Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA,
Fakultas MIPA, Universitas Negeri Yogyakarta, 2 Juni 2012
QUANTUM MECHANICAL IDEAL DIESEL ENGINE
E. Latifah1,2, A. Purwanto1
1
Laboratorium Fisika Teori dan Filsafat Alam, Jurusan Fisika, ITS, Surabaya
2
Jurusan Fisika, Universitas Negeri Malang (UM). Malang
Abstract
A cycle of quantum ideal diesel engine, based on quantum version of
thermodynamic processes, i.e., isentropic, isovolume and isobaric, had been designed.
By reformulation of the first law of thermodynamics, this engine was evaluated and its
efficiency was calculated. As a result, It is possible to resist the reduction in efficiency,
caused by increasing of the amount of energy levels of quantum system, by controlling
the compression of adiabatic process.
Keyword: quantum mechanics, ideal, diesel engine.
INTRODUCTION
As a device to convert heat energy into mechanical work, a thermodynamic heat engine
consists of an ideal gas, as a working substance, that expands and pushes a piston in a cylinder.
Quantum heat engines produce work using quantum matter as their working substance [1]. Heat
engine streams into study of quantum theory as a part of a consequency for more miniaturization of
devices, also heat engine. Very recently considerable progress has been made in understanding
foundational aspects of thermodynamics by addressing a new class of questions: whether there
exist additional fundamental limitations on thermal machines, arising specifically due to their size
[2, 3]. Further, the quantum nature of the working substance, can be expected to surpass the
maximum limit on the amount of work done by a classical thermodynamic cycle and then improves
the efficiency [1, 4]. Nowadays the physics of quantum heat engines is a rich field, some papers
describe one by studying the quantum thermodynamic processes, quantum generalization of force
or pressure to study quantum version of thermodynamic cycles [5-8]. While, other author stated
that the quantum efficiency can exceed the classical Carnot limit with quantum correlation [9].
Bender, et al., [10] provided a kind of cyclic Carnot heat engine employing a single
quantum-mechanical particle, as a working substance, confined to a potential well instead of gasfilled cylinder. The cycle consists of isothermal and adiabatic quantum processes that are close
analogues of the corresponding classical processes. By formulating 2-state quantum system the
efficiency is analogue for the classical Carnot efficiency. With a greater depth , the study for nstate of mechanical Carnot engine was done [5] and has result that the efficiency will be looser as
the greater of the number of eigenstates included, but it is possible to resist the reduction of
efficiency by controlling the expansion of isothermal. If it was difficult to control the amount of
eigenstate involved in the arrangement of the state of system of a quantum heat engine, the effort to
resist the decreasing of a quantum heat engine efficiency by studying its properties.
In this paper we design and evaluate, a cycle according to quantum diesel engine
employing a single quantum-mechanical particle, n-state, confined to a potential well. Rather than
having an ideal gas in a cylinder, we allow the walls of confining potential to play the role of the
piston by moving in and out. According to the one dimensional walls of confining potential
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E. Latifah and A. Purwanto/Quantum Diesel Engine
system, the quantities of pressure and volume are played by the force and the elongation or
compression of the moving wall. In this situation, it is natural to define the force on the wall of the
potential well as the negative derivative of the energy, dW   FdL .
By reformulating of the first law of thermodynamics , we defined adiabatic, isovolume and
isobaric quantum process. This formulation do not use the concept of temperature. It is represented
by the energy as given by the pure-state expectation value of Hamiltonian, that is, the ensemble
average of the energies of the quantum particle.
QUANTUM VERSION OF THERMODYNAMIC PROCESSES
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.
2
d 2
(1)

 E
2m dx 2
The infinite potential walls at x = 0 and x = L provide boundary conditions for the state functions
 0    L  0 . The normalized eigenfunctions n and its associated eigenvalues of energy En
have forms
2
 n 
n  x  
sin 
x ,
(2)
L
 L 
and
2
En  n2
2
.
2mL2
The expansion of state in these orthonormalized eigenfunctions (2) as
(3)

  x    ann  x  ,
n 1
(4)
with an are the coefficients of expansion.
The average of energy E, called the expectation value of Hamiltonian, can be given by

E   an En ,
2
(5)
n 1
and corresponds to probability of finding the value En, i.e. P(En)

E   P  En  En .
(6)
n 1
The coefficients an have the correct normalization

an

n 1
2
1.
(7)
The average energy (5), the eigenvalue of energy (3) can give the energy of the system as

