processes 关9–11兴 and decoherence phenomena 共see,
e.g., Zurek et al. 关12–14兴兲.
In order to characterize the intrinsic evolution of quantum
states, different information-theoretic entropy measures have
been discussed in the literature for both open and closed
quantum systems 关1,3,15–21兴. Among the best known examples are the von Neumann entropy, the Wehrl entropy, and
the joint entropy, introduced by Leipnik 关16兴. As discussed
by Anastopoulos and Halliwell 关19兴, the von Neumann entropy is particularly useful for studying environmentally induced effects in open quantum systems, because it is constant for unitary evolutions of pure states as typical of closed
systems. In contrast, Wehrl entropy 关19,20兴 and joint entropy
关16,21兴 can also be used to quantify the loss of information,
associated with temporally evolving pure quantum states
关22兴.
The objective of this paper is to explicitly illustrate this
fact by calculating the joint entropy for analytically tractable
quantum models. More precisely, we will compare the joint
entropy for the following three examples: 共i兲 the simplest
free Gaussian wave packet case 关22兴, 共ii兲 the evolution of
Gaussian wave packets in the presence of oscillating external
fields, and 共iii兲 the tunneling of a Gaussian wave packet
through a ␦ barrier.
Before discussing these examples in Sec. III, we shall
briefly review essential definitions and entropy concepts in
the next part.
number of spatial dimensions. Further, let f共t , x , p兲
= f共t , x1 , . . . , xd , p1 , . . . , pd兲 denote the non-negative, timedependent phase-space density function of this system. Assuming that f has been normalized to unity,
冕
the related 共dimensionless兲 Gibbs-Shannon entropy is defined by
S共t兲 = −
*Electronic address: dunkel@physik.hu-berlin.de
1050-2947/2005/71共5兲/052102共12兲/$23.00
1
N!
冕
dx dpf ln共hd f兲,
共1兲
where h = 2␲ប is the Planck constant. 共If boundaries are not
specified, then it is assumed throughout that integrals range
from −⬁ to +⬁.兲
If one is interested in generalizing the classical entropy
definition 共1兲 to quantum systems, then the main problem
consists in finding an appropriate quantum counterpart for
the classical phase-space density f. Different approaches to
this problem, which have become widely accepted nowadays, go back to alternative proposals made by von Neumann, Wehrl, and Leipnik 关16兴, respectively. To briefly illustrate the underlying ideas, let us consider a mixed quantum
system described by the normalized density matrix 共in
position-space representation兲
␳共t,x,x⬘兲 = 兺 w␣␺␣共t,x兲␺␣* 共t,x⬘兲,
␣
兺␣ w␣ = 1.
For example, in the case of the canonical equilibrium ensemble with temperature T, one usually assumes classical
probabilities
w␣ =
II. BASIC DEFINITIONS
Consider a classical 共nonquantum兲 system with d = DN degrees of freedom, where N is the particle number and D the
dx dpf共t,x,p兲 = 1,
exp共− ⑀␣/kBT兲
兺␣ exp共− ⑀␣/kBT兲
,
共2兲
where kB is the Boltzmann constant and ⑀␣ denotes the energy eigenvalues with normalized eigenfunctions ␺␣. In general, according to von Neumann’s definition, the quantum
analog of Eq. 共1兲 reads
052102-1
©2005 The American Physical Society
PHYSICAL REVIEW A 71, 052102 共2005兲
J. DUNKEL AND S. A. TRIGGER
SN共t兲 = − Sp共␳ ln ␳兲.
共3兲
A peculiar feature, which follows directly from definition
共3兲, is that, in the case of isolated systems, the von Neumann
entropy SN共t兲 is time independent for arbitrary nonequilibrium initial conditions—similar to the Gibbs entropy of classical isolated systems 关23兴.
In particular, for pure quantum states with w␣ = 0 for all
but one ␣, the von Neumann entropy vanishes identically—
i.e., SN共t兲 ⬅ 0 for all times t. Consequently, the von Neumann
entropy is useful, if one aims to quantify the specific gain
and loss of information associated with environmental effects 共change of the ẇ␣’s兲, but at the same time, SN共t兲 does
not capture the additional intrinsic variation of information,
as it already appears for temporally evolving pure quantum
states 共see examples discussed in Sec. III兲. For instance, one
possibility to obtain a time-dependent quantum entropy is to
consider a coarse-grained version of the density matrix ␳
关23兴.
An alternative description of the evolution of quantum
systems, leading to another quantum entropy definition, is
based on Wigner functions 关24,25兴. Given the matrix elements ␳共t , x , x⬘兲 of the statistical operator, the corresponding
Wigner function is defined by
f W共t,x,p兲 =
冕
冉
冊
dy eipy/ប
y
y
.
d ␳ t,x − ,x +
共2␲ប兲
2
2
冕
冉 冊冉 冊
dy eipy/ប
y
y
␺ t,x − ␺* t,x + .
共2␲ប兲d
2
2
By integrating f W over x or p, respectively, the quantum
probability densities are regained. Moreover, one can reconstruct the pure quantum state ␺ from f W; i.e., the quantum
system is completely described by its Wigner function
关8,26兴.
