Aerothermodynamics of high speed flows AERO 0033–1 Lecture 10: Quantum Mechanics and Statistical Physics (introduction) Thierry Magin, Greg Dimitriadis, and Johan Boutet [email protected]Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics Aerospace and Mechanical Engineering Department Faculty of Applied Sciences, University of Li` ege Room B52 +1/433 Wednesday 9am – 12:00pm February – May 2019 Magin (AERO 0033–1) Aerothermodynamics 2018-2019 1 / 80
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Aerothermodynamics of high speed flowsAERO 0033–1
Lecture 10: Quantum Mechanics and Statistical Physics (introduction)
1 The Dawn of the quantum theoryThe photoelectric effectThe Hydrogen atomic spectrumThe de Broglie wavesThe Bohr theory of the hydrogen atomHeisenberg’s uncertainty principle
2 The Schrodinger equation and a particle in a boxThe classical wave equationThe Schrodinger equationA particle in a boxPostulates of quantum mechanics
3 Rovibrational energy levels of diatomic moleculesHarmonic oscillatorRigid rotator
First Solvay conference in 1911 at the Hotel Metropole (Brussels)
Seated (L-R): W. Nernst, M. Brillouin, E. Solvay, H. Lorentz, E. Warburg, J. Perrin, W. Wien,M. Curie, and H. Poincare, standing (L-R): R. Goldschmidt, M. Planck, H. Rubens, A.
Sommerfeld, F. Lindemann, M. de Broglie, M. Knudsen, F. Hasenohrl, G. Hostelet, E. Herzen,J.H. Jeans, E. Rutherford, H. Kamerlingh Onnes, A. Einstein and P. Langevin
1 The Dawn of the quantum theoryThe photoelectric effectThe Hydrogen atomic spectrumThe de Broglie wavesThe Bohr theory of the hydrogen atomHeisenberg’s uncertainty principle
2 The Schrodinger equation and a particle in a box
3 Rovibrational energy levels of diatomic molecules
The Dawn of the quantum theory The photoelectric effect
The photoelectric effect
Hertz’s experiment (1886)UV light causes electrons to be emitted from a metallic surface:
Electron KE independent of incident radiation intensityThreshold frequency ν0 below which no electrons are ejectedAbove ν0, electron KE varies linearly with frequency ν
Einstein’s interpretation (1905)Planck’s hypothesis for the energy of electrons in the constituents ofradiating blackbody: E = hν, with h = 6.626 10−34 J sEinstein: the photoelectric effect is caused by absorption of quanta oflight (photons)
The Dawn of the quantum theory The photoelectric effect
Exercise
When lithium is irradiated with light, the kinetic energy of the ejectedelectrons is 2.935 ×10−19 J for λ = 300 nm and 1.280 ×10−19 J forλ = 400 nm. Calculate
1 The Planck constant,
2 The threshold frequency,
3 The work function of lithium,
for these data. The speed of light is 2.998× 108 m/s.
The Dawn of the quantum theory The Hydrogen atomic spectrum
The Hydrogen atomic spectrum
Every atom, when subjected to high temperatures or electricaldischarge, emits electromagnetic radiation of characteristic frequencies
Characteristic emission spectrum of an atom: certain discretefrequencies called linesHydrogen atomic spectrum: Lyman (1906), Balmer (1885), andPaschen (1908) series
The Balmer and Rydberg formulae for the frequency of the lines(1/λ = ν/c , RH = 109 677.581 cm−1)
The Dawn of the quantum theory The de Broglie waves
The de Broglie waves
de Broglie (1924): wave-particle duality of light and matterEinstein’s eq. for photons: λ = h
p , with p = mvde Broglie: both light and matter obey this equation
Experimental observationWhen a beam of X rays / electrons is directed at a crystalline substance, the beamis scattered in a definite manner characteristic of the atomic structure of the crystalDiffraction occurs because the interatomic spacings in the crystal are about thesame as the wavelength of the X-rays / electrons
The Dawn of the quantum theory The Bohr theory of the hydrogen atom
The Bohr theory of the hydrogen atom
Bohr (1911)Hydrogen atom: rather massive nucleus with one associated electronNucleus assumed to be fixed with electron revolving about it
Classical physics:Because the electron is constantly accelerated, it should emit electromagneticradiation and lose energyIt will spiral into the nucleus and a stable orbit is classically forbidden
Two nonclassical assumptions:Existence of stationary electron orbitsThe de Broglie waves of the orbiting electron must be in phase as the electronmakes one complete revolution as shown in case (a), as opposed to cases (b) and (c)
The Dawn of the quantum theory Heisenberg’s uncertainty principle
Heisenberg’s uncertainty principle
The position and the momentum of a particle cannot be specifiedsimultaneously with unlimited precision: ∆x∆p & h
e.g ., to measure the position of an electron within a distance ∆x , we can use light with a
wavelength λ ∼ δx . For the electron to be “seen” a photon must interact or collide in
some way with the electron. During the collision, some of the photon momentum
p = h/λ will be transferred to the electron, changing its momentum. A smaller
wavelength leads to a smaller ∆x and a larger ∆p.
