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Physics 452

Physics 452. Quantum mechanics II Winter 2012. Karine Chesnel. Physics 452 Homework. First homework assignment: Tuesday Jan 10 by 10pm. Assignment # 1: Problems 5.22, 5.23, 5.24 in the textbook. Second homework assignment: Thursday Jan 12 by 10pm. Phys 452.

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Physics 452

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  1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

  2. Physics 452 Homework First homework assignment: Tuesday Jan 10 by 10pm Assignment # 1: Problems 5.22, 5.23, 5.24 in the textbook Second homework assignment: Thursday Jan 12 by 10pm

  3. Phys 452 N particles ( ) Quantization of the energy for individual particles Total energy: How many ways we get the configuration ? Quantum statistical mechanics Thermal equilibrium, T

  4. Phys 452 • In-class example/ pb 5.23 • Textbook example Infinite square well Harmonic oscillator Quantum statistical mechanicsExample: 3 –particle system • For each type of particles: • List all the possible configurations • Determine the number of combinations of each configuration • Determine the probability of each configuration for a given energy

  5. Phys 452 Consider 3 distinguishable particles in a harmonic oscillator potential. If the total energy of the system is how many possibilities there are to get the configuration (1,1,1,0,…)? Quiz 1b A. 10 B. 4 C. 1 D. 6 E. 2

  6. Phys 452 Consider 3 fermions particles in a harmonic oscillator potential. If the total energy of the system is how many possibilities there are to get the configuration (1,1,1,0,…)? Quiz 1c A. 10 B. 4 C. 1 D. 6 E. 2

  7. Phys 452 Consider 3 bosons in a harmonic oscillator potential. If the total energy of the system is how many possibilities there are to get the configuration (1,1,1,0,…)? Quiz 1d A. 10 B. 4 C. 1 D. 6 E. 2

  8. Phys 452 • Identical fermions • Identical bosons Quantum statistical mechanics Statistical configuration number: • Distinguishable particle Work out example: harmonic oscillator, infinite square well

  9. Phys 452 Quiz 2a Let’s consider the Carbon atom: with 6 electrons To be distributed in the energy levels E1 and E2 What is the number of combinations ? A. 6! B. 15 C. 8! D. 70 E. 45

  10. Phys 452 Quiz 2b Let’s consider the Carbon atom: with 6 electrons to be distibuted in the shells (1s)(2s)(2p) What is the number of combinations ? A. 6! B. 15 C. 8! D. 70 E. 45

  11. Phys 452 Lagrange multipliers, using: Expressing Nn in terms of a and b Maximizing Q: Quantum statistical mechanics The most probable configuration

  12. Phys 452 • Distinguishable particle • Identical fermions • Identical bosons Quantum statistical mechanics Most probable occupation number:

  13. Phys 452 dimensionless b related to the temperature a related to chemical potential Quantum statistical mechanics Significance of a and b:

  14. Phys 452 Fermi surface “degeneracy” density of states Bravais k-space Volume in k-space of each individual state Quantum statistical mechanics Calculation of a and b: Case of ideal gas (free electron gas) One spherical shell

  15. Phys 452 Then using We can express Quantum statistical mechanics Calculation of a and b: Case of ideal gas : distinguishable particles

  16. Phys 452 Quantum statistical mechanics Case of ideal gas : distinguishable particles

  17. Phys 452 • Distinguishable particle • Identical fermions • Identical bosons Quantum statistical mechanics Most probable occupation number: Maxwell-Boltzmann statistic Fermi- Dirac statistic Bose-Einstein statistic

  18. Phys 452 Quiz 2c What is the maximum possible value for the density of occupation in case of fermions? A. B. C. 1 D. 0 E. undetermined

  19. Phys 452 if if Quantum statistical mechanics Fermi-Dirac distribution:

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