However, as well known, due the noncommutativity of
coordinates and momenta in quantum theory, the function f W
in general takes negative values. In order to obtain a nonnegative function, Husimi proposed to convolute f W with the
Gaussian kernel
G共x,p;x⬘,p⬘兲 =
冋
册
1
共x − x⬘兲2 共p − p⬘兲2
−
.
d exp −
共2␲␴ p␴x兲
2␴2x
2␴2p
If the uncertainty relation ␴ p␴x 艌 ប / 2 is fulfilled, then the
resulting convoluted Husimi function
f H共t,x,p兲 =
冕
dx⬘ dp⬘G共x,p;x⬘,p⬘兲f W共t,x⬘,p⬘兲
iប
⳵␺
= Ĥ␺,
⳵t
共5兲
is non-negative 关27兴. Choosing ␴ p and ␴x such that ␴ p␴x
= ប / 2 holds and replacing the classical distribution function f
in Eq. 共1兲 by the Husimi function f H, one obtains the so-
Ĥ = −
ប2 2
ⵜ + U共t,x兲.
2m
共6兲
According to the Born interpretation 关28兴, the functions
兩␺共t , x兲兩2 and 兩˜␺共t , p兲兩2, where
共4兲
In analogy to the classical distribution function f, the Wigner
function f W can be used to calculate quantum corrections for
classical thermodynamical quantities 共in the hightemperature limit兲. In the limit case h → 0, the Wigner function converges to the classical distribution function. For pure
states with ␳ = ␺␺*, the Wigner function can be represented
through the wave function in the form
f W共t,x,p兲 =
called Wehrl entropy SW共t兲. The properties of SW共t兲 have
been extensively studied by Anastopoulos and Halliwell in
the context of quantum Brownian motion 关19,20兴. Nevertheless, one should emphasize that, by construction, the Wehrl
entropy contains an additional free parameter 共␴x or ␴ p兲.
This implies that SW共t兲 cannot represent a “fundamental” entropy definition for quantum systems.
As already pointed out in the Introduction, the objective
of the present paper is to focus on the loss of information,
associated with the quantum evolution of pure states. According to the above discussion, neither the von Neumann
entropy SN 共which is constant for pure states兲 nor the Wehrl
entropy SW 共which contains an additional arbitrary parameter兲 provides satisfactory information-theoretic measures for
this purpose. Therefore, we shall pursue below another approach, which goes back to Leipnik 关16兴 and can be described as follows: Consider a normalized quantum state
␺共t , x兲 governed by the Schrödinger equation
˜␺共t,p兲 =
冕
dx e−ipx/ប
␺共t,x兲,
共2␲ប兲d/2
共7兲
give the quantum probability densities for position and momentum coordinates, respectively. Leipnik proposed to consider the product function 共joint distribution for pure states兲
f J共t,x,p兲 = 兩␺共t,x兲兩2兩˜␺共t,p兲兩2 艌 0.
共8兲
Similar to the Wigner function f W, also the function f J characterizes the evolution of quantum states in phase space,
even though it contains less information than f W. Inserting
Eq. 共8兲 into Eq. 共1兲 yields the joint entropy SJ共t兲 for the pure
state ␺共t , x兲, which can equivalently be rewritten in the form
SJ共t兲 = −
冕
dx兩␺共t,x兲兩2 ln兩␺共t,x兲兩2 −
− ln hd .
冕
dp兩˜␺共t,p兲兩2 ln兩˜␺共t,p兲兩2
共9兲
In the next section, the quantity SJ共t兲 is studied for three
simple model systems.
III. JOINT ENTROPY FOR ONE-DIMENSIONAL MODEL
SYSTEMS
As recently discussed in 关22兴, the joint entropy SJ共t兲 provides a useful time-dependent measure for the intrinsic information loss of temporally evolving, pure quantum states
共e.g., for the simplest case of free d-dimensional Gaussian
wave packets it was shown that SJ共t兲 grows monotonously
for t ⬎ 0兲. Furthermore, it was proposed in 关22兴 that this
“quantum trend,” driven by Heisenberg’s uncertainty principle, could be a rather universal property of pure quantum
systems. Here we shall pursue this idea further by also considering the influence of external potentials on the evolution
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PHYSICAL REVIEW A 71, 052102 共2005兲
TIME-DEPENDENT ENTROPY OF SIMPLE QUANTUM…
of initially maximal classical states 共MACS’s兲.
In Sec. III A we briefly review some essential results discussed in 关22兴, restricting ourselves to the case d = 1. Subsequently, the investigation will be extended to Schrödinger
problems with nonvanishing external potentials 共Secs. III B
and III C兲.
A. Free motion
For U ⬅ 0 and d = 1 the Schrödinger equation 共6兲 has the
normalized Gaussian wave packet solution 共see, e.g.,
关29,30兴兲
冉 冊
␣
␺ f 共t,x兲 =
2␲
1/4 i共k0x−␤k20t兲
e
冑␣ + i␤t
冋
exp −
册
共x − v0t兲2
, 共10兲
4共␣ + i␤t兲
where ␤ = ប / 共2m兲 and 冑␣ is the initial width of the wave
packet. The solution 共10兲 yields
⌬p共t兲 = ប/共2冑␣兲,
具p共t兲典 = p0 = បk0,
⌬x共t兲 = 冑␣ + ␤2t2/␣ .
具x共t兲典 = v0t = p0t/m,
共11兲
关Here, as usual, the standard deviation of an observable A is
defined by ⌬A共t兲 = 冑具A2共t兲典 − 具A共t兲典2, where 具·典 denotes the
expectation with respect to ␺.兴 From Eqs. 共11兲 one finds
immediately the time- dependent uncertainty relation 关29兴
⌬x共t兲⌬p共t兲 =
冑
ប
2
1+
␤ 2t 2 ប
艌 .