If we wish to locate any particle within a distance ∆x , then weintroduce atomically an uncertainty in its momentum ∆pThis uncertainty does not stem from poor measurement orexperimental technique, but it is a fundamental property of the act ofmeasurement itselfThe Heisenberg uncertainty principle is of no consequence formacroscopic bodies but it has very important consequences in dealingwith atomic and subatomic particles
2 The Schrodinger equation and a particle in a boxThe classical wave equationThe Schrodinger equationA particle in a boxPostulates of quantum mechanics
3 Rovibrational energy levels of diatomic molecules
The Schrodinger equation and a particle in a box The Schrodinger equation
The Schrodinger equation (1925)
The Schrodinger eq. is the fundamental equation of quantummechanics
This eq. cannot be derived, just as Newton’s laws are fundamentalpostulates of classical mechanicsIt gives a description of a quantum system evolving with time, such asatoms, molecules, and subatomic particlesThe Schrodinger equation was developed principally from the deBroglie hypothesisThe solutions to this eq. are called wave functions
The time-independent Schrodinger eq. for finding the wave functionof a particle
Even though we cannot derive the Schrodinger eq., we can at leastshow that it is plausible
The Schrodinger equation and a particle in a box A particle in a box
Interpretation of the wave function
Born’s interpretation of the wave functionψ∗(x)ψ(x)dx is the probability that the particle is located between x and x + dxThe probability that the particle is found outside of 0 ≤ x ≤ a is zero: ψ(x) = 0ψ(x) is assumed to be a continuous function ⇒ ψ(x = 0) = 0 and ψ(x = a) = 0
The wave functions must be normalizedThe particle is certain to be found in the region 0 ≤ x ≤ a:∫ a
0ψ∗(x)ψ(x)dx = 1
|B|2∫ a
0sin2 kx dx = 1
|B|2[
12x −
sin 2kx
4k
]a0
= 1 (2 sin2 kx = 1− cos2kx)
⇒ B =
(2
a
)1/2
ψn(x) =(
2a
)1/2sin nπx
a, n = 1, 2, . . .
Probability of finding a particle between x1 and x2Prob(x1 ≤ x ≤ x2) =
The Schrodinger equation and a particle in a box A particle in a box
Quantization of the energy
Quantized energy: En = h2n2
8ma2 n = 1, 2, . . .
The energy of the particle is quantized and the integer n is called aquantum numberQuantization arises naturally from the boundary conditions, beyond thestage of Bohr and Planck where the are introduced in an ad hocmanner
Schrodinger, Annalen der Physik 79, 361 (1926)“In this communication, I wish to show that the usual rules of quantization can bereplaced by another postulate (the Schrodinger equation) in which there occurs nomention of whole number. Instead, the introduction of integers arises in the samenatural way as, for example, in a vibrating string, for which the number of nodes isintegral. The new conception can be generalized, and I believe that it penetratesdeeply into the true nature of the quantum rules.”
