1 statistical physics 2. 2 topics l recap l quantum statistics l the photon gas l summary
TRANSCRIPT
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RecapRecap
/( )) ( )( ( ) E kTBg E dn E dE f E A eE g E dE
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. TheBoltzmann distribution describes how energyis distributed in an assembly of identical, but distinguishable particles.
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Quantum StatisticsQuantum Statistics
In quantum physics, particles are describedby 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 StatisticsQuantum Statistics
/
1( )
1BE E kTf E
e e
In 1924, the Indian physicist Bose derived the energy distribution function for indistinguishable mass-less particles that donot obey the Pauli exclusion principle.
The result was extended by Einstein to massive particles and is called the Bose-Einstein (BE) distribution
The factor e depends on the system understudy
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Quantum StatisticsQuantum Statistics
/
1( )
1FD E kTf E
e e
The corresponding result for particles thatobey the Pauli exclusion principle is called theFermi-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 StatisticsQuantum Statistics
/
1( )B E kT
f Ee e
The Boltzmann distribution can be written inthe form
Apart from the ±1 in the denominator, this isidentical to the BE and FD distributions.
The Boltzmann distribution is valid whene eE/kT >> 1. This can occur because of lowparticle densities and energies >> kT
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Quantum StatisticsQuantum Statistics
n
Comparison of Distribution Functions
For a system of two identical particles, 1 and2, one in state n and the other in state m, there are two possible configurations, asshown below
2 1
m
1 21st configuration
2nd configuration
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Quantum StatisticsQuantum Statistics
(1,2) (1) (2)nm n m
Comparison of Distribution Functions
The first configuration
n m1 2
is described by the wave function
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Quantum StatisticsQuantum Statistics
(2,1) (2) (1)nm n m
Comparison of Distribution Functions
The second configuration
n m2 1
is described by the wave function
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Quantum StatisticsQuantum Statistics
(2,1) (2) (1)nm n m
Comparison of Distribution Functions
If the particles were distinguishable, thenthe two wave functions
would be the appropriate ones to describethe system of two (non-interacting) particles
(1,2) (1) (2)nm n m
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Quantum StatisticsQuantum Statistics
Comparison of Distribution Functions
But since in general identical particles are not distinguishable, we must describe them usingthe symmetric or anti-symmetric combinations
1( ) ( ) ( ) ( )
21
( ) ( ) ( ) ( )
2 2
2
1 1
1 12
2
S n m n m
A n m n m
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Quantum StatisticsQuantum 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 StatisticsQuantum Statistics
Comparison of Distribution Functions
The probability that a particle occupies a given energy statesatisfies the inequality
FD B BEf f f All three functionsbecome the same when
E >> kT
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Quantum StatisticsQuantum Statistics
( ) ( ) ( )n E dE g E f E dE
Density of States
The number of particles with energy in therange E to E+dE is given by
Each function f(E) isassociated with adifferent densityof states g(E)
0
( )N n E dE
and the total number of particles N is given by
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Quantum StatisticsQuantum Statistics
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( ) , 4d
g E dE W d V p dph
Density of States
The number of states with energy in the rangeE to E + dE can be shown to be given by
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 themomentum of the particle
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The Photon GasThe Photon Gas
2
3
8( )
( )
VEg E dE dE
hc
Density of States for Photons
For photons, E = pc, and W = 2. (A photonhas two polarization states). Therefore,
Extra Credit: Derive this formuladue date: Monday after Spring Break
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The Photon GasThe Photon Gas
2
3 /
( ) ( ) ( )
8 1
( ) 1
BE
E kT
n E dE g E f E dE
VEdE
hc e
Distribution Function for Photons
The number of photons with energy between E and E + dE is given by
For photons = 0.
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The Photon GasThe Photon Gas
2
3 /0 0
8( ) /
( ) ( 1)E kT
E dEn E dE V
hc e
Photon Density of the Universe
The photon density is just the integral of n(E) dE / V over all possible photon energies
This yields approximately
38 / (2.40)kT hc
The photon temperature of the universe isT= 2.7 K, implying = 4 x 108 photons/m3
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The Photon GasThe Photon Gas
3
3 /
8( )
( ) ( 1)E kT
Eu E dE dE
hc e
Black Body Spectrum
This is the distribution first obtained by Max Planck in 1900 in his “act of desperation”
If we multiply the photon density n(E)dE/V by E, we get the energy density u(E)dE
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SummarySummary
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