1 / 14

Classical Statistics

Classical Statistics. What is the speed distribution of the molecules of an ideal gas at temperature T? Maxwell speed distribution This is the probabilitity of finding a particle with speed between v and v +dv And if we cared to calculate it we would find. Classical Statistics.

moorewilliam
Download Presentation

Classical Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Classical Statistics • What is the speed distribution of the molecules of an ideal gas at temperature T? • Maxwell speed distribution • This is the probabilitity of finding a particle with speed between v and v +dv • And if we cared to calculate it we would find

  2. Classical Statistics • Maxwell speed distribution

  3. Classical Statistics • In quantum mechanics we are generally interested in the energy • Maxwell’s velocity distribution can be converted into a statement about energy

  4. Classical Statistics • The factor is called the Maxwell-Boltzmann factor • FMB gives the probability that a state of energy E is occupied at temperature T • The E1/2 factor is not universal but rather specific to the molecular speed problem • We need one more element called the density of states g(E) • g(E) is the number of states available per unit energy interval

  5. Classical Statistics • We have then • This is useful because we can then calculate thermodynamic properties of interest

  6. Quantum Statistics • We would like to apply these same ideas using quantum mechanics • To give some idea of how quantum mechanics will change things consider the following example • Let a gas be made of only two particles A and B • Let there be three quantum states s=1,2,3

  7. Quantum Statistics • Maxwell-Boltzmann case • The two particles are distinguishable and two particles can occupy the same state • There are 9 distinct states shown on the right

  8. Quantum Statistics • Bose-Einstein case • The particles are identical bosons • Any number of particles can occupy the same state • Let B=A • There are 6 distinct states shown on the right

  9. Quantum Statistics • Fermi-Dirac case • The particles are identical fermions • No two particles can occupy the same quantum state (Pauli exclusion principle) • There are 3 distinct states

  10. Quantum Statistics • The number of particles with energy E for the three cases is given by

  11. Quantum Statistics • The three probability distributions with A = B1 = B2 = 1

  12. Fermi-Dirac Statistics • This is the case for electrons so let’s study it a bit more • The constant B1 can be written • You will hear a lot about the Fermi energy in solid state physics

  13. Fermi-Dirac Statistics • Comments • When E=EF , FFD = ½ • The Fermi energy EF is the energy of the highest occupied energy level at T=0 • As T→0, FFD=1 for E<EF and FFD=0 for E>EF • At/near T=0, the fermions will occupy the lowest energy states available (consistent with the Pauli exclusion principle) • As T increases, the can occupy higher energy states

  14. Fermi-Dirac Statistics • Define kBTF = EF

More Related