underpins%20thermodynamics,%20ideal%20gas%20(a%20classical%20physics%20model),%20ensembles%20of%20molecules,%20solids,%20liquids%20

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CHAPTER 9 Statistical Physics

• underpins thermodynamics, ideal gas (a classical
  physics model), ensembles of molecules, solids,
  liquids the universe
• 9.1 Justification for its need !
• 9.2 Classical distribution functions as examples
  of distributions of velocity and velocity2 in
  ideal gas
• 9.3 Equipartition Theorem
• 9.4 Maxwell Speed Distribution
• 9.5 Classical and Quantum Statistics
• 9.6 Black body radiation, Liquid Helium,
  Bose-Einstein
• condensates, Bose-Einstein
  statistics,
• 9.7 Fermi-Dirac Statistics

Ludwig Boltzmann, who spent much of his life
studying statistical mechanics, died in 1906 by
his own hand. Paul Ehrenfest, carrying on his
work, died similarly in 1933. Now it is our turn
to study statistical mechanics. Perhaps it will
be wise to approach the subject cautiously. -
David L. Goldstein (States of Matter, Mineola,
New York Dover, 1985)
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First there was classical physics with a cause
(or causes)
Newtons three force laws, first unification in
physics
Lagrange around 1790 and Hamilton around 1840
added significantly to the computational power of
Newtonian mechanics. Pierre-Simon de Laplace
(1749-1827) Made major contributions to the
theory of probability and well known clockwork
universe statement It should be possible in
principle to have perfect knowledge of the
universe. Such knowledge would come from
measuring at one time the position and velocities
of every particle of matter and then applying
Newtons law. As they are cause and effect
relations that work forwards and backwards in
time, perfect knowledge can be extended all the
way back to the beginning of the universe and all
the way forward to its end. So no uncertainty
principle allowed
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then there was the realization that one does not
always need to know the cause (causes), can do
statistical analyses instead

• Typical problem, flipping of 100 coins,
• One can try to identify all physical condition
  before the toss, model the toss itself, and then
  predict how the coin will fall down
• if all done correctly, one will be able to make a
  prediction on how many heads or tails one will
  obtain in a series of experiments
• Statistics and probabilities would just predict
  50 heads 50 tails by ignoring all of that
  physics,
• The more experimental trials, 100,000 coin
  tosses, the better this prediction will be borne
  out

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Speed distribution of particles in an ideal gas
in equilibrium, instead of analyzing what each
individual particle is going to do, one derives a
distribution function, determines the density of
states, and then calculates the physical
properties of the system (always by the same
procedures)
ltKEgt ltp2gt/2m
There is one characteristic kinetic energy (or
speed) distribution for each value of T, so we
would like to have a function that gives these
distribution for all temperatures !!!
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Path to statistical physics from classical to
quantum for bosons and fermions

• Benjamin Thompson (Count Rumford) 1753 1814
• Put forward the idea of heat as merely the
  kinetic energy of individual particles in an
  ideal gas, speculation for other substances.
• James Prescott Joule 1818 1889
• Demonstrated the mechanical equivalent of heat,
  so central concept of thermodynamics becomes
  internal energy of systems (many many particles
  at once)

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Beyond first or second year college physics
James Clark Maxwell 1831 1879, Josiah Willard
Gibbs 1839 1903, Ludwig Boltzmann 1844 1906
(all believing in reality of atoms, tiny minority
at the time) Brought the mathematical theories
of probability and statistics to bear on the
physical thermodynamics problems of their
time. Showed that statistical distributions of
physical properties of an ideal gas (in
equilibrium a stationary state) can be used to
explain the observed classical macroscopic
phenomena (i.e. gas laws) Gibbs invents notation
for vector calculus, the form in which we use
Maxwells equations today Maxwells
electromagnetic theory succeeded his work on
statistical foundation of thermodynamics so he
was a genius twice over.
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and then there came modern physics
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9.2 Maxwell Velocity and Velocity2 Distribution

• internal energy in an ideal gas depends only on
  the movements of the entities that make up that
  gas.
• Define a velocity distribution function .
• the probability of finding a
  particle with velocity
• between .
• where

is similar to the product of a wavefunction with
its complex conjugate (in 3D), from it we can
calculate expectation values (what is measured on
average) by the same integration procedure as in
previous chapters !!
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Maxwell Velocity Distribution

• Maxwell proved that the velocity probability
  distribution function is proportional to exp(-½
  mv2 / kT), special form of exp(-E/kT) the
  Maxwell-Boltzmann statistics distribution
  function.
• Therefore where C is a proportionality
  factor and ß (kT)-1. k Boltzmann constant,
  which we find everywhere in this field
• Because v2 vx2 vy2 vz2 then
• Rewrite this as the product
• of three factors.

