Physics 451

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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

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 5Last 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?

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

A.

B.

C.

D.

E.

Quiz 32b

Quantum mechanics

What is the spectroscopic symbol for Chlorine?

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

A.

B.

C.

D.

E.

Solids

Quantum mechanics

e-

What is

the wave function

of a valenceelectron

in the solid?

Solids

Quantum mechanics

e-

Basic Models:

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

Quantum mechanics

e-

e-

lz

ly

lx

Volume

Number of electrons:

Free electron gas

e-

3D infinite

square well

0

inside the cube

outside

Quantum mechanics

Free electron gas

e-

Separation of variables

Quantum mechanics

Free electron gas

Bravais

k-space

Quantum mechanics

Free electron gas

Fermi surface

Free electron density

Quantum mechanics

Bravais

k-space

Free electron gas

Fermi surface

Total energy contained inside the Fermi surface

Quantum mechanics

Bravais

k-space

Free electron gas

Fermi surface

Quantum mechanics

Solid Quantum pressure

Bravais

k-space

Solids

e-

Fermi surface

Bravais

k-space

Number of unit cells

Quantum mechanics

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

Solids

e-

Fermi surface

Bravais

k-space

Number of unit cells

Quantum mechanics

Solids

Bloch’s theorem

Quantum mechanics

Dirac comb

V(x)

Solids

Quantum mechanics

Circular periodic condition

V(x)

x-axis “wrapped around”

Solids

Quantum mechanics

Solving Schrödinger equation

V(x)

a

0

Solids

Quantum mechanics

Boundary conditions

V(x)

a

0

Solids
• Discontinuity of

Quantum mechanics

Boundary conditions at x = 0

V(x)

a

0

• Continuity of Y
Solids

Band structure

Quantum mechanics

Quantization of k:

Pb 5.18

Pb 5.19

Pb 5.21

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

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