Dimensionality and geometry effects on a quantum carnot engine efficiency Efectos

. - 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.


Introduction
A classical heat engine is a device that extracts energy  from a high temperature heat source, it generates work  with an amount of this energy and the rest is release into a low temperature drain.The efficiency  of a heat engine is defined by = /.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 Quantum Heat Engines (QHE) have been proposed theoretically [2][3][4][5][6][7][8][9][10][11][12][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][4][5].Based on this, in the present paper we further calculate the efficiency of a quantum 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.

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 , 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  that has periodicity every 2 in the  direction, thus the space where the particle moves are now on a cylinder that has length  and circumference 2.

Schrödinger equation and energy eigenvalues
We start from the time independent Schrödinger equation where ℏ is the Planck constant,  represents the potential,  is the wave function,  are the eigen energies obtained by the expectation value of the Hamiltonian [14], and ∇ 2 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  to denote the length of the different types of potential wells, i.e.,  = , .Once the energies  of each case is obtained (see Table 1), we calculated the force  exerted on the wall of the wells, which is defined as the negative derivative of the energy [3].
where the length  may vary.From table 1, we can see that each energy level state , is inversely proportional to the length of the well, i.e., , 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 ,  = 1,2,3, … are quantum numbers and zkl is the kth cero of the Bessel function of order one.

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.

Isothermal expansion.
Starting at the ground state, which corresponds to the potential well of length 1, we expand isothermally this length up to 2 and excite the second energy state of the system.In this process, a force 1 is applied and an amount of energy  is absorbed by the system.

Adiabatic expansion.
The expansion continues adiabatically up to 3 applying a force 2 on the wall, and the system remains in the second energy state.

Isothermal compression.
We compress isothermally the length of the well down to 4 until the system is back in the ground state.A force 3 is applied.4. Adiabatic compression.The compression down to 1, applying a force 4 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  done in a single cycle of the quantum Carnot engine [3].There is an associated force  to each of the four process, from these, we calculate the total work  done during a full cycle by evaluating the following integrals.We also calculate the energy  absorbed by the system during the isothermal expansion, which is given by Therefore, calculating  and  for the CPW and SPW, we finally calculate the efficiency  = / of each case.
Table 2. Efficiency obtained in each potential well as a function of its length.Here 11 ≈ 3.8317 and 01 ≈ 2.4048.

Results and Discussions
In the case of the SPW, we remained unchanged the non-periodical side on the  direction and the radius  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.

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 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.

Figure 1 .
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.

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