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Physics 451

Physics 451. Quantum mechanics I Fall 2012. Dec 3, 2012 Karine Chesnel. Homework. Quantum mechanics. Last two assignment HW 23 Tuesday Dec 4 5.9, 5.12, 5.13, 5.14 HW 24 Thursday Dec 6 5.15, 5.16, 5.18, 5.19. 5.21. Wednesday Dec 5 Last class / review. Periodic table.

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Physics 451

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  1. Physics 451 Quantum mechanics I Fall 2012 Dec 3, 2012 Karine Chesnel

  2. Homework Quantum mechanics • Last two assignment • HW 23 Tuesday Dec 4 • 5.9, 5.12, 5.13, 5.14 • HW 24 Thursday Dec 6 • 5.15, 5.16, 5.18, 5.19. 5.21 Wednesday Dec 5Last class / review

  3. Periodic table Quantum mechanics Hund’s rules • First rule: seek the state with highest possible spin S • (lowest energy) • Second rule: for given spin S, the state with highest possible • angular momentum L has lowest energy • Third rule: • If shell no more than half filled, the state with J=L-S • has lowest energy • If shell more than half filled, the state with J=L+S • has lowest energy

  4. Quiz 32a Quantum mechanics What is the spectroscopic symbol for Silicon? Si: (Ne)(3s)2(3p)2 A. B. C. D. E.

  5. Quiz 32b Quantum mechanics What is the spectroscopic symbol for Chlorine? Cl: (Ne)(3s)2(3p)5 A. B. C. D. E.

  6. Solids Quantum mechanics e- What is the wave function of a valenceelectron in the solid?

  7. Solids Quantum mechanics e- Basic Models: • Free electron gas theory • Crystal Bloch’s theory

  8. Free electron gas Quantum mechanics e- e- lz ly lx Volume Number of electrons:

  9. Free electron gas e- 3D infinite square well 0 inside the cube outside Quantum mechanics

  10. Free electron gas e- Separation of variables Quantum mechanics

  11. Free electron gas Bravais k-space Quantum mechanics

  12. Free electron gas Fermi surface Free electron density Quantum mechanics Bravais k-space

  13. Free electron gas Fermi surface Total energy contained inside the Fermi surface Quantum mechanics Bravais k-space

  14. Free electron gas Fermi surface Quantum mechanics Solid Quantum pressure Bravais k-space

  15. Solids e- Fermi surface Bravais k-space Number of unit cells Quantum mechanics

  16. Solids e- Pb 5.15: Relation between Etot and EF Pb 5.16: Case of Cu: calculate EF , vF, TF, and PF Fermi surface Bravais k-space Quantum mechanics

  17. Solids e- Fermi surface Bravais k-space Number of unit cells Quantum mechanics

  18. Solids Bloch’s theorem Quantum mechanics Dirac comb V(x)

  19. Solids Quantum mechanics Circular periodic condition V(x) x-axis “wrapped around”

  20. Solids Quantum mechanics Solving Schrödinger equation V(x) a 0

  21. Solids Quantum mechanics Boundary conditions V(x) a 0

  22. Solids • Discontinuity of Quantum mechanics Boundary conditions at x = 0 V(x) a 0 • Continuity of Y

  23. Solids Band structure Quantum mechanics Quantization of k: Pb 5.18 Pb 5.19 Pb 5.21

  24. Quiz 33 Quantum mechanics In the 1D Dirac comb model how many electrons can be contained in each band? A. 1 B. 2 C. q D. Nq E. 2N

  25. Solids Insulator: band entirely filled ( even integer) 2N electrons (2e in each state) Quantum mechanics Quantization of k: Band structure E Conductor: band partially filled N states Band Gap Semi-conductor: doped insulator N states Band Gap N states Band

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