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Advanced Studies in Theoretical Physics

A quantum Carnot engine in three-dimensions

Paul Bracken

Published Jan 1, 2014• 3 citations
Publisher

Abstract

A simple quantum engine is designed which makes use of the closed- form solution for the energy eigenvalues of the Schrodinger equation under an infinite three-dimensional spherical potential well. A reversible cycle is defined for this system and made to operate in such a way that equations of state can be formulated for it on each leg of its cycle. For a closed, reversible cycle, a quantum Clausius equation can be obtained.

DOI
10.12988/astp.2014.4568

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A simple quantum engine is designed which makes use of the closed- form solution for the energy eigenvalues of the Schrodinger equation under an infinite three-dimensional spherical potential well. A reversible cycle is defined for this system and made to operate in such a way that equations of state can be formulated for it on each leg of its cycle. For a closed, reversible cycle, a quantum Clausius equation can be obtained.
A cyclic thermodynamic heat engine runs most efficiently if it is reversible. Carnot constructed such a reversible heat engine by combining adiabatic and isothermal processes for a system containing an ideal gas. Here, we present an example of a cyclic engine based on a single quantum mechanical particle confined to a potential well. The efficiency of this engine is shown to equal the Carnot efficiency because quantum dynamics is reversible. The quantum heat engine has a cycle consisting of adiabatic and isothermal quantum processes that are close analogues of the corresponding classical processes.
We study two-dimensional quantum Carnot engines of spherical symmetry by considering the case of a particle on the surface of a sphere of changing radius. The Carnot cycle is built allowing the state of the system to change with the specific constrains discussed in Bender's work for Carnot cycles. After studying the Carnot cycle, we maximize the output power and efficiency of the system to show that as it happens in one dimension systems:(i) the efficiency can be optimized; being its optimal value independent of the parameters describing the system and that the optimal output power at the optimal efficiency is non-zero. (ii) The optimal efficiency of the spherical system is much bigger than that of the one-dimensional quantum well considered in Abe's work.
The potential well is a simple example that generally used to present an understanding of quantum mechanics. In this article, we used infinite spherical well model to evaluate the thermodynamic processes in a quantum Carnot engine. The energy of the particles depended on the value of n and l lead to complex calculations. For simplicity we used the φ100 and φ200 quantum states to determine work and efficiency of a quantum Carnot machine. The results obtained show that efficiency depends on the value of which is the ratio of RC and RB.
A cyclic thermodynamic heat engine,which combines adiabatic and isothermal processes and consists of classical ideal gas as working substance,runs most efficiently if it is reversible.In this paper,A new cyclic model of quantum Carnot heat engine whose working substance consists of non-interacting extreme relativistic particles confined to an infinite potential well,is set up.It is shown that the engine whose cycle consists of adiabatic and isothermal quantum processes are close analogues of the corresponding classical processes and that the efficiency of this engine is equal to the Carnot efficiency because quantum dynamics is reversible.Consequently,the expectation value of the Hamiltonian in the expression of the efficiency is similar to the temperature in that of the classical one.
In this paper,we consider three models of quantum heat engines based on Carnot cycle using three energy levels; (1) the ground state, (2) the degenerate state, and (3) the highest energy state. We investigate the variation in the transition state by selecting three different degenerated states. The result we obtained still analogous with the classical heat engine efficiency and also the previous Quantum Carnot Engine model, which only depends on the initial width and the final width of the potential well in isothermal expansion. Moreover, the effect of transition state generally can be accepted for multistate quantum heat engines with 3D systems in cubic potential.
The concept of a quantum heat pump is proposed as a convenient model in the thermodynamic interpretation of certain multilevel processes. The ideal quantum heat engine is defined as an idealization of realistic pumped multilevel systems in much the same way that the well-known Carnot cycle is an idealization of physically realizable, classical processes or engines. There is evidence that the conventional Carnot cycle can be operated only between reservoirs at absolute temperatures of identical sign. No such restriction applies, however, to the quantum heat engine. Thus it may be used to calibrate negative absolute temperatures by relating them directly to positive temperatures. Negative efficiencies or efficiencies greater than unity have particularly simple interpretations in the quantum-heat-engine model. An important application of these concepts is in the calculation of optical maser parameters.