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A multiple-state quantum Carnot engine based on single particle in one dimensional potential well is evaluated. The general forms of adiabatic and isothermal force as well as work are given. We apply them first to the simplest case of two-state system, and then to three-state and general n-state system. The first isothermal expansion starts from single ground state and cease to single highest state. In Addition to the simplest case, isothermal expansions may terminate not to highest state but an intermediate state but with the same of the total expansion. The result is that the efficiency of the multi-state machine could be enhanced by reducing the volume of isothermal expansion for the same of the total volume expansion.

As a device to convert heat energy into mechanical work, classically, heat engine consists of gas as working agents that expands and pushes a piston in a cylinder. A heat engine obtains its energy from high temperature heat reservoir T_{H}. A part of this captured energy is converted into mechanical energy and the others are transferred to the low temperature reservoir T_{C}. Namely, not all of energy drawn from the reservoir Q_{H} is converted into mechanical work W. Such that the efficiency of the machine, which is defined as the ratio between mechanical work W and energy absorbed by the machine Q_{H}, will be less than one

The main problem of a heat engine is none other than this efficiency which is generally small or low. Heat engine, working between high temperature reservoir and low temperature reservoir, will reach maximum efficiency if the process is reversible.

Mathematical model of an ideal heat engine was proposed by Sadi Carnot in 1824. It is reversible and has the highest possible efﬁciency for any engine operating between two given temperatures. For this reason, it serves as a model for real engines to emulate [_{H} while it is in contact with the high-temperature reservoir. Second, the gas continues to expand adiabatically in thermal isolation until its temperature drops to T_{C}. Third, the gas is compressed isothermally in contact with the low temperature reservoir. Fourth, the gas is compressed adiabatically until its temperature rises to T_{H}. As mentioned before, the interesting problem is the low efficiency of the engine.

One of the efforts to increase the efficiency of heat engine the application of quantum principles to the engine. The study of quantum engines started in 60’s [

Bender, et al., [

The purpose of this paper is further discussion of quantum Carnot heat engine employing a single quantum-mechanical particle conﬁned to a potential well with a multiplestate particle. In previous work [

This paper is organized as follows: In Section 2 we briefly review the simplest quantum mechanical Carnot engine based on one-dimensional potential well with two eigenstates. In Section 3, three level and generalization to n-level quantum system is discussed. Finally, discussion and conclusions will be given in Section 4.

Quantum heat engines (QHEs) [

We provided a kind of quantum heat engines, i.e. a cyclic Carnot heat engine employing a single quantummechanical particle, as a working substance, conﬁned to a potential well, instead of gas-ﬁlled cylinder. Rather than 2-state, we purpose further discuccion with 3-state and n-state of the particle.

The simplest quantum system is a particle mass m confined in a one-dimensional well of width L with infinite potential walls. The motion equation of this system is one-dimensional time independent Schrodinger equation.

The infinite potential walls at x = 0 and x = L provide boundary conditions for the state functions 0. The general solution of this equation is a linear combination of orthonormal eigenfunctions

and energy E,

where j_{n} and E_{n} have explicit forms as follows.

And, the coefficients a_{n} satisfy the orthonormal condition

From Equation (6), the energy (4) becomes

Now, if one of the infinite walls of the potential well, say the wall at x = L, is allowed to move an infinitesimal amount dL then the general wave function j (x), eigen functions j_{n} (x) and energy levels E_{n} all vary infinitesimally as function of L. In this situation, it is natural to define the force on the wall of the potential well as the negative derivative of the energy,

This force F exerts on the wall. Based on this force, several kinds of thermodynamical processes which have the quantum analogues to the classics can be defined. According to the force, work from L_{1} to L_{2} is given by

Heat engines are devices that extract energy from its environment in the form of heat and do useful work. The heart of every heat engine is its working substance. The operation of the heat engine is by subjecting the working substance of the engine to be a sequence of thermodynamics processes that forms a cycle. Carnot heat engine operates with isothermal and adiabatic processes. Classically, an adiabatic process is one in which the system is thermally isolated in such a way that heat cannot flow into or out of the system. An isothermal process is one in which as the piston moves, the system remain in equilibrium at all times. During the process, the system is in contact with a heat reservoir so that the temperature T of the gas in cylinder remains fixed. During the piston moves, the system does work both in adiabatic as well as in isothermal processes.

Now, the above classical thermodynamical system may be applied in a monatomic one-dimensional gas in the infinite potential well. One monatomic particle is as a working substance, a wall of infinite potential well at x = L as a piston can moves and the system remains in equilibrium at all times. Two processes for Carnot engine are adiabatic and isothermal.

In adiabatic process, there is no heat transfer from or into the system, and the potential wall changes as the wall moves. Since the system remains in equilibrium at all times, the absolute values of the expansion coefficients |a_{n}| must be constant. That is, we do not expect any transitions between states can occur during an adiabatic process.

