Journal of Modern Physics, 2012, 3, 1914-1917
http://dx.doi.org/10.4236/jmp.2012.312241 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Entropy of a Free Qu antum Particle
Jian-Ping Peng
Department of Physics, Shanghai Jiao Tong University, Shanghai, China
Received September 30, 2012; revised November 2, 2012; accepted November 10, 2012
ABSTRACT
The time-dependent entropy of a single free quantum particle in the non-relativistic regime is studied in detail for the
process started from a fully coherent quantum state to thermodynamic equilibrium with its surroundings at a finite tem-
perature. It is shown that the entropy at the end of the process converges to a universal constant, as a result of thermal
interaction.
Keywords: Entropy Generation; Quantum Thermodynamic Systems
1. Introduction
It is well-known that entropy, as the measure of “the
amount of uncertainty”, can not decrease in any sponta-
neous process according to the second law of thermody-
namics [1]. In a recent work [2], we studied the thermo-
dynamics of a single quasi-free massive quantum particle,
by performing statistics directly on the matter wave of the
particle. Taking into account the detailed configuration of
diffraction in real space and thermal interaction with the
surround space at a finite temperature, the complicated
behavior of the time-dependent internal energy is studied
for the whole process started from a fully coherent quan-
tum state to thermodynamic equilibrium with the sur-
rounding space. An expression for the entropy of the
particle is also shown in [2]. The purpose of the present
article is to present the detailed derivation of the expres-
sion of the time-dependent entropy for the particle and
study in more detail the physics in the irreversible proc-
ess. Numerical calculations confirm that the entropy in-
creases monotonically with time and the entropy gener-
ated in the whole process converges to a universal con-
stant. Although the system studied here is the simplest
quantum system at a finite temperature, it already shows
how a single quantum particle feels the temperature of its
surrounding space. In conventional quantum mechanics,
entropylike concepts are defined only for statistical de-
scription of ensembles of identical quantum systems. Our
results here confirm the conclusion that entropy is a physi-
cal observable that can be well-defined for each individ-
ual quantum system at finite temperatures [3].
2. Model Calculations
The system considered is a structureless quantum particle
of mass m and kinetic energy E0 initially at the origin,
moving along the x-axis in a space at a nonzero tempera-
ture T. The space here may be filled with electromagnetic
radiation just as the cosmic background in the universe.
In quantum mechanics, the particle is described by a
wave-packet sharply peaked at the de Broglie wavelength

12
0
2hmE   and the wave-packet propagates at


g
Vhm
group velocity

, with h being the Planck’s
constant. The matter wave front is assumed to be circular
with finite radius a0 which is large compared with the
wavelength, so that the shape and linear dimension of the
forward-going wave-front remains unchanged. Strictly
speaking, the particle is quasi-free in the model calcula-
tion, even though there is no interaction with other parti-
cles. If the radius a0 tends to be infinitely large, the re-
sults reduce to that of a free quantum particle. A point in
the central part of the wave-front generates forward-go-
ing semi-spherical waves, according to the Huygens prin-
ciple. A point at the edge of the wave-front is assumed to
generate out-going fully spherical waves and thus the
particle undergoes a kind of reflection. The kinetic en-
ergy associated with the forward-going wave-packet fol-
lows the form [2]

 
00
exp 2
k
Ex ExaL   , (1)
where x = Vgt representing its central position and L is a
temperature dependent parameter of dimension length
and is expected to be infinitely large as the temperature
tends to zero. This is just the energy for the source to
generate out-going fully spherical waves. In general, all
energy states are not equally likely. In principle, the par-
ticle may be in any of these diffracted states besides the
forward-going plane-wave state, i.e., the particle itself
constitutes automatically a thermodynamic system as a
C
opyright © 2012 SciRes. JMP

J.-P. PENG 1915
result of diffraction at the edge of its matter-wave front.
Thermal interaction between the particle’s system and
the surrounding space becomes possible and the space
here acts as the heat reservoir at constant temperature. As
time goes on, the probability decreases for the particle in
the quantum state described by forward-going wave-
packet. At the end of the process, the particle can only be
in a series of states diffracted at the edge and moves
equally to all directions. The partition function at a given
time is written in the form
 


fd
Z
tZtZt

, with

00
00
22
expexpexpexpe r
gg tr
Vt Vt
tE Et
aL aL




 
 
  

 

Z
(2)
f
representing contribution from the forward-going wave-front, and
 

0exp
00
0
expd ee
g
r
Vt Zt
dE E0 0
Z
Z
tPx EdPxx 
 

 



Z



(3)
from all spherical waves diffracted at the edge, respec-
tively. Here the notation