␣2
2
共12兲
This implies that at time t = 0 the solution 共10兲 is a MACS,
since only then does the left-hand side 共LHS兲 equal the RHS
in Eq. 共12兲. Using Eqs. 共10兲, 共7兲, and 共9兲, a straightforward
calculation of the joint entropy gives 关22兴
冋冑
SJf共t;d = 1兲 = ln
e
2
1+
册
␤ 2t 2
,
␣2
共13兲
system. Since Eqs. 共12兲 and 共13兲 are equivalent for free
Gaussian wave packets, the joint entropy contains the same
amount of information as the uncertainty relation; for more
complicated problems, this will not be the case anymore,
because in general for t ⬎ 0 the wave functions will essentially deviate from the initial Gaussian shape.
Although the above free wave packet example is rather
simple, it is helpful to elucidate some general properties of
the joint entropy. For more complicated problems with nonvanishing potentials U共t , x兲, an exact calculation of the joint
entropy becomes more difficult or in most cases—especially
in two or three spatial dimensions—even impossible. Therefore, when including external fields in the remainder, we
continue to concentrate on analytically tractable problems
with d = 1. The results will be compared with those just discussed for one-dimensional free wave packet.
B. Oscillating external fields
As slightly more complicated examples, which are still
exactly solvable, we next consider the time-periodic potentials
x⬘ = x
SJf共t;d = D兲 = D SJf共t;d = 1兲.
冉冊
SJ共t兲 艌 ln
e
,
2
共14兲
originally derived by Leipnik 关16兴 for arbitrary onedimensional one-particle wave functions. Another important
feature of the result 共13兲 is the monotonous increase of joint
entropy in time. In 关22兴 it was suggested that this is a general
property of quantum systems. Below we will demonstrate
that, at least asymptotically, this property also holds for more
complicated quantum systems.
Furthermore, according to Eqs. 共12兲 and 共13兲, the MACS
property is not conserved in time; the logarithmic divergence
of the joint entropy reflects a permanent loss of information,
representing an intrinsic property of the evolving quantum
共15a兲
Uc共t,x兲 = qE0x cos共⍀t兲.
共15b兲
These potentials describe, in the dipole approximation 关31兴,
the interaction between a quantum particle with electric
charge q and an oscillating, monochromatic electric field
with amplitude E0 and frequency ⍀. Analytic solutions of the
corresponding relativistic quantum wave equations were first
discussed by Gordon 关32兴 and Volkov 关33兴 about 70 years
ago. Recently, however, this problem has reattracted considerable theoretical interest 关34–38兴. Here we will use results
for the corresponding Schrödinger equation 共6兲, as derived
by Rau and Unnikrishnan 关34兴.
Before discussing the solutions, it is useful to transform to
rescaled dimensionless quantities
where e = 2.7182. . . is the Euler number. For completeness,
we mention that for a D-dimensional free wave packet one
finds, correspondingly,
It is worthwhile to note that Eq. 共13兲 is in agreement with
the following general inequality for the joint entropy:
Us共t,x兲 = qE0x sin共⍀t兲,
E⬘0 =
冉 冊
m⍀
ប
1/2
,
qE0
,
共M⍀3ប兲1/2
p⬘ =
p
,
共M⍀ប兲1/2
␺⬘共t⬘,x⬘兲 =
冉 冊
m⍀
ប
t⬘ = ⍀t,
−1/4
␺共t,x兲.
Formally, this corresponds to fixing characteristic units such
that ប = m = ⍀ = q = 1. Dropping for convenience the primes
again, the rescaled Schrödinger equation 共6兲 takes the simplified form
i
1 ⳵2
⳵␺
=−
␺ + Us/c共t,x兲␺ ,
2 ⳵x2
⳵t
共16兲
where the rescaled potentials read
Us共t,x兲 = E0x sin t,
共17a兲
Uc共t,x兲 = E0x cos t.
共17b兲
First, let us consider Us from Eq. 共17a兲. According to
formula 共19兲 of Ref. 关34兴, the time-dependent fundamental
solution of Eq. 共16兲 is given by
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PHYSICAL REVIEW A 71, 052102 共2005兲
J. DUNKEL AND S. A. TRIGGER
冋 冉
␾sp共t,x兲 = exp − iE20
冊册
冋 冉 冊册
1
3
sin 2t − sin t + t
8
4
冋 冉
␾cp共t,x兲 = exp − iE20 −
exp关− iE0x共1
− cos t兲 − iE0 p共sin t − t兲兴exp i px −
p2
t
2
. 共18兲
From this solution, one can construct special solutions via
superpositioning. As in the previous example, we would like
to consider solutions which are MAC states at initial time t
= 0. This can be achieved by setting
冉 冊 冕
␣
␺ 共t,x兲 =
2␲3
s
1/4
dp
2
e−␣共p − p0兲 ␾sp共t,x兲,
冉 冊
␣
2␲
冑␣ + it/2
再
再 冋 冉
⫻ exp − iE20
冋
2␣
␲共4␣2 + t2兲
再
⫻exp −
冉 冊
2␣
␲
册
冊册
共23兲
e
再
冋
冑␣ + it/2
关x + E0共t − cos t兲 − 2i␣ p0兴2
4共␣ + it/2兲
册
冎
i
⫻ exp − E20共t − cos t sin t兲 .
4
This gives the densities
兩␺ c共t,x兲兩2 =
冋
2␣
␲共4␣2 + t2兲
再
exp −
. 共20兲
From this solution one finds the two normalized quantum
probability densities
兩␺s共t,x兲兩2 =
p2
t
2
1/4 −␣ p20−iE0x sin t
⫻ exp −
关x + E0共t − sin t兲 − 2i␣ p0兴2
⫻ exp −
4共␣ + it/2兲
3
1
t + cos t − 1 sin t
4
4
冉 冊
␣
␺ 共t,x兲 =
2␲
c
共19兲
冎
冊 册冎
exp关− iE0x sin t
instead of Eq. 共18兲. Substituting indices s by c in Eq. 共19兲,
one then obtains
1/4 −␣ p20−iE0x共1−cos t兲
e
冋冉
冊册
− iE0 p共cos t − 1兲兴exp i px −
where the parameter p0 determines the initial momentum.