The Schrodinger equation and a particle in a box A particle in a box
The energy levels, wavefunctions (a), and
probability densities (b)for the particle in a box
Application: excitation of πelectrons in butadiene
H2C = CHCH = CH2 is assumedto be a linear moleculeLength a= 578 pm with 2 C=Cbonds (2×135 pm), 1 C-C bond(154 pm), 2 C radii (2×77 pm)Pauli exclusion principle: eachenergy state can hold only twoelectrons (with opposite spins)The four π electrons fill the firsttwo energy levelsFirst excited state (3rd level):
The Schrodinger equation and a particle in a box A particle in a box
A particle in a 3D box
The energy levels for a particle in a cube (a=b=c)
The energy levels of a particle in a cube are degenerate (same energyvalue for different index triplets) because of the symmetry introducedwhen a general rectangular box becomes a cube
The Schrodinger equation and a particle in a box Postulates of quantum mechanics
Postulates
The development of quantum mechanics is now sufficiently complete that we can reduce
the theory to a set of postulates. MacQuarrie and Simon have reviewed the following
postulates in their book Physical Chemistry, a molecular approach
Postulate 1. The state of a quantum-mechanical system is completelyspecified by a function ψ(x) that depends on the coordinate of theparticle. All possible information about the system can be derivedfrom ψ(x). This function, called the wave function or the statefunction, has the important property that ψ∗(x)ψdx is the probabilitythat the particle lies in the interval dx , located at the position x .
For simplicity of notation, we have assumed, that only one coordinate (x) is needed to
specify the position of one particle, as in the case of a particle in a one-dimensional box
The Schrodinger equation and a particle in a box Postulates of quantum mechanics
Postulates
Postulate 3. In any measurement of the observable associated withthe operator A, the only values that will ever be observed are theeigenvalues an, which satisfy an eigenvalue equation
Aψn = anψn.
Property: The eigenvalues of Hermitian operators are real and theireigen functions are orthonormal.
e.g., particle in 1D box: Hψn(x) = Enψn(x), with energy
yields to < A2 >< B2 > ≥ 14| < [A, B] > |2. The proof follows from the definition
σ2c =< (C− < C >)2 > and the following identity: [A− < A >, B− < B >] = [A, B].
There is an intimate connection between commuting operators and the UncertaintyPrinciple. If two operators A and B commute, then their eigen values a and b can bemeasured simultaneously to any precision. If two operators A and B do not commute,then their eigen values a and b cannot be measured simultaneously to any precision.
Gas composed of independent particles (perfect gas) thatare identical
Let us assume that the total energy of level I for a molecule is givenby its translational, rotational, vibrational and electroniccontributions: εI = εTI + εRI + εVI + εEI , I = 1, 2, . . .
Consider the following system of independent and identical particlesNumber of molecules:
∑I NI = N
Total energy:∑
I NI εI = EVolume: V
A macrostate NI is a population distribution N1,N2, . . . over theenergy levels
At some given instant, the molecules are distributed over the energy levels in adistinct way.In the next instant, due to molecular collisions, the populations of some levels maychange, creating a different set of NI ’s, and hence a different macrostate.
Different states can be associated to the same energy level εI when itis degenerate. The occupancy of an energy level with degeneracy aicompletely defines a microstate for a given macrostate.
Consider particles weakly interacting in a box (cube of volumeV = a3). Introducing the notation n2 = n2
x + n2y + n2
z , withnx , ny , nz = 1, 2, . . ., their translational energy is given byεT = h2/(8mV 2/3)n2
Show that the number of states with an energy less than ε isΓ = 4
3πV /h3(8mε)3/2
Compute the number of nitrogen molecules N0 in a volume of 1 cm3 attemperature T = 0 C and pressure p = 1 atm.Compute Γ assuming that ε = 10 < ε >, where < ε >= 3
2 kBT is theaverage kinetic energy of these particles.Show that in this case the great majority of states are emptyΓ/N0 1.
As consequence, it is possible to group the translational energy levels of similar energiesand to divide the energy scale into successive regions containing a range of energy valuessuch that the following inequality holds ai NI
Counting the number of microstatesTwo types of particles
Molecules and atoms with an odd number of elementary particles arecalled fermions and obey a Fermi-Dirac statistics.Molecules and atoms with an even number of elementary particles arecalled bosons and obey a Bose-Einstein statistics.
