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Physics 451

This document outlines the homework assignments (HW 23 and HW 24) for Quantum Mechanics taught by Karine Chesnel in Fall 2012. It includes topics such as Hund's rules, the wave function of valence electrons in solids, and various models like the free electron gas theory. The quiz section tests students on the spectroscopic symbols for Silicon and Chlorine. Additionally, it covers fundamental concepts like Bravais k-space, the Fermi surface, and the quantization of energy levels in solids. These principles are essential for understanding advanced quantum mechanics.

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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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