Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22.

Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925

Volumen 2 (1): 45-48 Enero-Marzo 2019 https://doi.org/10.37636/recit.v214548

ISSN: 2594-1925

Dimensionality and geometry effects on a quantum

carnot engine efficiency

Efectos de dimensionalidad y geometría en la eficiencia

del motor quantum carnot

Herrera Alcantar Hiram Kalid

, Carvajal García José Carlos

, Rosales Pérez Osvaldo

Villarreal-Sánchez Rubén Cesar

, Iglesias-Vázquez Priscilla Elizabeth

Facultad de Ciencias, Universidad Autónoma de Baja California, Ensenada, Baja California, México

Facultad de Ingeniería, Arquitectura y Diseño, Universidad Autónoma de Baja California, carretera

transpeninsular Ensenada-Tijuana 3917, colonia Playitas, Ensenada, Baja California, México

Autor de correspondencia: Priscilla Elizabeth Iglesias Vázquez, Facultad de Ciencias, Universidad

Autónoma de Baja California, Carretera Transpeninsular Ensenada-Tijuana 3917, Colonia Playitas,

Ensenada, Baja California, México. E-mail: piglesias@uabc.edu.mx.

Recibido: 21 de Julio del 2018 Aceptado: 15 de Enero del 2019 Publicado: 27 de Febrero del 2019

Resumen. - Calculamos la eficiencia de un ciclo de Carnot cuántico para una partícula confinada en dos

pozos de potencial infinitos diferentes, un pozo de potencial cilíndrico de radio variable y un pozo de

potencial bidimensional cuadrado con periodicidad en uno de sus lados. Encontramos que la eficiencia

depende directamente de la dimensionalidad y la geometría del pozo que confina a la partícula.

Palabras clave: Ciclo de Carnot; Motor térmico; Confinamiento cuántico.

Abstract. - We calculate the efficiency of a quantum Carnot cycle for a particle confined in two different

infinite potential wells, a cylindrical potential well of variable radius and a two-dimensional square

potential well with a periodicity in one of it sides. We find that the efficiency depends directly on the

dimensionality and geometry of the well that confined the particle.

Keywords: Carnot cycle; Heat engine; Quantum confinement.

1. Introduction

A classical heat engine is a device that extracts energy

𝑄𝐻 from a high temperature heat source, it generates

work 𝑊 with an amount of this energy and the rest is

release into a low temperature drain. The efficiency 𝜂

of a heat engine is defined by 𝜂= 𝑊/𝑄𝐻. It is well

known that the heat engine reaches the highest

possible efficiency following Carnot cycle model [1].

This cycle consists in a gas confined by a cylinder

with a movable piston. Although classical heat

engines have been extensively studied, it is of interest

to study the systems and processes that could

increase their efficiency. In recent years, with the

developments of nanotechnology and quantum

information processing, the study of quantum systems

began to attract more attention. Consequently, the

Revista de Ciencias Tecnológicas (RECIT). Volumen 2 (1): 45-48

ISSN: 2594-1925

Quantum Heat Engines (QHE) have been proposed

theoretically [2-13]. In QHE, rather than having a gas

confined in a cylinder with a movable piston, it is

considered a single particle confined by a quantum

potential well that walls play the role of the piston by

moving in and out. Current studies on QHE have

considered different type of potential wells, for

example, a single particle confined by one (1D), two

(2D) or three-dimensional (3D) infinite square

potential well [3-5]. Based on this, in the present paper

we further calculate the efficiency of a quantum

Carnot cycle considering a single particle confined by

two different types of potential wells, an infinite

cylindrical potential well and an infinite 2D square

potential well with periodicity, as to our best

knowledge, these cases have not been considered.

Comparison between these two cases enable us to

extend our understanding about the dimensionality

and geometry effects of quantum confinement on the

efficiency of a QHE.

2. Methodology

We consider a particle of mass m, confined by two

different types of quantum potential wells: an infinite

cylindrical potential well (CPW) of radius 𝑟, in this

case the particle is confined in the space inside the

CPW. Also, we consider an infinite 2D square

potential well (SPW) of length 𝑎 that has periodicity

every 2𝜋𝑅 in the 𝑦 direction, thus the space where the

particle moves are now on a cylinder that has length 𝑎

and circumference 2𝜋𝑅.