E   an n2
2
2
2
.
(8)
2mL2
According to this system, one particle in 1D well, the energy of system (8) can be interpreted as
the average kinetic energy of the system.
We suppose that one of infinite walls of potential well, say the wall at x=L, is allowed to
move an infinitesimal amount dL then the wave function (x), eigen functions n (x) and energy
eigenvalues En 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,
n 1
F 
dW
dL
F-2
(9)
Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA,
Fakultas MIPA, Universitas Negeri Yogyakarta, 2 Juni 2012
To define further how the quantum version of thermodynamic processes are. It is
convenient to use statistical interpretation of quantum mechanics [11], because statistical
interpretation of quantum mechanics can be a bridge between microscopic sense of quantum
mechanics and macroscopic sense of thermodynamics. The expectation value of Hamiltonian (6)
stated the average energy of the system can be the expression of internal energy

U   Pn En .
(10)
n 1
A system contacts thermally with a reservoir, by the principle of conservation energy, the internal
energy change is caused partly by work done by the system dW and partly by heat added into the
system from the reservoir dQ. It can be stated as the first law of thermodynamic,
(11)
dU  dQ  dW .
While from equation (10) we have a reformulation of the first law of thermodynamics (11)
(12)
dU    En dPn  Pn dEn  .
n
Due to the relationship S  kB i Pi lnPi between the entropy S and probabilities Pi , we can make
identification [12],
(13)
dQ   En dPn ,
n
and
dW   Pn dEn .
(14)
n
Equation (14) implies that the work performed corresponds to the change in the energy eigenvalues
En, and this is in accordance with the fact that (3) work can only be performed through a change in
the generalized coordinates.
Quantum Adiabatic Process
A classical adiabatic thermodynamic process can be formulated in terms of a microscopic
quantum adiabatic thermodynamic process. Because quantum adiabatic processes proceed low
enough such that the generic quantum adiabatic condition is satisfied, then the population
distributions remain unchanged, dPn = 0. According to equation (13), dQ = 0, there is no heat
exchange in a quantum adiabatic process, but work can still be nonzero according to equation (14).
A classical adiabatic process, however, does not necessarily require the occupation probabilities to
be kept invariant. For example, when the process proceeds very fast, and the quantum adiabatic
condition is not satisfied, internal excitations will likely occur, but there is no heat exchange
between the working substance and the external heat bath. This thermodynamic process is classical
adiabatic but not quantum adiabatic [7]. Thus it can be verified that a classical adiabatic process
includes, as a subset, a quantum adiabatic process; but the inverse is not valid [6].
In case of the system is in adiabatic process, the internal energy is converted into
mechanical energy,
dU  dW .
(15)
We use an assumption that the initial state of the system a square well of width L is a linear
combination of n-eigenstates as in equation (4). In this process, the size of the potential well
changes as the moving wall moves. There are no transition between state occured, it can be
represented the absolute values of the expansion coefficient an must remain constant.
The eigenstates n  x  equation (2) and the corresponding energy eigenvalues En (3), as the
wall at x=L, moves an infinitesimal amount dL, vary smoothly as functions of L. Each eigenvalue
of energy En decreases as the piston moves out (as L increases), so from equation (3) and (5) we
have the expectation value of Hamiltonian decreases as
2 2
2
E  L    an n 2
(16)
2
2mL
n
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E. Latifah and A. Purwanto/Quantum Diesel Engine
It is natural to define the force on the wall of the potential well can be derivative of the energy, .
According to equation (5), (6), (9), (14) and (15) this situation give us
2 2
2
,
(17)
F   an n2
3
mL
n
as the force of the system in an adiabatic process.
Quantum Isobaric Process
Isobaric is a process adding heat at constant pressure. In this model, quantity of presure is
played by force, so the force is constant,
F  constant .
(18)
The first law of thermodynamic will be
(19)
dU  dQ  dW .
Quantum Isochoric Process
In isochoric process, we add heat at constant volume, dL=0, thus no work done by the
system,
dW  0 .
(20)
And,
(21)
dQ  dU ,
The internal energy goes up by the head added to the system.
QUANTUM IDEAL DIESEL ENGINE
A diesel engine (also known as a compression-ignition engine) is an internal combustion
engine that uses the heat of compression to initiate ignition to burn the fuel, which is injected into
the combustion chamber invented by Rudolph Diesel in 1987. An idealized mathematical model is
assumed to have constant pressure during the first part of the combustion phase L2 to L3, while a
real physical diesel do have an increase in pressure during this period.
In this design, the ideal diesel engine is constructed by a cycle of four stages of quantum
version of thermodynamic processes. The cycle consists of adiabatic compression, isobaric
expansion, adiabatic expansion and isovolume quantum processes (12), as seen in figur 1. The
adiabatic processes, as in thermodynamic, are impermeable to heat; heat flows into the loop
through the left expanding isobaric process and some of it flows back out through the isovolume
process.
Figure 1. Quantum Ideal Diesel Cycle
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Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA,
Fakultas MIPA, Universitas Negeri Yogyakarta, 2 Juni 2012
Using the quantum version of adiabatic, isobaric and isovolume processes, we can now
construct the cycle of quantum ideal diesel engine. We consider a case of n-eigenstates of potential
well contribute to the wave function in the well.
We consider the following cyclic process. We start with a ground-stste wave function in
well of width L1. At this point the state is
  
