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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 >> 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 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 >> 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 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 >> kT, the B-E and F-D distributions can be approximated with the Boltzmann distribution

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