Transcript Statistical Physics
Statistical Physics 2
1
Topics
Recap Quantum Statistics The Photon Gas Summary 2
Recap
In classical physics, the number of particles with energy between
E
and
E
+
dE
, at temperature
T
, is given by (
f B d E
A
( )
e
dE
where
g(E)
is the
density of states
Boltzmann distribution describes how energy is distributed in an assembly of . The
identical
, but
distinguishable
particles.
3
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.
4
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 e
1
1
The factor study
e
depends on the system under 5
Quantum Statistics
The corresponding result for particles that obey the Pauli exclusion principle is called the
Fermi-Dirac (FD) distribution
f FD
e e
1
1
Particles, such as photons, that obey the Bose-Einstein distribution are called such as electrons, are called
fermions bosons
.
. Those that obey the Fermi-Dirac distribution, 6
Quantum Statistics
The Boltzmann distribution can be written in the form
f B
e e
1
Apart from the
±1
in the denominator, this is identical to the BE and FD distributions.
The Boltzmann distribution is valid when
e
e E/kT >> 1
. This can occur because of low particle densities and energies >> kT 7
Quantum Statistics
Comparison of Distribution Functions
For a system of two identical particles,
1 2
, one in state
n
and the other in state there are two possible configurations, as shown below
m
and ,
1 st configuration 1 2 2 nd configuration
2
n
1
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)
10
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 11
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
S
A
1 2
n
1 2
n
m
m
n
n
m
m
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Quantum Statistics
Comparison of Distribution Functions
The
symmetric
wave functions describe
bosons
1.
while the
fermions
. Using these wave functions one can deduce the following: 2.
A chance of finding other identical bosons in the same state A
anti-symmetric boson fermion
the same state ones describe in a quantum state
increases
in a quantum state
prevents
the any other identical fermions from occupying 13
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 and the total number of particles N is given by
N
0 Each function
f(E)
associated with a different density of states
g(E)
is 15
Quantum Statistics
Density of States
The number of
states E
to
E
+
dE
with energy in the range can be shown to be given by
W d
,
h
3
d
V
4 2
p dp
where
W V d
is called the
phase space volume
, is the degeneracy of each energy level, is the volume of the system and momentum of the particle
p
is the 16
The Photon Gas
Density of States for Photons
For photons,
E = pc
, and
W =
2
. (A photon has two polarization states). Therefore, 8
VE
2 (
hc
) 3
dE
Extra Credit
: Derive this formula due date: Monday after Spring Break 17
The Photon Gas
Distribution Function for Photons
The number of photons with energy between
E
and
E
+
dE
is given by
BE
8
VE
2 (
hc
) 3
e
For photons
= 0
.
1 1
dE
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The Photon Gas
Photon Density of the Universe
The photon density
n(E) dE / V
is just the integral of over all possible photon energies 0 / 0 8 2
E dE
(
hc
3 ) (
e
1) This yields approximately 8 3 (2.40) The photon temperature of the universe is
T= 2.7 K
, implying =
4 x 10 8 photons/m 3
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The Photon Gas
Black Body Spectrum
If we multiply the photon density
n(E)dE
/
V E
, we get the
energy density u(E)dE
by 8 (
hc
3 ) (
e E
3 1)
dE
This is the distribution first obtained by Max Planck in 1900 in his “act of desperation” 20
Summary
Particles come in two classes:
bosons fermions
.
and 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
Boltzmann distribution , the B-E and F-D distributions can be approximated with the 21