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

4. Quantum statistical mechanics Example: 3 –particle system Phys 452 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 Textbook example Infinite square well In-class example/ pb 5.23 Harmonic oscillator

5. Quiz 1b Phys 452 A. 10 B. 4 C. 1 D. 6 E. 2 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,…)?

6. Quiz 1c Phys 452 A. 10 B. 4 C. 1 D. 6 E. 2 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,…)?

7. Quiz 1d Phys 452 A. 10 B. 4 C. 1 D. 6 E. 2 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,…)?

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

9. Quiz 2a Phys 452 Let’s consider the Carbon atom: with 6 electrons To be distributed in the energy levels E 1 and E 2 What is the number of combinations ? A. 6! B. 15 C. 8! D. 70 E. 45

10. Quiz 2b Phys 452 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. Quantum statistical mechanics Phys 452 The most probable configuration Lagrange multipliers, using: Maximizing Q: Expressing N n in terms of  and 

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

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

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

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

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

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

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

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