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Statistical Physics
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Topics

Recap

Quantum Statistics

The Photon Gas

Summary
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Recap
In classical physics, the number of particles
with energy between E and E + dE, at
temperature T, is given by
n( E )dE  g ( E ) f B ( E )dE  Ag ( E )e  E / kT dE
where g(E) is the density of states. The
Boltzmann distribution describes how energy
is distributed in an assembly of identical,
but distinguishable particles.
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Quantum Statistics
In quantum physics, particles are described
by wave functions. But when these overlap,
identical particles become indistinguishable
and we cannot use the Boltzmann distribution.
We therefore need new energy distribution
functions.
In fact, we need two: one for particles that
behave like photons and one for particles that
behave like electrons.
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Quantum Statistics
In 1924, the Indian physicist Bose derived
the energy distribution function for
indistinguishable mass-less particles that do
not obey the Pauli exclusion principle.
The result was extended by Einstein to
massive particles and is called the
Bose-Einstein (BE) distribution
f BE ( E ) 
1
 E / kT
e e
1
The factor e depends
on the system under
study
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Quantum Statistics
The corresponding result for particles that
obey the Pauli exclusion principle is called the
Fermi-Dirac (FD) distribution
f FD ( E ) 
1
 E / kT
e e
1
Particles, such as photons, that obey the
Bose-Einstein distribution are called bosons.
Those that obey the Fermi-Dirac distribution,
such as electrons, are called fermions.
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Quantum Statistics
The Boltzmann distribution can be written in
the form
1
f B ( E )   E / kT
e e
Apart from the ±1 in the denominator, this is
identical to the BE and FD distributions.
The Boltzmann distribution is valid when
e eE/kT >> 1. This can occur because of low
particle densities and energies >> kT
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Quantum Statistics
Comparison of Distribution Functions
For a system of two identical particles, 1 and
2, one in state n and the other in state m,
there are two possible configurations, as
shown below
1st
configuration
2nd configuration
1
2
2
1
n
m
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Quantum Statistics
Comparison of Distribution Functions
The first configuration
1
2
n
m
is described by the wave function
 nm (1, 2)   n (1) m (2)
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Quantum Statistics
Comparison of Distribution Functions
The second configuration
2
1
n
m
is described by the wave function
 nm (2,1)   n (2) m (1)
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Quantum Statistics
Comparison of Distribution Functions
If the particles were distinguishable, then
the two wave functions
 nm (1, 2)   n (1) m (2)
 nm (2,1)   n (2) m (1)
would be the appropriate ones to describe
the system of two (non-interacting) particles
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Quantum Statistics
Comparison of Distribution Functions
But since in general identical particles are not
distinguishable, we must describe them using
the symmetric or anti-symmetric combinations
1
S 
 n (1) m (2)  n (2) m (1)
2
1
A 
 n (1) m (2)  n (2) m (1)
2
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Quantum Statistics
Comparison of Distribution Functions
The symmetric wave functions describe bosons
while the anti-symmetric ones describe
fermions. Using these wave functions one can
deduce the following:
1. A boson in a quantum state increases the
chance of finding other identical bosons in
the same state
2. A fermion in a quantum state prevents any
other identical fermions from occupying
the same state
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Quantum Statistics
Comparison of Distribution Functions
The probability that a
particle occupies a
given energy state
satisfies the inequality
f FD  f B  f BE
All three functions
become the same when
E >> kT
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Quantum Statistics
Density of States
The number of particles with energy in the
range E to E+dE is given by
n( E )dE  g ( E ) f ( E )dE
and the total number of particles N is given by

N   n( E )dE
0
Each function f(E) is
associated with a
different density
of states g(E)
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Quantum Statistics
Density of States
The number of states with energy in the range
E to E + dE can be shown to be given by
d
g ( E )dE  W 3 , d   V 4 p 2 dp
h
where d is called the phase space volume,
W is the degeneracy of each energy level,
V is the volume of the system and p is the
momentum of the particle
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The Photon Gas
Density of States for Photons
For photons, E = pc, and W = 2. (A photon
has two polarization states). Therefore,
8VE
g ( E )dE 
dE
3
(hc)
2
Extra Credit: Derive this formula
due date: Monday after Spring Break
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The Photon Gas
Distribution Function for Photons
The number of photons with energy between
E and E + dE is given by
n( E )dE  g ( E ) f BE ( E )dE
 8 VE 2  
1 

 dE
3   E / kT
1 
 (hc)   e
For photons  = 0.
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The Photon Gas
Photon Density of the Universe
The photon density is just the integral of
n(E) dE / V over all possible photon energies
8 E 2 dE
   n( E )dE / V  
0
0 (hc)3 (e E / kT  1)
This yields approximately


  8  kT / hc  (2.40)
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The photon temperature of the universe is
T= 2.7 K, implying  = 4 x 108 photons/m3
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The Photon Gas
Black Body Spectrum
If we multiply the photon density n(E)dE/V by
E, we get the energy density u(E)dE
8 E
u ( E )dE 
dE
3
E / kT
(hc) (e
 1)
3
This is the distribution first obtained by
Max Planck in 1900 in his “act of desperation”
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Summary




Particles come in two classes: bosons and
fermions.
A boson in a state enhances the chance to
find other identical bosons in that state.
A fermion in a state prevents other identical
fermions from occupying the state.
When identical particles become
distinguishable, typically, when they are well
separated and when E >> kT, the B-E and F-D
distributions can be approximated with the
Boltzmann distribution
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