It is clear from Equation (7) that the eigen energy E_{n} depends on L and the energy (4) will changes during an adiabatic process. The energy increases if the system compresses and decreases if the system expands. When the piston moves out, then the energy decreases and the energy that is lost equals the mechanical work done by the force F (9), which is given by

where a_{n}(L_{i}) is the n-th coefficient at initial state with a width L_{i}.

Applying the force (11) into Equation (10), we obtain a work along adiabatic process from L_{1} as the initial volume to L_{2} as the final volume

In isothermal process, the system is in contact with a heat source so that the temperature T of the gas in the cylinder is fixed. It implies that the transition between states occur so that the internal energy of the gas during isothermal process remains constant. In other word, the expansion coefficients a_{n} in general change but the orthonormal condition (5) remains satisfied. This implies that during the process the width of the system satisfies

where L_{i} is the initial width.

The force of the piston is constrained by Equation (12) and has a form as follows.

According to the force, the work along isothermal process from L_{1} as the initial volume, to L_{2} as the final volume is given by

A work of the system is equal to the absorbed energy.

Using the two quantum adiabatic processes and two quantum isothermal processes above, we can construct a Carnot heat engine. The simplest case of the particle in the system is the particle has only two states i.e. ground state (n = 1) and the first excited state (n = 2). It means the orthonormal condition (5) becomes simpler form.

The four-step cyclic quantum process of the quantum Carnot engine is illustrated as a diagram between the force and the width as in _{A}. It means the coefficients a_{1} = 1 and a_{2} = 0. The energy of the system is the lowest value. The wall at L_{A} expands isothermally and the particle may excite and in general the state of particle is combination of two possible states j = a_{1}j_{1} + a_{2}j_{2}. Along the expansion, the relation of widths (13) has a simple form

.

It gives, in turn, the maximum expansion L_{B} = 2L_{A} when the state of particle is purely excited state a_{2} = 1 and a_{1} = 0. If the expansion is maximum then the total work, that is the work done to expand from L_{A} to 2L_{A} is given by

The second expansion from L_{B} is adiabatic expansion with a single state j_{2}. Different from isothermal process, there is no definit limit of the maximum width in the adiabatic process. Then we assume the piston moves from L_{B} to L_{C} = aL_{B} where the real number a is bigger than one, then the work during the isothermal expansion is given by

Next, the system is compressed isothermally from L_{C} to L_{D}. At the beginning of compression the system is purely in the first excited state, a_{1} = 0 and a_{2} = 1. Along the compression the transistion occurs, the system is in the mixed state and the energy remains constant. In order the process occurs in close loop then at width L_{D} the system should be at a single ground state. It means a_{1} (L_{D}) = 1, a_{2} (L_{D}) = 0. Since a_{1} (L_{C}) = 0, a_{2} (L_{C}) = 1 then from (13) we obtain a maximum compression L_{D} = L_{C}/2. Hence, the isothermal work during compression from L_{C} to L_{D} = L_{C}/2 is given by

The last process is adiabatic compression from L_{D} to L_{A}. During the compression the system is in single ground state. The work of the compression is given by

The mechanical work W done in single cycle of the quantum heat engine is represented by the area of the closed loop in _{B} = 2L_{A}, L_{C} = aL_{B} and L_{D} = L_{C}/2 the total mechanical work is the sum of W_{AB}, W_{BC}, W_{CD}, and W_{DA}. The total work is

and the efficiency is the ratio between W and W_{AB} and given by

Formulation can be generalized, as stated qualitatively by Bender [_{1} then the maximum expansion is nL_{1}. We study more detail for the case of n-state, and first study for 3-state.

We assume the system may have three eigen states j_{1}, j_{2}, j_{3} associated with the eigen energies E_{1}, E_{2} and E_{3}. As in case of two-state, we start with the initial state a ground state j_{1} with energy E_{1} of width L_{A}. The expansion coefficiens of the initial condition are a_{1} = 1, a_{2} = a_{3} = 0. Then, the system isothermally expands and does transition to all three possible states. In general the system is in a linear combination of three states j = a_{1}j_{1} + a_{2}j_{2} + a_{3}j_{3} and satisfies orthonormal condition

Using a_{1}_{ }(L) = 1, a_{2}_{ }(L) = a_{3}_{ }(L) = 0, Equation (13) becomes

yielding maximum expansion L = L_{C} = 3L_{A} which is occurred when a_{1}_{ }(L_{B}) = a_{2}_{ }(L_{B}) = 0 and a_{3}_{ }(L_{B}) = 1. Hence, the work until maximum expansion is given by

The second expansion from L_{B} to L_{B} = aL_{C} occurs in highest state j_{3}. The adiabatic work (12) becomes

For the isothermal compression from L_{C} to L_{D} the initial conditions of the system are a_{1} = a_{2} = 0 and a_{3} = 1 and then does transition to the linear combination of three possible states. Relation (13) reduces to the form

yielding a maximum compression to L_{D} = L_{C}/3 with the system in a single ground state. The associated work to maximum compression is given by

From relation among L_{A}, L_{B}, L_{C} and L_{D} we obtain work of the adiabatic compression in the form of initial width L_{A} of the whole process

Then, the total work is given by

and the efficiency is the same with a case of two states

We are also interested in other case. We consider first the multiply factor a that is the same for two cases, two-states and three-states, a = L_{C}/L_{B}. But, two previous cases are different at total expansion from initial width, 2a for two-states and 3a for three-states.