Is the product of the three functions gx, gy gz
which are just for one variable (1D) each
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Maxwell Velocity Distribution

• g(vx) dvx is the probability that the x component
  of a gas molecules velocity lies between vx and
  vx dvx.
• if we integrate g(vx) dvx over all of vx and
  set it equal to 1, we get the normalization
  factor
• The mean value (expectation value) of vx

Full Widths at Half Maximum e-0.5 0.607 g(0)
That is similar to the expectation value of
momentum in the square wells
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Maxwell Velocity2 Distribution

• The mean value of vx2, also an expectation value
  that is a simple function of x

This is not zero because it is related to kinetic
energy, remember the expectation value of p2 was
also not zero
It relates the human invented energy scale (at
the individual particle level) to the absolute
temperature scale (a physical thing)
1.3806488(13)10-23 J?K-1
8.6173324(78)10-5 eV?K-1
gas constant R divided by Avogadros number NA
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Maxwell Velocity2 Distribution

• The results for the x, y, and z velocity2
  components are identical.
• The mean translational kinetic energy of a
  molecule
• Equipartion of the kinetic energy in each of 3
  dimension a particle may travel, in each degree
  of freedom of its linear movement
• this result can be generalized to the
  equipartition theorem

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9.3 Equipartition Theorem

• Equipartition Theorem
• For a system of particles (e.g. atoms or
  molecules) in equilibrium a mean energy of ½ kT
  per system member is associated with each
  independent quadratic term in the energy of the
  system member.
• That can be movement in a direction, rotation
  about an axis, vibration about an equilibrium
  position, , 3D vibrations in a harmonic
  oscillator
• Each independent phase space coordinate
• degree of freedom

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Equipartition Theorem

• In a monatomic ideal gas, each molecule has
• There are three degrees of freedom.
• Mean kinetic energy is 3(1/2 kT) 3/2 kT
• In a gas of N helium atoms, the total internal
  energy is
• CV 3/2 N k
• For the heat capacity for 1 mole
• The ideal gas constant R 8.31 J/K

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As predicted, only 3 translational degrees of
freedom
2 more (rotational) degrees of freedom
2 more (vibrational) degrees of freedom plus
vibration, which also adds two times 1/2 kBT
discrepancies due to quantized vibrations, not
due to high particle density
We get excellent agreement for the noble gasses,
they are just single particles and well isolated
from other particles
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Molar Heat Capacity

• The heat capacities of diatomic gases are
  temperature dependent, indicating that the
  different degrees of freedom are turned on at
  different temperatures.
• Example of H2

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The Rigid Rotator Model

• For diatomic gases, consider the rigid rotator
  model.
• The molecule rotates about either the x or y
  axis.
• The corresponding rotational energies are ½ Ix?x2
  and ½ Iy?y2.
• There are five degrees of freedom, three
  translational and two rotational. (I is
  rotational moment of inertia)

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Two more degrees of freedom, ½ Ix?x2 and ½ Ix?x2
Two more degrees of freedom, because there are
kinetic and potential energy, both are
quadratic (both have variables that appear
squared in a formula of energy is a degree of
freedom, ½ m (dr/dt)2 and ½ ? r2
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Using the Equipartition Theorem

• In the quantum theory of the rigid rotator the
  allowed energy levels are
• From previous chapters, the mass of an atom is
  largely confined to its nucleus
• Iz is much smaller than Ix and Iy. Only rotations
  about x and y are allowed at reasonable
  temperatures.
• Model of diatomic molecule, two atoms connected
  to each other by a massless spring.
• The vibrational kinetic energy is ½ m(dr/dt)2,
  there is kinetic and potential energy ½ ? r2 in a
  harmonic vibration, so two extra degrees of
  freedom
• There are seven degrees of freedom (three
  translational, two rotational, and two
  vibrational for a two-atom molecule in a gas).

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not that simple
six degrees of freedom
according to classical physics, Cv should be 3 R
6/2 kBT NA for solids and independent of the
temperature
We will revisit this problem when we have learned
of quantum distributions, concept of phonons,
which are quasi-particle that are not restricted
by the Pauli exclusion principle
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Maxwells speed (v) distribution
Slits have small widths, size of it defines dv (a
small speed segment of the speed distribution)
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9.4 Maxwell Speed Distribution ?v?

