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Revista de Ciencias Tecnológicas

Dimensionality and geometry effects on a quantum carnot engine efficiency

H. K. Herrera-Alcantar, José Carlos Carvajal García, Osvaldo Rosales Pérez, Rubén Cesar Villarreal-Sánchez, Priscilla Elizabeth Iglesias Vázquez

Published Feb 27, 2019• 0 citations
Publisher

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.

DOI
10.37636/recit.v214548

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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.
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In their work [J. Phys. A 33, 4427 (2000)], Bender, Brody, and Meister have shown by employing a two-state model of a particle confined in the one-dimensional infinite potential well that it is possible to construct a quantum-mechanical analog of the Carnot engine through changes of both the width of the well and the quantum state in a specific manner. Here, a discussion is developed about realizing the maximum power of such an engine, where the width of the well moves at low but finite speed. The efficiency of the engine at the maximum power output is found to be universal independently of any of the parameters contained in the model.
Carnot cycle consists of isothermal and adiabatic processes which are reversible. Using analogy in quantum mechanics, these processes can be well explained by replacing variables in classical process with a quantum system. Quantum system which is shown in this paper is a particle that moves under the influence of a cubic potential which is restricted only to the state of the two energy levels. At the end, the efficiency of the system is shown as a function of the width ratio between the initial conditions and the farthest wall while expanding. Furthermore, the system efficiency will be considered 1D and 2D cases. The providing efficiencies are different due to the influence of the degeneration of energy and the degrees of freedom of the system.
In (1) a thermodynamical-statistical mechanical approach is presented for a two energy level quantum system for a particle in a box with infinite potential walls. It is shown that classical statistical mechanical efficiency of a Carnot cycle 1- Tc/Th holds for this quantum system. In this note we argue that the Carnot cycle efficiency holds in general if one uses delta E = delta Q - delta Work with delta Q/T = delta S if E and S are state variables as they are. One does not need to perform the two energy level specific integrals presented in (1) because the Carnot efficiency result is guaranteed for any number of energy levels.
In this article, we study the quantum Carnot engine using a one-dimensional infinite well potential model in which two particles are non-interacting, indistinguishable and have the same mass. We find that the efficiency of this model is the same as that of the one-dimensional infinite potential well model in which there is one particle. Moreover, efficiency can be measured only when L_C/L_A≥2, where L_A and L_C are the potential well widths during isothermal and adiabatic expansion, respectively.
Carnot model of heat engine is the most efficient cycle consisting of isothermal and adiabatic processes which are reversible. Although ideal gas usually used as a working fluid in the Carnot engine, Bender used quantum particle confined in 1D potential well as a working fluid. In this paper, by following Bender we generalize the situation to 2D symmetric potential well. The efficiency is express as the ratio of the initial length of the system to the final length of the compressed system. The result then is shown that for the same ratio, 2D potential well is more efficient than 1D potential well.