^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

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.

A deformed quantum mechanics with a generalized Heisenberg Uncertainty (GUP) has been introduced by Kemp et al. [

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 [

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.

The general form one-dimensional Schrodinger equation is as follows

with operator

where

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

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

and energy

which when we take

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 [

The change of the energy during the moving is given by

The above equation is analogous to the first law of thermodynamics. The term

For practical reason, we choose the system with two-level energy state

Let us consider first the isoenergetic process. The isoenergetic process analogous to isothermal process in classical thermodynamics, so

Because the initial state entirely to

By using (11), we get

As noted earlier, that during the isoenergetic process, the total energy remains constant. Then we get

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

Similar with isoenergetic process, we can calculate the heat exchanged from

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

We obtain that work along two isoentropic process cancel each other, that is

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

with

Then we plot the graph between the efficiency versus the width of potential as

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

At

Schrodinger limit as

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

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 [

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 [

This work is supported by LPPM ITS.

A.Purwanto,H.Sukamto,B. A.Subagyo, (2015) Quantum Carnot Heat Engine Efficiency with Minimal Length. Journal of Modern Physics,06,2297-2302. doi: 10.4236/jmp.2015.615234