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Singel particle energy from Bolztman

Boltzmann  number of 1-particle state >>number of particles. Singel particle energy from Bolztman. How many 1-particle states? Remember the sphere used to explain degeneracy?.

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Singel particle energy from Bolztman

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  1. Boltzmann number of 1-particle state >>number of particles Singel particle energy from Bolztman How many 1-particle states? Remember the sphere used to explain degeneracy? number of 1-particle states with an energy lower than e =number of lattice points enclosed by the sphere in the positive octant:

  2. Many particles Boltzman energy The average energy of N particles is N times the average energy of one particle

  3. yi1 yi2 yi3 yi2 yi2 yi3 yi4 yi5 yi6 NON Bolztmann ensembles Fermions: 2 particles cannot occupy the same quantum stateoccupation number nk can only be 0 or 1 P1,2y=-y antisymmetric Bosons: 2 particles can occupy the same quantum stateoccupation number nk can have any value P1,2y=+y symmetric

  4. There are many ways of counting distributions, Partition functions for F or B choose Grand-Canonical Ensemble

  5. Summing with occupation numbers

  6. Fermions Fermi-Dirac

  7. Bosons from our outstanding math background … we recognize the series Bose-Einstein Combining Bose-Einstein with the Fermi-Dirac we obtain a gral. Eq.

  8. Grand-Canonical equation

  9. G-canonical equations II Keep in mind that even though we started with non-interacting particles, there wavefunctions are symmetrized they are NOT INDEPENDENT

  10. High T low r limit for n Boltzmann number of 1-particle state >>number of particles 

  11. High T low r limit for E

  12. Examples: Photon gas, an electromagnetic field in thermal equilibrium with its container Example To describe the state of the field, we need to know how many n are in each oscillator Photons are bosons  n=0,1,2,3…..

  13. Cont.

  14. cont The average number of photons in a state j… Planck distribution

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