The integral 共19兲 can be calculated analytically, yielding
␺s共t,x兲 =
1
1
sin 2t + t
8
4
兩␺˜ c共t,p兲兩2 =
冉 冊
2␣
␲
册
共24兲
1/2
冎
2␣关x − p0t + E0共1 − cos t兲兴2
,
4␣2 + t2
1/2
exp关− 2␣共p − p0 + E0 sin t兲2兴,
共25兲
1/2
yielding
冎
2␣关x − p0t + E0共t − sin t兲兴2
,
4␣2 + t2
具x共t兲典 = p0t − E0 + E0 cos t,
具p共t兲典 = p0 − E0 sin t,
1/2
共26兲
Obviously, each of the two densities in Eq. 共21兲 corresponds
to a Gaussian with oscillating mean value. More precisely,
we find
while, for the standard deviations, Eqs. 共22b兲 are regained.
In summary, if identical MACS initial conditions are considered, then—compared with the free wave packet case—
the presence of the monochromatic oscillating fields is neither reflected by the uncertainty relation nor by the joint
entropy 共in spite of the fact that the joint probability funcf
tions f c/s
J and f J are different兲.
具x共t兲典 = 共p0 − E0兲t + E0 sin t,
C. Tunneling through a ␦-peak potential
兩˜␺s共t,p兲兩2 =
exp兵− 2␣关p − p0 + E0共1 − cos t兲兴2其.
共21兲
具p共t兲典 = 共p0 − E0兲 + E0 cos t,
and, furthermore,
⌬x共t兲 =
冉
1
t2
4␣ +
2
␣
冊
共22a兲
As third example, consider the Schrödinger equation 共6兲
with d = 1 and time-independent potential
U共x兲 = U0␦共x兲,
1/2
,
⌬p共t兲 =
1
2 冑␣
.
共22b兲
Thus, after reinstating quantities ប, m, ⍀, and q one arrives
at exactly the same uncertainty relation as found earlier for
the free particle; see Eq. 共12兲. Moreover, by explicitly calculating the joint entropy on the basis of Eq. 共21兲, one can
show that Ss/c
J 共t兲 coincides with the free-particle result 共13兲.
It comes as no surprise that an analogous result is obtained, if one considers Uc from Eq. 共17b兲. According to
formulas 共22兲 and 共23兲 in 关34兴, for Eq. 共17b兲 one must use
the fundamental solution
U0 ⬎ 0,
共27兲
corresponding to an infinitely high and infinitely thin barrier
at x = 0. Similar to before, we take as initial condition a
Gaussian MACS
␺共0,x兲 =
冉 冊 冋
1
2␲␣
1/4
exp ik0x −
册
共x − x0兲2
,
4␣2
共28兲
where it is further assumed that
具x共0兲典 = x0 ⬍ 0,
具p共0兲典 = បk0 ⬎ 0,
and
052102-4
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PHYSICAL REVIEW A 71, 052102 共2005兲
TIME-DEPENDENT ENTROPY OF SIMPLE QUANTUM…
FIG. 1. Tunneling of a MAC state through the potential barrier U共x兲 = U0␦共x兲: Evolution of the spatial probability density 兩␺共t , x兲兩2 for
“large” initial momentum p0. Solid lines correspond to the “exact” solution 共31兲 and dashed lines to the approximate result 共32兲. Time t is
measured in units ␶ = m␣ / ប and x in units 冑␣, where 冑␣ determines the initial width of the wave packet. The remaining parameter values are
x0 = −10, k0 = p0 / ប = 5, and U0 = 1 with units given by 关x0兴 = 冑␣, 关k0兴 = 1 / 冑␣, and 关U0兴 = ប冑␣ / ␶.
冏 冏 冏冑 冏
x0
=
⌬x共0兲
x0
␣
Ⰷ1
共29b兲
hold. The conditions 共29a兲 mean that the maximum of wave
packet approaches the barrier from the left, while Eq. 共29b兲
states that, at time t = 0, the particle is 共almost兲 surely localized on the left-hand side of the barrier. For fixed initial
position x0 ⬍ 0 and initial momentum p0 = បk0 ⬎ 0, the time
of the collision with the barrier can be defined as
1. Evolution of the spatial density
If the condition 共29b兲 is satisfied, then one finds the following integral representation for the related time-dependent
solution 共see the Appendix for details of the calculation兲:
␺共t,x兲 ⯝
冉 冊冕
␣
2␲3
⫻
兩x0兩 m兩x0兩
tc =
=
.
p0/m
បk0
共30兲
As we shall see below, the quantity tc can be used to identify
peculiarities in the entropy curves SJ共t兲.
冋
1/4
⬁
−⬁
2
d ke−␣共k0 − k兲 ei共k0−k兲x0−i␤k
2t
册
2␬
k
exp共ikx兲 + ⌰共− x兲
sin共kx兲 .
k − i␬
k − i␬
共31兲
Here ⌰共x兲 denotes the unit-step function, defined in Eq.
共A28兲, and the parameter
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PHYSICAL REVIEW A 71, 052102 共2005兲
J. DUNKEL AND S. A. TRIGGER
FIG. 2. Tunneling through the ␦ barrier for “moderate” initial momentum p0 = 1 共units and all other parameters are chosen as in Fig. 1兲.
