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: jppeng@sjtu.edu.cn

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

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