N96770 Statistical Mechanics class on 11292002 - PowerPoint PPT Presentation

Title: N96770 Statistical Mechanics class on 11292002


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N96770 ???????
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(41X01??)
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OUTLINES

• Fermi-Dirac Bose-Einstein Gases

• Microcanonical Ensemble

• Grand Canonical Ensemble

Reference K. Huang, Statistical Mechanics, John
Wiley Sons, Inc., 1987.
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Quick Review
is a vector and a state of a system.
is an eigenvector of the position operators of
all particles in a system.
is the wave function of the system in the state
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a complex number and a function of time
n a set of quantum numbers
the probability associated with n
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Ideal Gases
Two types of a system composed of N identical
particles
Fermi-Dirac system
The wave functions are antisymmetric under an
interchange of any pair of particle coordinates.
Particles with such characteristics are called
fermions.
Examples electrons, protons.
Bose-Einstein system
The wave functions are symmetric under an
interchange of any pair of particle coordinates.
Particles with such characteristics are called
bosons.
Examples deuterons (2H), photons.
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Microcanonical Ensemble
the number of states of a system having an energy
eigenvalue that is between E and E?E.
N(E)
A state of an ideal system can be specified by a
set of occupation numbers np so that there are
np particles having the momentum p in the state.
total energy
total number of particles
np 0, 1, 2, for bosons
np 0, 1 for fermions
level (energy eigenvalue)
h Plancks constant
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g4
g3
g2
g1
cell
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For Fermions
The number of particles in each of the gi subcell
of the i-th cell is either 0 or 1.
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For Bosons
Each of the gi subcell of the i-th cell can be
occupied by any number of particles.
Entropy
It can be shown that
the set of occupation numbers that maximizes
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(for bosons)
? chemical potential
(for fermions)
where
kB Boltzmanns constant
It can be shown that (by using Stirlings
approximation)
(for bosons)
(for fermions)
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Grand Canonical Ensemble
Partition function for ideal gases
where
the occupation numbers np are subject to the
condition
the number of states corresponding to np is
for bosons and fermions
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Consider the grand partition function Z,
n 0, 1, 2, for bosons
n 0, 1 for fermions
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(for bosons)
(for fermions)
Equations of state
(for bosons)
(for fermions)
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Now let V ? 8, then the possible values of p
become continuous.
Equations of state for ideal Fermi-Dirac gases
Equations of state for ideal Bose-Einstein gases
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Let
and
Then equations of state for ideal Fermi-Dirac
gases become
where
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And equations of state for ideal Bose-Einstein
gases become
where