 
00
2exp2
E
Px aLxaL  
1B kT


is used as usual
with kB being the Boltzmann constant. The constant Z0 is
defined as the non-zero real solution of the transcend
equation 00, and an approximate value
Z0 = 1.25643 is used in our numerical calculations. The
probability density function is defined as

and the step length is chosen to be

00 .
The time is scaled as
0 0
2dZaL E  
rc

ttt

exp21 0ZZ

, where

0c
is the temperature dependent characteristic time.
2tmaLh
The expectation value of energy of the particle or its
internal energy Ux(t) for the coordinate is
     







20
00000 00
0
0000 00
0
expdexp 4expexp 2
expexpexp2 expexp
eee
,
g
rrr
Vt
xEE gg
ttt
rr
E
EdPxEdPx xVtaLEVtaL
Zt
tZ ZEEtZ Z
Zt ZZt




1
Ut Zt







 


ln
(4)
which evolves depending on the temperature and the
particle’s initial energy in a complicated form. In general,
a quantum particle absorbs or gives out heat continuously
when its initial energy E0 is less or more than kBT/2, in-
dicating exchange of energy between the particle and its
surrounding space in the whole process. The limiting
value of the internal energy for the freedom in the x-di-
rection is kBT/2, regardless of its initial energy [2]. The
limit is reached within several tc and the overall decay or
increase in internal energy does not follow the simple
exponential form.
We use the well-known definition of entropy of a dis-
crete system
B
ii
i
Skpp  (5)
where pi is the probability in the ith state and the sum
goes over all states accessible to the system. The defini-
tion is perfectly unambiguous for systems of any size and
there is no restriction to equilibrium situations: the prob-
ability will be time-dependent if the system evolves dy-
namically. The entropy of the particle is thus a function
of the probability distribution and is not fluctuating since
it has nothing to do with the state in which it happens to
be. We first assume that the forward-going wave-front
representing a single quantum state makes little contribu-
tion to the entropy. The entropy is then determined by all
those states diffracted at the edge. In quantum mechanics,
the spreads in energy and time are related by the uncer-
tainty relation 2π.Eth

 To prepare the original
state with energy E0, the uncertainty in time is estimated
to be 0 2πth E

Vt , corresponding to an uncertainty in
position of the particle g




. Therefore, waves dif-
fracted at the edge should be indistinguishable when the
wave-front moves forward within about one wavelength.
Please note that the analysis here does not mean the en-
tropy is physically related to the principle of uncertainty
in quantum mechanics. In fact, as shown below, a dif-
ferent choice of the indistinguishable length leads to an
unimportant additive constant in entropy. We choose  as
the shortest step and rewrite the partition function due to
diffraction at the edge of the wave-front in discrete form
0
0
e
g
l
Vt
Z
f
dl
l
Zt Af




, (6)
where


exp
l
f
Al

 and

0
2
A
aL  . According
to Equation (5), the entropy is then

J.-P. PENG 1916
 
0
0
ln
e
g
l
Vt

0
e,
l
Z
f
l
d
Af
Zt


BZf
xd l
l
d
k
St Af
Zt




  (7)
where the subscript x represents the coordinate and d
diffracted states at the edge. This expression is exact and
must be used within the time
g
tV

. At a later time
g
tV

, by changing summation into integration and
with the help of the exponential integral function [4], we
obtain
 







00
0
exp
.
d
Z
Zt
EiZ
 

00
0
exp e
1ln
exp ee
r
rr
t
r
xd d
B
tt
r
d
tZ
St Zt
k
tZEiZ
ZZ t



 


(8)
Note that the constant term


ln
A

is eliminated
because it is an additive constant depending on how to
define a distinguishable state and in thermodynamics we
are interested only in the entropy generation during the
process of evolution. Moreover, the factor
A

in the
discrete form of the partition function Equation (6) cor-
responds to a common weight factor for all states in the
problem and has no physical consequence in calculating
the average value of a quantity except the entropy.
Similar discussion applies also to the case that the con-
tribution from the forward-going wave-front is included.
The entropy of the particle is expressed as
  




0
e
ln
l
0
0
e
ln ,
g
l
Vt
Z
f
l
Af
Zt


BZf
xl
l
ff
B
k
St Af
Zt
Zt Zt
kZt Zt







(9)
which ensures that the entropy starts from zero and re-
mains positive later. At a time
g
tV



ln
, by eliminating
the
A

term once again and changing the summa-
tion into integration, the time-dependent entropy of the
particle for the x-coordinate can be obtained
  

0
exp
1
1
ln
elnd exp
r
xx
B
tZl rr
St Ut Zt
kZ
ll ttE








0exp. r
t
t


 