Here the transmission 共tunneling兲 probability approximately equals the reflection probability. As a consequence, for t ⬎ tc the joint entropy
is much larger than in the case of “large” and “small” initial momenta; compare Fig. 4.
␬=−
mU0
ប2
␺0共t,x兲 ⯝
␣
2␲
1/4
冋
⫻ exp −
characterizes the height of the barrier. Unfortunately, because
of the prefactors
k
c1共k兲 =
,
k − i␬
冉 冊冉
冋 冉
1
␣ + i␤t
the integral 共31兲 cannot be solved exactly. With regard to
subsequent 共numerical兲 calculations of the joint entropy, it
turns out as useful to make an additional approximation by
replacing in Eq. 共31兲 the k-dependent functions c1/2共k兲 with
the constants c1/2共k0兲. These approximations are reasonable
for sufficently large parameter values ␣ in the Gaussian exponential of the integral 共31兲, and one then obtains the approximate solution
1/2
eik0x0
k0 − i␬
册再
冊 册冎
共x − x0兲2 + 4i␣k20␤t
4共␣ + i␤t兲
⫻␬ exp −
2␬
c2共k兲 =
,
k − i␬
冊
x0x + 2i␣k0x
−1
␣ + i␤t
k0 + i⌰共− x兲
.
共32兲
Figures 1–3 show the evolution of the spatial probability
density 兩␺共t , x兲兩2 for three different choices of the quotient
k0 / 兩␬兩, characterizing the ratio between initial momentum
and barrier height. In each diagram solid lines were numerically calculated using the “exact” solution 共31兲, while dashed
lines correspond to the approximate solution ␺0 from Eq.
共32兲. As one can deduce from these figures, ␺0 provides indeed a useful approximation of the more exact solution 共31兲.
Moreover, one observes that the deviation between Eqs. 共31兲
and 共32兲 decreases for k0 Ⰷ 兩␬兩; e.g., for k0 / 兩␬兩 = 5 共see Fig. 1兲,
052102-6
PHYSICAL REVIEW A 71, 052102 共2005兲
TIME-DEPENDENT ENTROPY OF SIMPLE QUANTUM…
FIG. 3. Tunneling through the ␦ barrier for “small” initial momentum p0 = 0.5 共units and all other parameters are chosen as in Figs. 1 and
2兲. Here the reflection probability is much larger than the tunneling probability. Moreover, one can see that the deviation of the approximate
solution 共32兲 from the “exact” solution 共31兲 becomes more significant, if the initial momentum is small compared to the effective barrier
height 兩␬兩.
the dashed and solid lines are nearly indistinguishable.
˜␺ 共t,p兲 =
0
冉 冊
␣
2 ␲ប2
3
2. Joint entropy
In order to calculate the joint entropy SJ共t兲 via Eq. 共9兲,
one requires the Fourier-transformed wave function ˜␺共t , p兲.
For the wave ␺共t , x兲 from Eq. 共31兲 it is difficult to find an
explicit formula for ˜␺共t , p兲. Hence, it is also rather difficult
to calculate SJ共t兲 from the integral representation 共31兲. However, as we have seen in the preceding section, the approximate solution ␺0 from Eq. 共31兲 provides a useful estimate of
the true solution. Therefore, instead of Eq. 共31兲, we will restrict ourselves below to considering the joint entropy for the
approximate solution 共32兲.
For ␺0共t , x兲 from Eq. 共32兲 the related momentum wave
function ˜␺0共t , p兲 can be easily calculated from Eq. 共7兲, and
one finds
1/4 i共p0−p兲x0/ប
e
p0 − iប␬
冋
␣共p − p0兲2 + ip2␤t
ប2
⫻ exp −
再
册
⫻ 2p0 + iប␬关Erf共␥−兲 − 1兴
冉
+ iប␬ exp
冎
冊
2ipx0 4␣ pp0
−
关Erf共␥+兲 + 1兴 ,
ប
ប2
共33兲
where Erf共x兲 denotes the error function, defined in Eq.
共A17兲, and the abbreviations
052102-7
␥−共t,p兲 =
បx0 + 2p␤t + 2i␣共p0 − p兲
2ប冑␣ + i␤t
,
PHYSICAL REVIEW A 71, 052102 共2005兲
J. DUNKEL AND S. A. TRIGGER
maximum for the three non-solid curves in Fig. 4—i.e.,
冏 冏
d 2S J
dt2
FIG. 4. MACS tunneling through the barrier U0␦共x兲: timedependent joint entropy for different initial momenta p0 = បk0,
where 关k0兴 = 1 / 冑␣. The curves were obtained by numerical integration of Eq. 共9兲 for the approximate solution 共32兲. Parameter values
and units were chosen as in Figs. 1–3; i.e., we have set x0 = −10 and
U0 = 1, where 关x0兴 = 冑␣, 关U0兴 = ប冑␣ / ␶ and ␶ = m␣ / ប. For comparison, we also plotted the joint entropy 共13兲 for the spreading of free
wave packet 共solid line兲. The three nonsolid curves exhibit locally
maximal slopes in the vicinity of the collision time tc = 兩x0兩 / p0. Also
note the appearance of the plateaulike region 共arrow兲 for small initial momenta.
␥+共t,p兲 =
បx0 − 2p␤t + 2i␣共p0 + p兲
2ប冑␣ + i␤t
have been introduced. On the basis of formulas 共32兲 and
共33兲, it is now straightforward to numerically calculate the
related joint entropy SJ共t兲 from Eq. 共9兲. In Fig. 4, one can see
examples of entropy curves, based on the same parameter
values as used in Figs. 1–3. Analogous to preceding diagrams, the representation in Fig. 4 refers to the characteristic
unit system defined by m = ប = 冑␣ = 1, where 冑␣ is the initial
width of the wave packet. For example, we simply have ␤
= 1 / 2, ␬ = −U0, etc., in these units. Furthermore, the related
unit time reads ␶ = m␣ / ប and the collision time tc, defined in
Eq. 共30兲, reduces to tc = 兩x0兩 / p0.
