Modern Physics 1

Fall 2007

Test 5

Chapter 9. Quantum Statistics

FAX solutions to 361/593-2184 by Mon. 2pm, Dec. 10, 2007

Questions:

Q1. Different Statistics

Discuss the basic assumptions of Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics. How do they differ, and what are their similarities?

Q2. Terminology

Define, discuss, or explain each of the following terms: (a) boson, (b) fermion, (c) B-E condensation, (d) Pauli exclusion principle, (e) Einstein temperature, (f) Fermi temperature, (g) Fermi energy, (h) equipartition of energy.

Q3. Specific Heat Capacity

Discuss the basic assumptions of Einstein’s theory of specific heat. How do they differ from the classical assumptions? And what does Einstein’s result become in the limit of high and low temperatures?

Q4. Free-Electron Gas

Discuss the basic assumptions and conclusions of the free-electron gas theory of metal. What happens at low temperatures and at high temperatures?

Problems:

P1. Maxwell-Boltzmann Statistics

Two moles of monatomic helium gas are placed in a container at 0^oC and 1 atmospheric pressure. (a) What is the volume of the gas? (b) What is the total thermal energy of the gas? (c) What is the average kinetic energy of one molecule of the gas? (d) What is the root mean square speed of a molecule of the gas? (e) What is the most probable speed of a molecule of the gas?

P2. Enumerated Statistics

Consider a system of 3 particles with a total energy of 4E in a system of 5 equally spaced energy states with energies 0, E, 2E, 3E, and 4E. Enumerate the states (for example, let state (a,b,0,c,0) represent particle a with energy 0, b with energy E and c with energy 3E or let state (0,2,1,0) represent two identical particles each with energy E and one particle with energy 2E, etc.) and list the probability p(nE) of finding a particle with energy nE, where n is an integer 0, 1, 2, 3, 4. Do this for (a) Maxwell-Boltzmann statistics, (b) Bose-Einstein statistics, and (c) Fermi-Dirac statistics.

P3. Bose-Einstein Statistics

(a) Find the average energy per photon for photons in thermal equilibrium with a cavity at temperature T. (b) Calculate the average photon energy in electron volts at T = 6000 K. (c) What is the wavelength of such a photon? What is its color? Hint: Two useful integrals are and .

P4. Fermi-Dirac Statistics

The Fermi energy of aluminum is 11.63 eV. (a) Assuming that the free electron model applies to aluminum, calculate the number of free electrons per unit volume at low temperatures. (b) Determine the valence of aluminum by dividing the answer found in part (a) by the number of aluminum atoms per unit volume as calculated from the density and the atomic weight. Note that aluminum has a density of 2.70 g/cm³.