Entropy of a Free Quantum Particle

doi:10.4236/jmp.2012.312241

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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

Email: [email protected]

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



2hmE



 and the wave-packet propagates at





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]







exp 2

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

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

 





tZtZt



, with



expexpexpexpe r

Vt Vt

tE Et

aL aL









 

 



 



 



(2)

representing contribution from the forward-going wave-front, and

 



0exp

expd ee

Vt Zt

dE E0

tPx EdPxx



 



 







Z







(3)

from all spherical waves diffracted at the edge, respec-

tively. Here the notation











2exp2

Px aLxaL









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







ttt



exp21 0ZZ



, where



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

   

 



00000 00

0000 00

expdexp 4expexp 2

expexpexp2 expexp

eee

rrr

xEE gg

ttt

EdPxEdPx xVtaLEVtaL

tZ ZEEtZ Z

Zt ZZt







Ut Zt















 





(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

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



the shortest step and rewrite the partition function due to

diffraction at the edge of the wave-front in discrete form

Zt Af







, (6)

where





exp



 and





. According

to Equation (5), the entropy is then

J.-P. PENG

1916

 







BZf

xd l

St Af







  (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



. At a later time



, by changing summation into integration and

with the help of the exponential integral function [4], we

obtain

 



exp

EiZ

 



exp e

1ln

exp ee

xd d

St Zt

tZEiZ

ZZ t





 





(8)

Note that the constant term







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



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

  



ln ,





BZf

St Af

Zt Zt

kZt Zt













(9)

which ensures that the entropy starts from zero and re-

mains positive later. At a time







, by eliminating

the



term once again and changing the summa-

tion into integration, the time-dependent entropy of the

particle for the x-coordinate can be obtained

  



exp

elnd exp

tZl rr

St Ut Zt

ll ttE













0exp.

















(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





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 

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



and thus the entropy as a function of the scaled

time





tt E

shows no direct dependence on



Although the entropy evolves in a complicated from, it

tends to reach its limit within a time about





5tt



The limit can be calculated exactly from Equation (10)

and is of the form



11e

21e

1.27

xB Z

CZEiZ

Sk Z











 





 (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]

2π

ln B

mk T





ln 22

 , (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







, 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.