Quantum Carnot Heat Engine Efficiency with Minimal Length


In this paper, the effects of the minimum lengths () to the efficiency of a quantum heat engine are considered. A particle in infinite one-dimensional potential well is used as the “working substance”. We obtain quantized energy of particle in the presence of minimal length, and then we do the isoenergetic cycle. We calculate heat exchanged between the system and reservoir, and then we get the efficiency of the engine. We observe that the minimum length increases efficiency of the engine at the small width of the potential well.

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Purwanto, A. , Sukamto, H. and Subagyo, B. (2015) Quantum Carnot Heat Engine Efficiency with Minimal Length. Journal of Modern Physics, 6, 2297-2302. doi: 10.4236/jmp.2015.615234.

Received 12 November 2015; accepted 28 December 2015; published 31 December 2015

1. Introduction

A deformed quantum mechanics with a generalized Heisenberg Uncertainty (GUP) has been introduced by Kemp et al. [1] [2] . As a consequence, there exist smallest distance limitations in spacetime, known as minimal lengths. This minimal lengths change quantum mechanics that have been established. As an example, there has been calculated Schrodinger equation in the presence of minimal length [3] [4] , the effect of the minimal length on the energy spectrum of Coulomb potential [5] [6] , Casimir effect [7] -[10] , and Dirac Oscillator [11] -[14] .

The minimum length also affects the quantum thermodynamics, quantum generalization of the classical thermodynamics, for instance, quantum heat engine. In the quantum thermodynamics, there is isoenergetic process that is analogous to the isothermal process; and isoentropic process that is analogous to adiabatic process in classical thermodynamics. The cycle composed of two isoenergetic and two isoentropic trajectories is called isoenergetic cycle [15] . The efficiency of quantum heat engine has been calculated in [15] -[17] . The results show that the efficiency depends only on the expansion parameter.


The point is that the width of the potential well has no effect on the value of efficiency. In this paper, we compute the effect of the minimum length on the quantum heat engine efficiency.

This paper is organized as follows. In Section 2 we derive quantized particle energy in infinite one-dimen- sional potential well in the presence of minimal length. In Section 3 we determine inward and outward heat through the system by isoenergetic and isoentropic process, and then we compute the efficiency of Carnot Quantum heat engine with two-level state. Finally, in Section 4 we present a discussion of our results and our conclusions.

2. Schrodinger Equation with Minimal Length

The general form one-dimensional Schrodinger equation is as follows


with operator. In order to incorporate minimal lengths in our equation, we used literature [3] about the position space representations as follows


where is a small parameter. With the representation above, we obtain Schrodinger equation with minimal lengths as follows


We choose one-dimensional infinite potential well as a simple model, with potential energy


So, particle in potential well can be described by one-dimensional time independent Schrodinger as follows


The equation can be solved by first determine the roots of equation


And we get


We only have two boundary condition. It is impossible to find solutions of the equation by using all four roots. So, in order to obtain exact energy particle that can be applied to boundary conditions, we only use two roots. Then we propose the solution as follows


By applying the boundary conditions and nornalization condition, we obtain quantized wave functions as follows


and energy


which when we take, we have ordinary quantized energy in infinite one-dimensional potential well without minimal length


3. Isoenergetic Cycle Process with Minimum Length

The system is assumed to be driven by reversible quasi-static process. That means the wall is moved very slowly by an applied external forced [15] . Because we work on quantum thermodynamics, it is necessary to introduced the ensemble average energy of the system as


The change of the energy during the moving is given by


The above equation is analogous to the first law of thermodynamics. The term analogous to internal energy, analogous to heat exchanged, and to the work done.

For practical reason, we choose the system with two-level energy state. The Carnot cycle is shown as Figure 1.

Let us consider first the isoenergetic process. The isoenergetic process analogous to isothermal process in classical thermodynamics, so. According to reference [15] , the heat exchanged along trajectory to given by


Because the initial state entirely to and final state entirely to then we get relation as follows

Figure 1. Carnot circle for two-level system.


By using (11), we get


As noted earlier, that during the isoenergetic process, the total energy remains constant. Then we get, that makes. So during the first isoenergetic process, the heat flows from environment to system with


The work done to the system, can be obtain by.

At the second, we arrive at isoentropic process. For isoentropic process, the probability is unchanged through von Neumann entropy


The heat exchange during isoentropic process equal to zero. As a Figure 1, we expand the width of the potential well, from to. If the width of the potential well is changed, then so does the total energy. Which means that it is not necessary to change the quantum state of the system during the isoentropic process, the state still on. So the work can be calculated as follows


Similar with isoenergetic process, we can calculate the heat exchanged from to. The heat exchanged from the system to the environment along this process is given by


The last path along the cycle is isoentropic process, which return fully to the initial condition. The work performed during this process from to is given by


We obtain that work along two isoentropic process cancel each other, that is. Therefore, the efficiency of the cycle can be expressed by


By substituting Equation (18) and Equation (21), we obtain the explicit analytical expression




Then we plot the graph between the efficiency versus the width of potential as Figure 2. From Figure 2, we obtain that the efficiency value depends on the initial value of potential width. We get interesting result that the efficiency value increase above classical result with the decreasing the width of potential. The efficiency is also affected by the size of minimal length. If we approximate, the value of efficiency (24) would be


And at large width of potential well, the efficiency value approaches classical result.

At Figure 3, we plot the graph with variations. We take and using L’hopital theorem, we get

Figure 2. The efficiency versus initial potential width, with and variations.

Figure 3. The efficiency versus initial potential width, with and variations.

Schrodinger limit as


Efficiency value returns to the quantum engine efficiency without the presence of minimal length.

4. Discussion and Conclusion

In this work, we have studied the consequences of the minimal length on the quantum thermodynamics. This minimal length modifies Schrodinger equation to be fourth order differential equation. We choose periodic solutions in order to obtain the exact solutions. After that, we calculate the efficiency of heat engine with procedure in Reference [15] . We obtain for the width of potential smaller than, the efficiency as (26). But for the width greater than, the efficiency approaches to classical result (27).

We conclude that the minimal length affects the efficiency of the quantum heat engine at small size of potential well. This effect can be explained by considering the particle as a ball-point having a finite size which is of order of the minimal length [1] .


This work is supported by LPPM ITS.

Conflicts of Interest

The authors declare no conflicts of interest.


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