• Maxwell velocity distribution
• Where
• lets turn this into a speed distribution.
• F(v) dv the probability of finding a particle
  with speed
• between v and v dv.
• One cannot derive F(v) dv (i.e. a distribution of
  a scalar entity) simply from f(v) d3v (the
  velocity distribution function, i.e. a
  distribution of vectors and their components), we
  need idea of phase space for this derivation

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Maxwell Speed Distribution

• Idea of phase space, to count how many states
  there are
• Suppose some distribution of particles f(x, y, z)
  exists in normal three-dimensional (x, y, z)
  space.
• The distance of the particles at the point (x, y,
  z) to the origin is
• the probability of finding a particle
  between .

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Maxwell Speed Distribution

• Radial distribution function F(r), of finding a
  particle between r and r dr sure not equal to
  f(x,y,z) as we want to go from coordinates to
  length of the vector, a scalar
• The volume of any spherical shell is 4pr2 dr.
• now replace the 3D space coordinates x, y, and z
• with the velocity space coordinates vx, vy, and
  vz
• Maxwell speed distribution
• It is only going to be valid in the classical
  limit, as a few particles would have speeds in
  excess of the speed of light.

note speed distribution function is different to
velocity distribution function, but both have the
same Maxwell-Boltzmann statistical factor
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Maxwell Speed Distribution

• The most probable speed v, the mean speed ,
  and the root-mean-square speed vrms are all
  different.

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Maxwell Speed Distribution

• Most probable speed (at the peak of the speed
  distribution), simply plot the function, take the
  derivative and set it zero, derive the
  consequences
• Average (mean) speed, will be an expectation
  value that we calculate from on an integral on
  the basis of the speed distribution function

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average (mean) of the square of the speed, will
be an expectation value that we calculate from
another integral on the basis of the speed
distribution function
We define root mean square speed on its basis
which is of course associated with the mean
kinetic energy
We can also calculate the spread (standard
deviation) of the speed distribution function in
analogy to quantum mechanical spreads
Note that sv in proportional to
So now we understand the whole function, can make
calculations for all T
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Straightforward turn speed distribution into
kinetic energy (internal energy of ideal gas)
distribution
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So we recover the equipartition theorem for a
mono-atomic gas
Number of particles with energy in interval E and
E dE
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9.5 Needs for Quantum Statistics

• If molecules, atoms, or subatomic particles are
  fermions, i.e. most of matter, in the liquid or
  solid state, the Pauli exclusion principle
  prevents two particles with identical wave
  functions from sharing the same space. The
  spatial part of the wavefunction can be identical
  for two particles in the same state, but the spin
  part f the wavefunction has to be different to
  fulfill the Pauli exclusion principle.
• If the particles under consideration are
  indistinguishable and Bosons, they are not
  subject to the Pauli exclusion principle, i.e.
  behave differently
• There are only certain energy values allowed for
  bound systems in quantum mechanics.
• There is no restriction on particle energies in
  classical physics.

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Classical physics Distributions

• Boltzmann showed that the statistical factor
  exp(-ßE) is a characteristic of any classical
  system in equilibrium (in agreement with
  Maxwells speed distribution)
• quantities other than molecular speeds may
  affect the energy of a given state (as we have
  already seen for rotations, vibrations)
• Maxwell-Boltzmann statistics for classical
  system ß (kBT)-1
• The energy distribution for classical system
• n(E) dE the number of particles with energies
  between E and E dE.
• g(E) the density of states, is the number of
  states available per unit energy range.
• FMB gives the relative probability that an energy
  state is occupied at a given temperature.

A is a normalization factor, problem specific
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Classical / quantum distributions

• Characteristic of indistinguishability that makes
  quantum statistics different from classical
  statistics.
• The possible configurations for distinguishable
  particles in either of two (energy or anything
  else) states
• There are four possible states the system can be
  in.

State 1 State 2
AB
A B
B A
AB
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Quantum Distributions

• If the two particles are indistinguishable
• There are only three possible states of the
  system.
• Because there are two types of quantum mechanical
  particles, two kinds of quantum distributions are
  needed.
• Fermions
• Particles with half-integer spins, obey the Pauli
  principle.
• Bosons
• Particles with zero or integer spins, do not obey
  the Pauli principle.