Quantum mechanical property: indistinguishability of particles
The number of microstates W in a given macrostate N1,N2, . . .follows Pauli’s exclusion principle
For fermions, only one molecule may be in any given degenerate state at anyinstant (C(ai ,NI ) combination of ai states taken NI times without repetition)
W (NI ) =∏I
ai !
(ai − NI )!NI !
For bosons, the number of molecules that can be in any given degenerate state atany instant is unlimited (C(ai + NI − 1,NI ) combination of ai states taken NI timeswith repetition)
The total number of microstates of the system is defined as
Ω =∑
over all sets of NIsuch that
∑I NI=N
and∑
I NI εI =E
W (NI )
The most probable macrostate NI is that macrostate which has themaximum number of microstates. It is derived to be
NI =ai
eαeβεI ± 1
where quantities α and β are Lagrange multipliers associated with theconstraints (+1 for fermions and -1 for bosons)
That constitutes the thermodynamic equilibrium of the system.For N large, it can be shown that only the largest term in the summakes any effective contribution to Ω and one has:
The values of α and β can be obtained from the constraints∑I NI = N and
∑I NI εI = E
We investigate the limiting case for which |eαeβεI | 1, the physicalmeaning of this assumption will be discussed later. The Boltzmanndistribution is derived as NI = aie
−αe−βεI .Notice that this result is based on indistinguishability of particles as opposed to a purely
classical derivation based on Boltzmann statistics.
Introducing the partition function Q =∑
I aie−βεI , quantity α is
given by α = exp(Q/N) and the following equation is obtained fromthe conservation of particles
Boltzmann’s relationUsing the sparse population property ai NI shown in a previousexercise and the formula ln(1 + x) ∼ x for |x | 1, one obtainslnW =
∑I NI (ln ai
NI+ 1). The number of microstates is given by
ln Ω = lnWmax = N(ln QN + 1) + βE
All possible microstates of a system corresponding to given values of N, E , and the εI ’sare a priori equally probable. Since Ω = Wmax , in the Boltzmann limit, this system spendmost of its time in the macrostate corresponding to the Boltzmann distribution.
Boltzmann postulated a functionalrelationship between the entropy of asystem and the number of microstates
S = kB ln Ω
= kBN(ln QN + 1) + kBβE
where kB is Boltzmann’s constant
The entropy is additive: S1 + S2 =kB ln Ω1 + kB ln Ω2 = kB ln(Ω1Ω2)
In the Boltzmann limit, let us examine the number of ways W inwhich a distribution can occur for values NI = N∗I + ∆NI around theBoltzmann distribution N∗I .
Assuming that |∆NI | N∗I , expand ln(N∗I + ∆NI ) neglecting thirdorder terms.Show that the number of microstate satisfies the following relation
ln WWmax
∼ − 12
∑I
(∆NI
N∗I
)2
N∗I
Consider 1 cm3 of nitrogen gas at 1atm pressure and 0 C. Supposethat the average deviation |∆NI |/N∗I is 0.1%, compute the value of theratio W /Wmax .
We note that the right-hand side is negative showing that the extremum is a maximum.
In a system with a large number of particles, even a small deviation in the distribution
numbers from the Boltzmann values N∗I leads to an enormous reduction in the number of
Show that the Lagrange multiplier is given in the Boltzmann limit bythe following expression
α = ln
[VN
(2πmkBT
h2
)3/2]
+ ln(Q int
).
The internal partition function Q int is not small. This equation shows a posteriori thatthe condition |eαeβεI | 1 for the Boltzmann limit reduces to
V
N
(2πmkBT
h2
)3/2
1.
This requirement would be violated and the gas degenerate, for example, when theparticle mass is very small and the number density is sufficiently large, as in the case of anelectron gas in metals. It is also violated for normal molecular values of the mass m andnumber density N/V when the temperature T is very low. In this situation the particlestend to crowd together in the lower energy states.
