3Recap In classical physics, the number of particles with energy between E and E + dE, attemperature T, is given bywhere g(E) is the density of states. TheBoltzmann distribution describes how energyis distributed in an assembly of identical,but distinguishable particles.
4Quantum Statistics In quantum physics, particles are described by wave functions. But when these overlap,identical particles become indistinguishableand we cannot use the Boltzmann distribution.We therefore need new energy distributionfunctions.In fact, we need two: one for particles thatbehave like photons and one for particles thatbehave like electrons.
5Quantum Statistics In 1924, the Indian physicist Bose derived the energy distribution function forindistinguishable mass-less particles that donot obey the Pauli exclusion principle.The result was extended by Einstein tomassive particles and is called theBose-Einstein (BE) distributionThe factor ea dependson the system understudy
6Quantum Statistics The corresponding result for particles that obey the Pauli exclusion principle is called theFermi-Dirac (FD) distributionParticles, such as photons, that obey theBose-Einstein distribution are called bosons.Those that obey the Fermi-Dirac distribution,such as electrons, are called fermions.
7Quantum Statistics The Boltzmann distribution can be written in the formApart from the ±1 in the denominator, this isidentical to the BE and FD distributions.The Boltzmann distribution is valid whenea eE/kT >> 1. This can occur because of lowparticle densities and energies >> kT
8Quantum Statistics 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 below121st configuration212nd configuration
9Quantum Statistics Comparison of Distribution Functions The first configuration12is described by the wave function
10Quantum Statistics Comparison of Distribution Functions The second configuration21is described by the wave function
11Quantum Statistics Comparison of Distribution Functions If the particles were distinguishable, thenthe two wave functionswould be the appropriate ones to describethe system of two (non-interacting) particles
12Quantum Statistics Comparison of Distribution Functions But since in general identical particles are notdistinguishable, we must describe them usingthe symmetric or anti-symmetric combinations
13Quantum 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 stateA fermion in a quantum state prevents any other identical fermions from occupying the same state
14Quantum Statistics Comparison of Distribution Functions The probability that aparticle occupies agiven energy statesatisfies the inequalityAll three functionsbecome the same whenE >> kT
15Quantum Statistics Density of States The number of particles with energy in therange E to E+dE is given byand the total number of particles N is given byEach function f(E) isassociated with adifferent densityof states g(E)
16Quantum Statistics Density of States The number of states with energy in the rangeE to E + dE can be shown to be given bywhere 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 themomentum of the particle
17The Photon Gas 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
18The Photon Gas Distribution Function for Photons The number of photons with energy betweenE and E + dE is given byFor photons a = 0.
19The Photon Gas Photon Density of the Universe The photon densityr is just the integral ofn(E) dE / V over all possible photon energiesThis yields approximatelyThe photon temperature of the universe isT= 2.7 K, implying r = 4 x 108 photons/m3
20The Photon Gas Black Body Spectrum If we multiply the photon density n(E)dE/V byE, we get the energy density u(E)dEThis is the distribution first obtained byMax Planck in 1900 in his “act of desperation”
21Summary 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