State 1 State 2
XX
X X
XX
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Multiply each state with its number of
microstates for distinguishable particles sum
it all up and you get the distribution of
classical physics particles Ignore all
microstates for indistinguishable particles sum
it all up, that would be the distribution for
bosons Ignore all microstates and states that
have more than one particle at the same energy
level, - sum it all up, that would be the
distribution of fermions Serway et al, chapter 10
for details Realize, there must be three
different distribution functions !!
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Quantum Distributions
36
Classical and Quantum Distributions
For photons in cavity, Planck, A 1, a 0
E is quantized in units of h if part of a bound
system
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Quantum Distributions
If all three normalization factors 1, just for
comparison
has to do with specific normalization factor

• The exact forms of normalization factors for the
  distributions depend on the physical problem
  being considered.
• Because bosons do not obey the Pauli exclusion
  principle, more bosons can fill lower energy
  states (are actually attracted to do so)
• All three graphs coincide at high energies the
  classical limit.
• Maxwell-Boltzmann statistics may be used in the
  classical limit when particles are so far apart
  that they are distinguishable, can be tracked by
  their paths

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When there are so many states that there is a
very low probability of occupation
also if the particles are heavy (macroscopic),
i.e. a bunch of classical physics particle,
Bohrs correspondence principle again
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Anything to do with solids, when high probability
of occupancy of energy states, e.g. electrons in
a metal, which are fermions
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anything to do with liquids, when high
probability of occupancy of energy
states Bose-Einstein condensate for at 2.17 K
superfluidity (explained later on)
https//www.youtube.com/watch?v2Z6UJbwxBZI
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degeneracy of the first exited state in H atom
n l ml ms up ms down
2 0 0 1/2 -1/2
2 1 1 1/2 -1/2
2 1 0 1/2 -1/2
2 1 -1 1/2 -1/2
g functions, density of states, how many states
there are per unit energy value, in other words
the degeneracy if we talk about a hydrogen atom
g functions are problem specific !!
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revisited
Einsteins assumptions in 1907, atoms vibrate
independently of each other he used
Maxwell-Boltzmann statistics because there are so
many possibly vibration states that only a few of
the available states will be occupied, (and the
other distribution functions were not known at
the time)
(starting from zero point energy, due to
uncertainty principle)
A. Einstein, "Die Plancksche Theorie der
Strahlung und die Theorie der spezifischen
Wärme", Annalen der Physik 22 (1907) 180190
i.e. at high temperatures is approaches the
classical value of 2 degrees of freedom with ½ kT
each times 3 vibration direction (Bohrs
correspondence principle once more)
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To account for different bond strength, different
spring constants
hetero-polar bond in diamond much stronger than
metallic bond in lead and aluminum, so much
larger Einstein Temperature for diamond (1,320 K)
gtgt 50-100 K for typical metals
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Peter Debye lifted the assumption that atoms
vibrate independably, similar statistics, Debye
temperature TD even better modeling with
phonons, which are pseudo-particle of the boson
type
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Blackbody Radiation

• Blackbody Radiation
• Intensity of the emitted radiation is
• Use Bose-Einstein distribution because photons
  are bosons with spin 1 (they have two
  polarization states)
• For a free particle in terms of momentum in a 3D
  infinitely deep well
• E pc hf so we need the equivalent of this
  formulae in terms of momentum (KE p2 / 2m)

now our particles are measles
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Phase space again
Density of states in cavity, we can assume the
cavity is a sphere, we could alternatively assume
it is any kind of shape that can be filled with
cubes
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Bose-Einstein Statistics

• The number of allowed energy states within
  radius r of a sphere is
• Where 1/8 comes from the restriction to positive
  values of ni and 2 comes from the fact that there
  are two possible photon polarizations.
• Resolve Energy equation for r, and substitute
  into the above equation for Nr
• Then differentiate to get the density of states
  g(E) is
• Multiply the Bose-Einstein factor in

For photons, the normalization factor is 1, they
are created and destroyed as needed
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Bose-Einstein Statistics

• Convert from a number distribution to an energy
  density distribution u(E).
• For all photons in the range E to E dE
• Using E hf and dE (hc/?2) d?
• In the SI system, multiplying by c/4 is required.

and world wide fame for Satyendra Nath Bose 1894
1974  !
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Liquid Helium

• Has the lowest boiling point of any element (4.2
  K at 1 atmosphere pressure) and has no solid
  phase at normal pressure.
• The density of liquid helium as a function of
  temperature.

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Liquid Helium

• The specific heat of liquid helium as a function
  of temperature
• The temperature at about 2.17 K is referred to as
  the critical temperature (Tc), transition
  temperature, or lambda point.
• As the temperature is reduced from 4.2 K toward
  the lambda point, the liquid boils vigorously. At
  2.17 K the boiling suddenly stops.
• What happens at 2.17 K is a transition from the
  normal phase to the superfluid phase.