E
(10)
In fact, time-dependent entropy of a quantum system
has been studied by Gheorghiu-Svirschevski using an
extended Liouville-von Neumann equation [5]. Unfortu-
nately, we failed to obtain an expression for the entropy
using this theory to compare with Equation (10). The
reason is that the system here is not described by stan-
dard plane-wave with infinite spacial extension. The pre-
sent work starts from a circular matter wave pulse with
finite radius a0, its time evolving is governed by the Huy-
gens-Fresnel principle. Furthermore, every point at the
edge of a wave-front is assumed to generate continuously
out-going spherical secondary wavelets. In principle,
such a system can be studied with the path integral for-
mulation of quantum theory, but not the Schroedinger
formulation.
In Figure 1 we plot the numerical results for the en-
tropy of the particle determined by Equation (10) as a
function of the scaled time for different values of 0

.
For 02E


, the entropy starts from zero and increases
with increasing


c
tt E and shows little dependence on
the exact value of 0

. For 0 250E

, Equation
(10) may become negative numerically, indicating that it
is inexact in the initial stage. Fortunately, it starts to be
positive at a latter time


1tt 050E
c. For

, the
entropy starts to be non-negative at

1tt 
E
c and then
increases monotonically with time in accordance with the
second law of thermo-dynamics. Mathematically, Equa-
tion (10) reduces to Equation (8) in the limit of large
0

and thus the entropy as a function of the scaled
time


tt E
c0
shows no direct dependence on

.
Although the entropy evolves in a complicated from, it
tends to reach its limit within a time about


5tt
c

.
The limit can be calculated exactly from Equation (10)
and is of the form


0
0
00
0
ln
11e
ln
21e
1.27
Z
xB Z
B
CZEiZ
Sk Z
k






 


 (11)



where Ei the exponential integral function [4]. Note that
the limit is universal with no regards to the unknown
Figure 1. The entropy of a single quantum particle for the
x-coordinate as a function of the scaled time for different
values of
 E0. The curves may become inexact for (t/tc) < 1,
as described in the text.

J.-P. PENG 1917
parameter L, the temperature of the surrounding space
and the initial state of the particle. In a textbook of statis-
tical physics, it is well-known that the entropy per parti-
cle in an ideal gas of N particles in a volume V and at
constant temperature T is expressed as the Sackur-Tet-
rode formula [6]
2
2π
ln B
B
mk T
k
h


a
53
ln 22
BB
SN
kk
NV
 , (12)
showing dependences on the temperature and the mass of
the particle. It should be pointed that Equation (11) is the
entropy per freedom generated in the process started
from an initial coherent quantum state to final thermo-
dynamic equilibrium with its surroundings, before we
call it a particle in conventional statistical physics, i.e., it
moves equally to all directions. For a real free particle
0



, independent motion is allowed in the perpen-
dicular direction. Therefore, the total entropy per particle
generated in the whole process of decoherence should be
3Sx() 3.81kB.
3. Conclusion
In conclusion, we have derived an expression for the
time-dependent entropy of a single non-relativistic quan-
tum particle freely moving in a space at constant nonzero
temperatures. The entropy increases monotonically with
time in accordance with the second law of thermody-
namics. Although the initial state of the particle and the
temperature of the surrounding space play important
roles in the process, the total entropy generated tends to
be a universal constant.
REFERENCES
[1] L. D. Landau and E. M. Lifshitz, “Statistical Physics,”
3rd Edition, Pergamon Press Ltd., Oxford, 1980.
[2] J. P. Peng, “Temperature Dependent Motion of a Massive
Quantum Particle,” Journal of Modern Physics, Vol. 3,
2012, pp. 610-614. doi:10.4236/jmp.2012.37083
[3] G. P. Beretta, “Entropy and Irreversibility for a Single
Isolated Two Level System: New Individual Quantum
States and New Nonlinear Equation of Motion,” Interna-
tional Journal of Theoretical Physics, Vol. 24, No. 2,
1985, pp. 119-134. doi:10.1007/BF00672647
[4] I. S. Gradshteyn and I. M. Rizhik, “Tables of Integrals,
Series, and Products,” 7th Edition, Elservier Inc., London,
2007.
[5] S. Gheorghiu-Svirschevski, “Nonlinear Quantum Evolu-
tion with Maximal Entropy Production,” Physical Review
A, Vol. 63, 2001, Article ID: 022105.
[6] M. Plischke and B. Bergersen, “Equilibrium Statistical
Physics,” 2nd Edition, World Scientific Publishing Co.
Pte. Ltd., 2003.