Now we consider a case of the expansion from initial witdh L_{A} not to L_{B} = 3L_{A} but to L_{B} = 2L_{A}. When the system expands isothermally and comes to rest at L_{B} = 2L_{A} then the work of the system is

At L_{B} = 2L_{A} the system still has a linear combination of all three possible states, and the coefficients a_{1}, a_{2} and a_{3} satisfy a relation

Equation (30) and orthonormal condition (22) form a line of condition of the system during adiabatic expansion from L_{B} = 2L_{A} to L_{C} (

When the system expands from 2L_{A} to L_{C}, its state is a fixed definite state at a line state _{C} = 3aL_{A}_{ }then the work (13) becomes

The next process is isothermal compression from L_{C} = 3aL_{A} to L_{D}. Since the expansion from L_{B} to L_{C} is adiabatic then the coefficients do not change. It implies that condition (30) prevails also at L_{C}. It further implies, from relation (13) that the maximum compression from L_{C} is is L_{D} = L_{C}/2, and the work is given by

At wall of width L_{C}/2 the system is at ground state, and then expands adiabatically until L_{A}. Since L_{D} = 3aL_{A}/2 then its work of the compression from L_{D} to L_{A} is

The total work of the cycle is

and the efficiency of the heat engine W/W_{AB} is given by

It is straightforward to generalize the three-state quantum Carnot engine to n-state. For n-state with initial state is single ground state and width is L_{A}, the isothermal expansion can be maximum of width nL_{A}. During the expansion the system does transition to all possible n states and it is as a liniear combination of all n-state. The coefficients of the expansion satisfy an orthonormal condition

At the maximum width the system back to a single state, the highest excited state, n^{th}-state. The work of the expansion is

The next expansion is adiabatic expansion from nL_{A} to naL_{A}. During the expansion, the state is in single n-state. The work is

that depend on the ratio of final and initial width of the adiabatic expansion.

The isothermal compression occurs with condition (13) of the form

then the maximum compression that leads to cyclic process is L_{D} = L_{C}/n. The work is

The adiabatic compression occurs in single ground state and its work is negative of adiabatic expansion (38). Then the total of cyclic work is

with the efficiency is equal to efficiency of 2 or 3 states (28).

The second scenario of system with n states is the isothermal expansion not to maximum nL_{A} but kL_{A} where integer k is smaller than n. When the system expands isothermally and comes to rest at L_{B} = kL_{A} then the work of the system is

At L_{B} = kL_{A} the system still has a linear combination of all possible states, and the coefficients a_{1}, a_{2}, , a_{n} also satisfy a relation

Equation (42) and orthonormal condition (36) make intersection and give a condition of the system along adiabatic expansion from L_{B} = kL_{A} to L_{C}.

When the system expands from kL_{A} to L_{C} its state is a fixed definite state at a point at hypervolume of intersecttion of orthonormal condition (36) and coefficient of maximum expansion. The coefficient number of force is given condition (42) and expansion from L_{B} = kL_{A} to L_{C} = naL_{A} is performed by the work

The next process is isothermal compression, with the initial width L_{C} = naL_{A}. Condition (42) yields a maximum compression L_{D} = L_{C}/k where at this width the system is at single a ground state. The work is given by

At wall of width L_{C}/k the system is at ground state, and then expands adiabatically until L_{A}. The work of compression is given by

The total work of the cycle is

and the efficiency of the heat engine W/W_{AB} depends on the first expansion factor k and total expansion na

In the perspective of energy, since L_{C} = aL_{B}, then the efficiency (28) which is also satisfied by system of nstate is the same to

Whereas, the efficiency (47) for the total expansion naL_{A} but intermediate expansion kL_{A} is the same to

where.

We have elaborated the quantum mechanical Carnot engine [^{2}/n^{2}. In other statement, if the too many number of state of system, causing decreasing efficiency, cannot be control, the efficiency can be enhanced by controlling its isothermal expansion.

The maximum isothermal expansion L_{B} of the n states system is nL_{A}. It is occurred when both initial and final state are single state, ground state for initial state and the highest state for the final state. However, if the initial width in the state of k then the maximum expansion is nL_{A}/k.

The authors give thank to the Theoretical Physics Group at University of Indonesia for its hospitality during Conference on Theoretical Physics and Nonlinoear Phenomena 2010. This work was support of Indonesia Ministry of Education.