The joint entropy curves SJ共t兲 in Fig. 4 were obtained by
numerically integrating Eq. 共9兲 with the computer software
Mathematica 关39兴. The three nonsolid lines refer to the tunneling through the ␦ barrier. For comparison, we also the
plotted the joint entropy 共13兲 for the spreading of free wave
packet 共solid line兲. The main observations in Fig. 4 are the
following
共i兲 For t → 0, all four curves converge to the value SJ共0兲
= ln共e / 2兲 = 0.306. . .; compare Eq. 共13兲. This reflects the fact
that we have chosen special initial conditions, corresponding
to MAC states.
共ii兲 For t ⬍ tc all three nonsolid curves in Fig. 4 run below
the solid 共free wave packet兲 entropy curve SLf共t兲. This reflects
the fact that, at the early stages of the tunneling process, the
␦ barrier hinders the spreading of the wave packet. In particular, for small initial momenta a plateaulike region appears, indicated by the arrow 关but it should also be kept in
mind that for k0 Ⰶ 兩␬兩 the approximate solution 共32兲 becomes
less reliable; see Fig. 3兴.
共iii兲 At t ⬇ tc the slope of the joint entropy exhibits a local
= 0,
t⬇tc
冏 冏
d 3S J
dt3
⬍ 0.
t⬇tc
共iv兲 For t ⬎ tc the curve for “p0 = 1,” corresponding to a
moderate initial momentum value, clearly dominates the
other curves. Roughly speaking, this can be explained by the
fact that for moderate initial momenta the tunneling probability approximately equals the reflection probability 共compare Fig. 2兲, thus leading to maximum uncertainty.
共v兲 Each of the shown curves increases monotonously in
time. However, the appearance of the plateaulike region 共arrow兲 at low initial momenta seems to indicate that confinement effects due to interactions or external potentials might
also lead to a temporary decrease of the joint entropy on
short time scales. Of course, in such a scenario, Leipnik’s
inequality 共14兲 constitutes a lower bound for SJ共t兲.
IV. SUMMARY
We have studied the joint entropy for explicit timedependent solutions of three simple one-dimensional
Schrödinger problems: 共i兲 the spreading of a free Gaussian
wave packet, 共ii兲 the motion of a wave packet in a monochromatic electromagnetic field, and 共iii兲 the tunneling of a
wave packet through a ␦ barrier. The second example can be
considered as an open quantum system, because in this case
the Hamiltonian is explicitely time dependent. In contrast,
examples 共i兲 and 共iii兲 correspond to closed systems. As initial
conditions maximal classical states have been used. MACS’s
minimize the Heisenberg uncertainty as well as the joint entropy. A quantum system that has been in a MACS at time
t = 0 inevitably evolves into a non-MACS at times t ⬎ 0. This
intrinsic property of quantum systems is, e.g., reflected by a
monotonous increase of the joint entropy. Most likely, this
quantum trend also manifests itself for other type of initial
wave packets and external potentials, as well as in manyparticle systems.
In order to be able to calculate the joint entropy for the
tunneling process through a ␦ peak potential 关example 共iii兲兴,
an approximate analytic solution of the corresponding timedependent Schrödinger problem was derived 共Appendix兲. By
means of this solution, it could be shown that the slope of the
corresponding joint entropy exhibits a local maximum in the
vicinity of the collision time. Based on this observation one
may conclude that, on many occasions, interactions tend to
speed-up the joint entropy increase 共i.e., the loss of phasespace information兲 in quantum systems. With respect to
many-particle systems, an interesting question to be answered in the future is whether or not such quantum trends
are important for the relaxation to thermodynamic equilibrium.
ACKNOWLEDGMENTS
The authors are grateful to M. Fedorov, S. Hilbert, and O.
Weckner for valuable discussions. One of the authors
共S.A.T.兲 wishes to thank A. P. Delis, P. B. Vlassuk, and A. A.
052102-8
PHYSICAL REVIEW A 71, 052102 共2005兲
TIME-DEPENDENT ENTROPY OF SIMPLE QUANTUM…
Zarankin for stimulating support and The Netherlands Organization for Scientific Research 共Grant No. 047.016.020兲.
APPENDIX: SOLVING SCHRÖDINGER’s EQUATION
FOR THE ␦ BARRIER
We wish to find the time-dependent solution ␺共t , x兲 for the
one-dimensional Schrödinger problem
冋
册
␺共0,x兲 =
冉冑 冊 冋
1
exp ik0x −
␲␭
共x − x0兲
2␭2
2
␾k⬘共0+兲 − ␾k⬘共0−兲 =
册
.
共A2兲
冉冊
1
冑␲ ␭
冋
exp −
册
共x − x0兲2
,
␭2
冕
⬁
1=
冕
冕
共A4兲
⬁
冕
dx ␺*共0,x兲x␺共0,x兲 = x0 ,
共A5a兲
⬁
dx ␺*共0,x兲
⳵␺
共0,x兲 = បk0 ,
⳵x
共A5b兲
and the initial width of the wave packet reads
⌬x共0兲 ⬅ 冑具x2共0兲典 − 具x共0兲典2 =
␭
␺共t,x兲 =
2m
+
共A11c兲
⬁
−
dx ␾+k 共x兲␾k⬘共x兲 = 0.