Statistical physics Mixture of inert perfect gases
Energy of chemical bonds
Formation enthalpy
Energy is released in the gas or absorbed by the gas when chemicalreactions occurA common level from which all the energies are measured can beestablished by means of a formation enthalpy hFiNotice that assigning a formation enthalpy to species is an arbitraryconvention, solely a difference of energy can be measured in a chemicalreactor (1st law of thermodynamics)This formation enthalpy can be fixed for instance at 0 K
Formation entropy
There is no need for a formation entropyThe entropy of a perfect crystal at 0 K is exactly equal to zero (3rd lawof thermodynamics)Notice that the semi-classical expression of the entropy presented inthis introduction does not satisfy this property
Statistical physics Mixture of inert perfect gases
Boltzmann distribution for a mixture of perfect gases
The most probable macrostate of gas mixture can be obtained in theBoltzmann limiting case following a method similar to the onedeveloped for the single species gas
Ni ,I = ai ,I e−αi−βmih
Fi e−βεi,I
Boltzmann distribution for equilibrium population Ni ,I of energy levelsI of species i Ni ,I
Ni=
1
Qiai ,I exp(
−εi ,IkBT
)
Energy level I of species i : εi,IDegeneracy of level εi,I of species i : ai,IPartition function of species i : Qi =
Statistical physics Mixture of inert perfect gases
Exercise1 Compute the specific heat ratio for:
A monoatomic gas such as He,Air at ambient temperature.
Molecular mass data: MN = 14× 10−3kg /mol, MO = 16× 10−3kg /mol, and
MHe = 4× 10−3kg /mol.
2 Explain why your voice changes when you breath helium.
The speed of a sound wave through helium will then be much higher. So by inhaling
helium and using it as the source of the perceived sound, the frequency of the voice
changes but neither the pitch, since your vocal chords are vibrating at the same speed as
when you are using air, nor the configuration of the vocal tract. So while the base
frequency of the chords remains the same, the frequency of the sound heard by others is
increased.
3 Give bounds for the specific heat of a nitrogen gas at the exit of thenozzle of the Longshot facility assuming that the nitrogen moleculesare not dissociated and their vibrational energy mode partially excited.
Some high-temperature effects in nitrogen can be simulated by means of cold helium.
Statistical physics Mixture of reacting perfect gases
Mixture of reacting perfect gases
Gas mixture of species i ∈ S = R∪ EIndependent chemical components (elements / charge) i ∈ EDependent chemical components i ∈ R
A convenient set of chemical reactions, sufficient to compute theequilibrium composition, is obtained by writing formation of thedependent components in terms of elements
Xi ∑j∈E
ν ijXj , i ∈ R,
Notice that these reactions do not correspond to elementary processes
Stoichiometric matrix ν ij , i ∈ R, j ∈ E , for 8-species carbon dioxide
Statistical physics Mixture of reacting perfect gases
A library for high enthalpy and plasma flows at VKI
Quantities relevant to engineering design for hypersonic flows
Heat flux & shear stress to the surface of a vehicleTheir prediction strongly relies on completeness and accuracy of thenumerical methods & physico-chemical models
Why a library for physico-chemical models?
Implementation common to several CFD codesNonequilibrium models, not satisfactory today, are regularly improvedBasic data are constantly updated(spectroscopic constants, cross-sections,. . .)
Constraints for the library implementationHigh accuracy of the physical models
Laws of thermodynamics must be satisfiedValidation based on experimental data
Statistical physics Mixture of reacting perfect gases
MUTATION++ library
MUTATION++ library: MUlticomponent Transport AndThermodynamic properties / chemistry for IONized gases written inC++
⇒ Breeding of the following libraries: Pegase (VKI), Chemkin(Livermore), and EGlib (Ecole Polytechnique)
Modules1 Thermodynamic properties2 Chemistry3 Transport properties4 Energy / chemistry exchange terms
New features versus the fortran Mutation libraryFunctionality rebuilt into a modern, object oriented, extensible framework (J.B.Scoggins)Improved methodology when solving equilibrium compositionsEfficient calculation of costly model parameters using lookup tables defined a priorior during initialization based on error constraints