2 Topics l Recap l Quantum Statistics l The Photon Gas l Summary
3 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.
4 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.
5 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 e depends on the system under study
6 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.
7 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 e e E/kT >> 1. This can occur because of low particle densities and energies >> kT
8 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 21 12 1 st configuration 2 nd configuration
9 Quantum Statistics Comparison of Distribution Functions The first configuration 12 is described by the wave function
10 Quantum Statistics Comparison of Distribution Functions The second configuration 21 is described by the wave function
11 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
12 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
13 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: 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
14 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
15 Quantum Statistics Density of States The number of particles with energy in the range E to E+dE is given by Each function f(E) is associated with a different density of states g(E) and the total number of particles N is given by
16 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 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 the momentum of the particle
17 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
18 The Photon Gas Distribution Function for Photons The number of photons with energy between E and E + dE is given by For photons = 0.
19 The Photon Gas 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 The photon temperature of the universe is T= 2.7 K, implying = 4 x 10 8 photons/m 3
20 The Photon Gas 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
21 Summary l Particles come in two classes: bosons and fermions. l A boson in a state enhances the chance to find other identical bosons in that state. l A fermion in a state prevents other identical fermions from occupying the state. l 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