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 , 5.16, 5.18, Wednesday Dec 5 Last class / review

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

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

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

Solids Quantum mechanics e-e- What is the wave function of a valence electron in the solid?

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

Free electron gas Quantum mechanics e-e- e-e- lzlz lyly lxlx Volume Number of electrons:

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

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

Free electron gas Quantum mechanics Bravais k-space

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

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

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

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

Solids Quantum mechanics e-e- Bravais k-space Fermi surface Pb 5.15:Relation between E tot and E F Pb 5.16:Case of Cu: calculate E F, v F, T F, and P F

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

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

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

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

Solids Quantum mechanics V(x) Boundary conditions 0 a

Solids Quantum mechanics V(x) Boundary conditions at x = 0 0 a Continuity of  Discontinuity of

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

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

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