冕
⬁
dk关a+共k兲␾+k 共x兲 + a−共k兲␾−k 共x兲兴e−iE共k兲t/ប .
冑2
共A12兲
The restriction to non-negative k values suffices here, be±
共x兲 holds. Furthermore, at time t = 0 the
cause ␾±k 共x兲 = ± ␾−k
condition
共A5c兲
.
␺共0,x兲 =
冕
⬁
dk关a+共k兲␾+k 共x兲 + a−共k兲␾−k 共x兲兴
共A13兲
0
must be satisfied. Consequently, the coefficients a±共k兲 are
determined by projection onto the initial condition:
The constant energy of the wave packet is given by
具Ĥ共t兲典 =
冕
0
−⬁
ប2k20
共A11b兲
−
−⬁
−⬁
具p共0兲典 ⬅ − iប
dx ␾−k 共x兲␾k⬘共x兲 = ␦共k − k⬘兲,
Hence, one can expand the solutions of the time-dependent
Schrödinger problem 共A1兲 in the form
Furthermore, we have at initial time t = 0
冕
共A10兲
共A11a兲
⬁
−⬁
具x共0兲典 ⬅
mU0
.
ប2
+
−⬁
dx ␺ 共0,x兲␺共0,x兲.
␬=−
dx ␾+k 共x兲␾k⬘共x兲 = ␦共k − k⬘兲,
−⬁
共A3兲
*
共A9兲
For k , k⬘ ⬎ 0 the solutions 兵␾±k 其 satisfy the orthonormality
relations
which is normalized to unity:
⬁
␬
,
k
␩共k兲 = arctan
This corresponds to the initial Gaussian probability density
兩␺共0,x兲兩2 =
2mU0
␾k共0兲,
ប2
one finds that for the phase of the symmetric solutions
共A1兲
where U0 ⬎ 0 and the initial condition is fixed as
1/2
共A8c兲
Note that symmetric and antisymmetric solutions are labeled
by “⫹” and “⫺,” respectively. From the matching condition
1. The problem
⳵
ប2 ⳵2
iប ␺ = −
+ U0␦共x兲 ␺ ⬅ Ĥ␺ ,
⳵t
2m ⳵x2
ប 2k 2
⬎ 0.
2m
E共k兲 =
冉 冊
ប2
U0
exp −
+
4m␭2 冑␲␭
␭
x20
2
.
共A6兲
a±共k兲 =
冕
⬁
dx ␾±k 共x兲␺共x,0兲.
共A14兲
−⬁
2. Superposition of eigenfunctions
The eigenfunctions of the related stationary Schrödinger
eigenvalue problem
Ĥ␾共x兲 = E␾共x兲
共A7兲
read 共see, e.g., p. 147 in Ref. 关29兴兲
␾+k 共x兲
=
1
冑␲ cos关k兩x兩 + ␩共k兲兴,
␾−k 共x兲 =
with eigenvalues given by
1
冑␲
sin共kx兲,
This follows from the orthonormality relations 共A11兲. Using
the special initial condition 共A2兲, one finds
where
共A8a兲
b+共k兲 ⬅
冋冑
a+共k兲 = b+共k兲 − b+共− k兲,
共A15a兲
a−共k兲 = b−共k兲 − b−共− k兲,
共A15b兲
␭
2 ␲共k2 + ␬2兲
冉
冋
⫻ k − i␬ Erf
共A8b兲
and
052102-9
册 冋
1/2
exp i共k0 − k兲x0 −
x0 + i共k0 − k兲␭2
冑2␭
册冊
,
␭2
共k0 − k兲2
2
册
共A16a兲
PHYSICAL REVIEW A 71, 052102 共2005兲
J. DUNKEL AND S. A. TRIGGER
b−共k兲 ⬅ i
冉冑冊 冋
␭
1/2
exp i共k0 − k兲x0 −
2 ␲
册
␭2
共k0 − k兲2 .
2
I−共0,x兲 =
冑 冕
␲
z
2
共A17兲
dt e−t .
0
Equations 共A15兲 and 共A16兲 further imply
a±共k兲 = − a±共− k兲.
共A18兲
4 冑␲ ␭
− exp −
The error function in Eq. 共A16a兲 is defined by
2
ប
,
2m
x0 ⬍ 0,
␺共t,x兲 = I+共t,x兲 + I−共t,x兲,
共A20兲
where
2
dk a+共k兲␾+k 共x兲e−i␤k t ,
冕
2
dk a−共k兲␾−k 共x兲e−i␤k t .
共A21b兲
By using Eq. 共A15b兲 and inserting the explicit expression for
␾−k 共x兲, we find
冕
冑 冕
冑 冕
1
I−共t,x兲 =
冑␲
=
2 ␲
␲
共A16b兲
= i
⬁
dk 关b−共k兲 − b−共− k兲兴sin共kx兲e
dkb−共k兲sin共kx兲e−i␤k
冉冑 冊 冕
冋
2 ␲
1/2
3
2t
冉冑冊
␭
b+共k兲 ⯝
冑k2 + ␬
冑 冕
1
I+共t,x兲 =
␲
=
册
␭
共k0 − k兲2 − i␤k2t .
2
2
The integral 共A22兲 can be calculated:
冋
− exp −
where, in particular,
册 冉
1/2
4 ␲共␭2 + 2i␤t兲
再 冋
x,y 苸 R,
␭2k20
exp ik0x0 −
2
1 共x0 − x + ik0␭2兲2
2
␭2 + 2i␤t
1 共x0 + x + ik0␭2兲2
2
␭2 + 2i␤t
冋
exp i共k0 − k兲x0 −
2
⬁
共A24兲
册
册冎
冊
册
␭2
共k0 − k兲2 .
