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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

Physics 451

Quantum mechanics I

Fall 2012

Dec 3, 2012

Karine Chesnel

homework
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

periodic table
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
Quiz 32a

Quantum mechanics

What is the spectroscopic symbol for Silicon?

Si: (Ne)(3s)2(3p)2

A.

B.

C.

D.

E.

quiz 32b
Quiz 32b

Quantum mechanics

What is the spectroscopic symbol for Chlorine?

Cl: (Ne)(3s)2(3p)5

A.

B.

C.

D.

E.

solids
Solids

Quantum mechanics

e-

What is

the wave function

of a valenceelectron

in the solid?

solids1
Solids

Quantum mechanics

e-

Basic Models:

  • Free electron gas theory
  • Crystal Bloch’s theory
free electron gas
Free electron gas

Quantum mechanics

e-

e-

lz

ly

lx

Volume

Number of electrons:

free electron gas1
Free electron gas

e-

3D infinite

square well

0

inside the cube

outside

Quantum mechanics

free electron gas2
Free electron gas

e-

Separation of variables

Quantum mechanics

free electron gas3
Free electron gas

Bravais

k-space

Quantum mechanics

free electron gas4
Free electron gas

Fermi surface

Free electron density

Quantum mechanics

Bravais

k-space

free electron gas5
Free electron gas

Fermi surface

Total energy contained inside the Fermi surface

Quantum mechanics

Bravais

k-space

free electron gas6
Free electron gas

Fermi surface

Quantum mechanics

Solid Quantum pressure

Bravais

k-space

solids2
Solids

e-

Fermi surface

Bravais

k-space

Number of unit cells

Quantum mechanics

solids3
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

solids4
Solids

e-

Fermi surface

Bravais

k-space

Number of unit cells

Quantum mechanics

solids5
Solids

Bloch’s theorem

Quantum mechanics

Dirac comb

V(x)

solids6
Solids

Quantum mechanics

Circular periodic condition

V(x)

x-axis “wrapped around”

solids7
Solids

Quantum mechanics

Solving Schrödinger equation

V(x)

a

0

solids8
Solids

Quantum mechanics

Boundary conditions

V(x)

a

0

solids9
Solids
  • Discontinuity of

Quantum mechanics

Boundary conditions at x = 0

V(x)

a

0

  • Continuity of Y
solids10
Solids

Band structure

Quantum mechanics

Quantization of k:

Pb 5.18

Pb 5.19

Pb 5.21

quiz 33
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

solids11
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