2
sin 
x
 L1 
L1


 i  1 L  L 
1
(22)
and the expectation value of Hamiltonian,
Ei 
2
2
(23)
2mL12
Step 1: The wall is compressed L1 to L2 adiabatically, the population distributions of everu
eigenstates remain unchanged, so the wave function is only the groundstate and the average energy
along this process is given by
E1 2 
2
2
2mL2
.
(24)
The force to the wall, as stated in equation (15) with n=1 is
2 2
.
F12 
3
(25)
mL
Step 2 : The process is isobaric expansion. Starting in L2, some initial quantities according
to this process are the state function
 
2
(26)
2i 3 
sin  x  ,
L2
 L2 
The energy
E2i 3 
2
2
2mL22
,
(27)
And the force
F2i3 
2
2
(28)
mL32
Along isobaric process, there are transition between eigenstates, the probabilities (the square
absolute of expansion coefficients) are no longer constant, thus the equations (26), (27), (28) would
be,

23   ann  x  ,
(29)
n 1
the wave function as a superposition of n-eigenstates. The energy is
2 2
2
.
E23   an n 2
2
2mL
n
(30)
And the force is
F23   an n2
2
n
2
2
mL3
.
Quantum version of this isobaric process gives rice the constant force along the process,
F2i3  F23 .
Thus, from equation (28) and (31), we have
L3   an  L  n2 L32
2
(31)
(32)
(33)
n
It is a convenient to use a supposition the expansion factor along isobaric expansion is 3 , so that
(34)
L3  3 L2 .
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E. Latifah and A. Purwanto/Quantum Diesel Engine
and from equation (3) we have 33   an  L  n2 and 3 has maximum value an  L   1 or the state
2
2
n
of particle only in highest excited state.
Step 3: Adiabatic expansion. Suppose that at the end of isobaric expansion, the wave
function consists of of n-eigenstates. So, we have the expectation value of Hamiltonian,
2 2
2
.
(35)
E34   an n 2
2
2mL
n
The force asserted is
F34   an n2
2
2
n
2
.
3
mL
(36)
Step 4: Finally, we continue to isovolume process. The system does not do work. The
expectation value of Hamiltonian is given by
E41   an n 2
2
n
2
2
(37)
2
A
2mL
The force becomes decrease and reaches to the initial value of the cycle.
The four-step cyclic quantum process that we have just described is illustrated in figure 1.
The diesel cycle is drawn in the (F,L)-plane, which is the one-dimensional version of the (P,V)plane. The area of the closed loop represents the mechanical work W done in a single cycle of the
quantum heat engine. To calculate the work done W, we evaluate the following integral
Wtot 
L2

L3
L4
L1
L2
L3
L4
F1 2 dL   F23 dL   F3 4 dL   F4 1dL
L1
(38)
The last term of equation (38) is zero. and by assumption L3  3 L2 and L1  1 L2 , we get the total
work,
2 2 
1 3
(39)
Wtot 
3 3  3  2  32 
2 
2mL2 
1 1 
The effort to calculate the efficiency of this quantum diesel engine reaches to evaluate the
head added into the system along isobaric expansion. The first law of thermodynamic can give us
information about the heat added into the system,
QH  dU  dW
 E2f3  E2i 3  W23 .


2
2
2mL22
(40)
 33  3
The efficiency of an engine is defined to be

Wtot
.
QH
(41)
Using equation (39) and (40) we can write the formula of efficiency (41) as
  1
 32   3  1
.
312
(42)
It was known that 33   an  L  n2 , or  3 increases by the multiplicity of the eigenstates
2
n
participating in the system. The second term of (42) has denominator, namely, compression factor
of adiabatic compression.
F-6
Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA,
Fakultas MIPA, Universitas Negeri Yogyakarta, 2 Juni 2012
CONCLUSION
By reformulating the first law of thermodynamic in fashion of probability interpretation of
quantum mechanic, we define quantum version of adiabatic, isobaric and isovolume processes.
This processes have been constructed to be a quantum mechanical ideal diesel engine.
We have shown that more eigen states mixed up with the system of quantum diesel engine,
the efficiency would be decrease under certain condition. If the multiplicity of eigenstates cannot
be avoided, we can resist the decreasing of efficiency by controlling the adiabatic compression.
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