2
dk关b+共k兲 − b+共− k兲兴e−i␤k
0
冋
冉 冊册
⫻ cos k兩x兩 + arctan
2t
−⬁
⫻ exp −
共A23兲
Using this approximate expression, we can rewrite I+共t , x兲
from Eq. 共A21a兲. Inserting Eq. 共A15a兲 and the explicit expression for ␾+k 共x兲, we find
−i␤k2t
冑 冕
1
␲
⫻
共A22兲
冋冑
Ⰷ 1.
共A25兲
dk sin共kx兲
␭
k + i␬
1/2
2 ␲
⬁
⫻ exp i共k0 − k兲x0 −
I−共t,x兲 =
2␭
lim Erf共x + iy兲 = − 1,
−⬁
␭
x0
x→−⬁
−⬁
⬁
1
=
dk关b−共k兲 − b−共− k兲兴sin共kx兲e−i␤k
0
1
冏冑 冏
k0 ⬎ 0,
we obtain in the limit case 共A23兲 from Eq. 共A16a兲
⬁
0
⬁
.
This means that 共i兲 the wave packet approaches the barrier
from the left and 共ii兲 the spatial probability is initially concentrated in the region x ⬍ 0. By virtue of
共A21a兲
0
I−共t,x兲 ⬅
册冎
In the following we shall consider the limit case
共A19兲
the general solution 共A12兲 can be written as
⬁
共x0 + x兲2
− ik0x
2␭2
册
3. Limit case 円x0円 š ␭
␤⬅
冕
共x0 − x兲2
+ ik0x
2␭2
Unfortunately, because of the error function in Eq. 共A16a兲,
the remaining integral I+共t , x兲 cannot explicitly be solved.
Therefore, we shall next try to find a convenient integral
representation of the solution ␺共t , x兲 by considering a special
limit case.
Introducing the abbreviation
I+共t,x兲 ⬅
exp −
冋
共A16b兲
Erf共z兲 =
冉 冊再 冋
1/2
1
冋
⬁
␬
k
,
dk关b+共k兲 − b+共− k兲兴e−i␤k
0
k cos共k兩x兩兲 − ␬ sin共k兩x兩兲
冑k2 + ␬2
册
2t
2t
.
Since the integrand is an even function with respect to k, we
obtain
I+共t,x兲 =
冑 冕
1
2 ␲
⫻
=
,
冋
␲
dk关b+共k兲 − b+共− k兲兴e−i␤k
−⬁
k cos共k兩x兩兲 − ␬ sin共k兩x兩兲
冑 冕
1
⬁
冑k2 + ␬2
⬁
−⬁
dk b+共k兲e−i␤k
2t
冋
册
k cos共k兩x兩兲 − ␬ sin共k兩x兩兲
Inserting Eq. 共A25兲 for b+共k兲, we have
052102-10
2t
冑k2 + ␬2
册
.
PHYSICAL REVIEW A 71, 052102 共2005兲
TIME-DEPENDENT ENTROPY OF SIMPLE QUANTUM…
I+共t,x兲 ⯝
冉冑 冊 冕
␭
2 ␲
1/2
3
⬁
dk
−⬁
冋
k + i␬
关k cos共k兩x兩兲
k2 + ␬2
− ␬ sin共k兩x兩兲兴exp i共k0 − k兲x0 −
A共k,x兲x⬍0 =
册
␭2
共k0 − k兲2 − i␤k2t .
2
Formally, the above results can be summarized as follows:
␺共t,x兲 ⯝
共A26兲
Combining Eqs. 共A22兲 and 共A26兲 we get
␺共t,x兲 ⯝
冉冑 冊 冕
册
␭
1/2
2 ␲
3
冋
⬁
dk A共k,x兲exp i共k0 − k兲x0 −
−⬁
冉冑 冊 冕
冋
␭
1/2
2 ␲3
⬁
dk ei共k0−k兲x0−共␭
2/2兲共k − k兲2−i␤k2t
0
−⬁
册
2␬
k
exp共ikx兲 + ⌰共− x兲
sin共kx兲 ,
k − i␬
k − i␬
⫻
␭2
共k0
2
2␬
k
exp共ikx兲 +
sin共kx兲.
k − i␬
k − i␬
共A27兲
where the step function ⌰共x兲 is defined by
− k兲2 − i␤k2t ,
⌰共x兲 ⬅
再
1, x 艌 0,
0, x ⬍ 0.
冎
共A28兲
Finally, we can reparametrize
where
␭ = 冑2␣ ,
k cos共k兩x兩兲 − ␬ sin共k兩x兩兲
+ i sin共kx兲
A共k,x兲 =
k − i␬
which allows us to rewrite the solution 共A27兲 as
k关cos共k兩x兩兲 + i sin共kx兲兴 + ␬关sin共kx兲 − sin共k兩x兩兲兴
=
.
k − i␬
␺共t,x兲 ⯝
␣
2␲3
⫻
For x 艌 0 we have 兩x兩 = x and, thus,
A共k,x兲x艌0 =
冉 冊冕
冋
1/4
⬁
−⬁
dk ei共k0−k兲x0−␣共k0 − k兲
2−i␤k2t
册
2␬
k
exp共ikx兲 + ⌰共− x兲
sin共kx兲 ,
k − i␬
k − i␬
共A29兲
k
exp共ikx兲,
k − i␬
whereas for x ⬍ 0 we have 兩x兩 = −x, yielding
where ␣ and ␤ play the same role as, e.g., in Eq. 共10兲. Note
that in the limit case ␬ → 0, the solution 共A29兲 correctly reduces to the free-particle solution 共10兲.
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