1 statistical physics 2. 2 topics l recap l quantum statistics l the photon gas l summary

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Statistical Physics Statistical Physics 22

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TopicsTopics

Recap

Quantum Statistics

The Photon Gas

Summary

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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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/

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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/

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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/

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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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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(1,2) (1) (2)nm n m


The first configuration

n m1 2

is described by the wave function

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(2,1) (2) (1)nm n m


The second configuration

n m2 1

is described by the wave function

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(2,1) (2) (1)nm n m


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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But since in general identical particles are not distinguishable, we must describe them usingthe symmetric or anti-symmetric combinations

1( ) ( ) ( ) ( )

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( ) ( ) ( ) ( )

2 2

2

1 1

1 12

2

S n m n m

A n m n m

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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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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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( ) ( ) ( )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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( ) , 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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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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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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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

1 statistical physics 2. 2 topics l recap l quantum statistics l the photon gas l summary

Documents

quantum state

boltzmann distribution

state slide

quantum physics

distinguishable particles

boseeinstein distribution

wave function slide

number of particles