2.1 Schrödinger equation and energy eigenvalues

We start from the time independent Schrödinger

equation

where ℏ is the Planck constant, 𝑉 represents the

potential, 𝜓 is the wave function, 𝐸 are the eigen

energies obtained by the expectation value of the

Hamiltonian [14], and ∇

is the Laplace operator in the

corresponding coordinates of each of the potential

wells, i.e., in cylindrical coordinates for the CPW and

in 2D Cartesian coordinates for the SPW. We shall use

the symbol 𝑆 to denote the length of the different types

of potential wells, i.e., 𝑆 = 𝑟, 𝑎. Once the energies 𝐸

of each case is obtained (see Table 1), we calculated

the force 𝐹 exerted on the wall of the wells, which is

defined as the negative derivative of the energy [3].

where the length 𝑆 may vary. From table 1, we can see

that each energy level state 𝐸𝑘,𝑙 is inversely

proportional to the length of the well, i.e., 𝐸𝑘,𝑙

decreases as the length of the of the well increases,

and vice versa, in this sense we can imagine that the

walls of the potential well can move like a piston in a

classical thermodynamics system [3].

Table 1. Energies obtained in each potential well. Here 𝑘, 𝑙 =

1,2,3, … are quantum numbers and zkl is the kth cero of the

Bessel function of order one.

2.2 Quantum Carnot cycle

The authors of Ref. 3 calculated the efficiency of a

quantum Carnot cycle by using a single particle

confined by a 1D infinite square potential well. Using

the procedure described in Ref. 3, we further

investigate the efficiency of a quantum Carnot cycle

by considering different type of potential well that

have not been reported. The quantum analogue of

classical Carnot cycle consists of four processes

described below and illustrated in figure 1.

1. Isothermal expansion. Starting at the ground

state, which corresponds to the potential well of length

𝑆1, we expand isothermally this length up to 𝑆2 and

excite the second energy state of the system. In this

process, a force 𝐹1 is applied and an amount of energy

𝑄𝐻 is absorbed by the system.

Revista de Ciencias Tecnológicas (RECIT). Volumen 2 (1): 45-48

ISSN: 2594-1925

2. Adiabatic expansion. The expansion continues

adiabatically up to 𝑆3 applying a force 𝐹2 on the wall,

and the system remains in the second energy state.

3. Isothermal compression. We compress

isothermally the length of the well down to 𝑆4 until

the system is back in the ground state. A force 𝐹3 is

applied.

4. Adiabatic compression. The compression

continues down to 𝑆1, applying a force 𝐹4 on the wall.

During this process the system remains in the ground

state.

The area of the closed loop in figure 1 represents the

work 𝑊 done in a single cycle of the quantum Carnot

engine [3]. There is an associated force 𝐹 to each of

the four process, from these, we calculate the total

work 𝑊 done during a full cycle by evaluating the

following integrals.

Figure 1. Illustration of a four-step quantum Carnot cycle, where

S denote the length of the different types of potential wells and

F is the force exerted on the wall of the wells.

We also calculate the energy 𝑄𝐻 absorbed by the

system during the isothermal expansion, which is

given by

Therefore, calculating 𝑊 and 𝑄𝐻 for the CPW and

SPW, we finally calculate the efficiency 𝜂 =

𝑊/𝑄𝐻 of each case.

Table 2. Efficiency obtained in each potential well as a function

of its length. Here 𝑧

≈ 3.8317 and 𝑧

≈ 2.4048.

3. Results and Discussions

In the case of the SPW, we remained unchanged the

non-periodical side on the 𝑥 direction and the radius

𝑅 of the periodicity on the other side was varied. For

the CPW, the parameter that was varied was the radius

of the cylinder. For each case, the procedure indicated

in the methodology section was developed. The

efficiencies obtained are shown in Tables 2 and 3.

Table 3. Efficiency of each potential well as a function of its

energy level states

4. Conclusions

It was found, from the efficiencies shown above, that

the efficiency of the quantum Carnot cycle depends of

the length of the potential well. It should be noted that

the importance of this work relies in the fact that the

efficiency changes as a function of the geometry and

dimension of the potential well that confined the

single particle, this could help for future works to find

a QHE with a higher efficiency and possible

applications such as those proposed in Ref. 11, 12 and

13, where possible applications are proposed for a

Revista de Ciencias Tecnológicas (RECIT). Volumen 2 (1): 45-48 
         
48 
ISSN: 2594-1925 
 
QHE. As a future work, other type of thermodynamic 
cycles such as the Otto cycle or Stirling cycle can be 
analyzed  to  determine  how  the  dimensionality  and 
geometry affects their efficiencies. 
 
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