Thermal conductivity goes to infinity at lambda
point, so no hot bubbles can form while the
liquid is boiling,
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Liquid Helium

• The rate of flow increases dramatically as the
  temperature is reduced because the superfluid has
  an extremely low viscosity.
• Creeping film formed when the viscosity is very
  low and some helium condenses from the gas phase
  to the glass of some beaker.

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Liquid Helium

• Liquid helium below the lambda point is part
  superfluid and part normal.
• As the temperature approaches absolute zero, the
  superfluid approaches 100 superfluid.
• The fraction of helium atoms in the superfluid
  state
• Superfluid liquid helium is referred to as a
  Bose-Einstein condensation.
• not subject to the Pauli exclusion principle
  because (the most common helium atoms are bosons
• all particles are in the same quantum state

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https//www.youtube.com/watch?v2Z6UJbwxBZI
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9.7 Fermi-Dirac Statistics

• EF is called the Fermi energy.
• When E EF, the exponential term is 1.
• FFD ½
• In the limit as T ? 0,
• At T 0, fermions occupy the lowest energy
  levels.
• Near T 0, there is no chance that thermal
  agitation will kick a fermion to an energy
  greater than EF.

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Fermi-Dirac Statistics
T gt 0

• T 0

• As the temperature increases from T 0, the
  Fermi-Dirac factor smears out.
• Fermi temperature, defined as TF EF / k.
  .

T gtgt TF
T TF

• When T gtgt TF, FFD approaches a decaying
  exponential of the Maxwell Boltzmann statistics.

At room temperature, only tiny amount of fermions
are in the region around EF,i.e. can contribute
to elecric current,
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Classical Theory of Electrical Conduction

• Paul Drude (1900) showed on the basis of the idea
  of a free electron gas inside a metal that the
  current in a conductor should be linearly
  proportional to the applied electric field, that
  would be consistent with Ohms law.
• His prediction for electrical conductivity
• Mean free path is .
• Drude electrical conductivity

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Classical Theory of Electrical Conduction
From Maxwells speed distribution

• According to the Drude model, the conductivity
  should be proportional to T-1/2.
• But for most metals is very nearly proportional
  to T-1 !!
• This is not consistent with experimental results.
  
• l and t make only sense for a realistic
  microscopic model, so whole approach abandoned,
  but free electron gas idea kept, just a different
  kind of gas

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All condensed matter (liquids and solids)
problems are statistical quantum mechanics
problems !! Quantum condensed matter physics
problems are typically low temperature problems
Ideal gasses can be modeled classically,
because they have very low matter densities
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Quantum Theory of Electrical Conduction

• The allowed energies for electrons are
• The parameter r is the radius of a sphere in
  phase space.
• The volume is (4/3)pr 3.
• The exact number of states upto radius r is
  .

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Quantum Theory of Electrical Conduction

• Rewrite as a function of E
• At T 0, the Fermi energy is the energy of the
  highest occupied level.
• Total of electrons
• Solve for EF
• The density of states with respect to energy in
  terms of EF

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Quantum Theory of Electrical Conduction

• At T 0,
• The mean electronic energy
• Internal energy of the system
• Only those electrons within about kT of EF will
  be able to absorb thermal energy and jump to a
  higher state. Therefore the fraction of electrons
  capable of participating in this thermal process
  is on the order of kT / EF.

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Quantum Theory of Electrical Conduction

• In general,
• Where a is a constant gt 1.
• The exact number of electrons depends on
  temperature.
• Heat capacity is
• Molar heat capacity is

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Quantum Theory of Electrical Conduction

• Arnold Sommerfeld used correct distribution n(E)
  at room temperature and found a value for a of p2
  / 4.
• With the value TF 80,000 K for copper, we
  obtain cV 0.02R, which is consistent with the
  experimental value! Quantum theory has proved to
  be a success.
• Replace mean speed in Eq (9,37) by Fermi
  speed uF defined from EF ½ uF2.
• Conducting electrons are loosely bound to their
  atoms.
• these electrons must be at the high energy
  level.
• at room temperature the highest energy level is
  close to the Fermi energy.
• We should use

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Quantum Theory of Electrical Conduction

• Drude thought that the mean free path could be no
  more than several tenths of a nanometer, but it
  was longer than his estimation.
• Einstein calculated the value of l to be on the
  order of 40 nm in copper at room temperature.
• The conductivity is
• Sequence of proportions.

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• Rewrite Maxwell speed distribution in terms of
  energy.
• For a monatomic gas the energy is all
  translational kinetic energy.
• where