Statistical Physics 2

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Statistical Physics 2
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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
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
The factor ea 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
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
Apart from the 1 in the denominator, this
is identical to the BE and FD distributions. The
Boltzmann distribution is valid when ea eE/kT gtgt
1. This can occur because of low particle
densities and energies gtgt 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
1
2
1st configuration
2
1
2nd configuration
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Quantum Statistics
Comparison of Distribution Functions
The first configuration
1
2
is described by the wave function
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Quantum Statistics
Comparison of Distribution Functions
The second configuration
2
1
is described by the wave function
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Quantum Statistics
Comparison of Distribution Functions
If the particles were distinguishable, then the
two wave functions
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
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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
• A boson in a quantum state increases the chance
  of finding other identical bosons in the same
  state
• 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
All three functions become the same when E gtgt kT
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Quantum Statistics
Density of States
The number of particles with energy in the range
E to EdE is given by
and the total number of particles N is given by
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
where dG 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,
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
For photons a 0.
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The Photon Gas
Photon Density of the Universe
The photon densityr is just the integral of n(E)
dE / V over all possible photon energies
This yields approximately
The photon temperature of the universe is T 2.7
K, implying r 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
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 gtgt kT, the B-E and F-D distributions can be
  approximated with the Boltzmann distribution