Title: N96770 Statistical Mechanics class on 11292002
1N96770 ???????
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(41X01??)
2OUTLINES
- Fermi-Dirac Bose-Einstein Gases
Reference K. Huang, Statistical Mechanics, John
Wiley Sons, Inc., 1987.
3Quick 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
4 a complex number and a function of time
n a set of quantum numbers
the probability associated with n
5Ideal 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.
6Microcanonical 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
7g4
g3
g2
g1
cell
8For Fermions
The number of particles in each of the gi subcell
of the i-th cell is either 0 or 1.
9For 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
10(for bosons)
? chemical potential
(for fermions)
where
kB Boltzmanns constant
It can be shown that (by using Stirlings
approximation)
(for bosons)
(for fermions)
11Grand 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
12Consider the grand partition function Z,
n 0, 1, 2, for bosons
n 0, 1 for fermions
13(for bosons)
(for fermions)
Equations of state
(for bosons)
(for fermions)
14Now 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
15Let
and
Then equations of state for ideal Fermi-Dirac
gases become
where
16And equations of state for ideal Bose-Einstein
gases become
where