Physics 451

Quantum mechanics I

Fall 2012

Nov 30, 2012

Karine Chesnel

Test 3Last day Today!

Quantum mechanics

• HW 23 Tuesday Dec 4

• HW 24 Thursday Dec 6

Homework

Periodic table

Quantum mechanics

Filling the shells

1 2 2 3 3 4 ...s s p s p s2 2 6

s: “sharp”p: “principal”d: “diffuse”f: “fundamental”

Periodic table

Quantum mechanics

Filling the shells

Periodic table

Quantum mechanics

Spectroscopic symbol

1 2 2 3 3 4 ...s s p s p s2 1S

JL

J L S

, 1,....,J L S L S L S

Periodic table

Quantum mechanics

Hund’s rules2 1S

JL

• 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=IL-SI

has lowest energy If shell more than half filled, the state with J=L+S

has lowest energy

Periodic table

Quantum mechanics

1 2 2 3 3 4 ...s s p s p s

2 1SJL

Quiz 31Quantum mechanics

What is the spectroscopic symbol for Silicon ?

A.

B.

C.

D.

E.

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

21S

32P

30P

42S

42D

Periodic table

Quantum mechanics

Filling the shells

1 2 2 3 3 4 ...s s p s p s2 2 6

s: “sharp”p: “principal”d: “diffuse”f: “fundamental”

Periodic table

Quantum mechanics

Filling the shells

Quantum mechanics

Periodic table

Quantum mechanics

Spectroscopic symbol

1 2 2 3 3 4 ...s s p s p s2 1S

JL

J L S

, 1,....,J L S L S L S

Periodic table

Quantum mechanics

1 2 2 3 3 4 ...s s p s p s

2 1SJL

Periodic table

Quantum mechanics

Hund’s rules2 1S

JL

• 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 33Quantum mechanics

What is the spectroscopic symbol for Silicon ?

A.

B.

C.

D.

E.

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

21S

32P

30P

42S

42D

Quiz 33Quantum mechanics

What is the spectroscopic symbol for Chlorine ?

A.

B.

C.

D.

E.

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

21S

23/2P

30P

42S

42D

SolidsQuantum mechanics

e-

What is the wave function

of a valence electron 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 x y zV l l l

Number of electrons: .N q

Free electron gas

22 ( )

2H V r

m

Quantum mechanics

( , )r t

e-

3D infinitesquare well

, ,V x y z 0 inside the cube

outside

22

2E

m

Free electron gas

Quantum mechanics

e-

22

2E

m

Separation of variables

( , ) ( ) ( ) ( )x y zr t x y z 2

2

2 i i i iEm

8( , ) sin sin sinyx z

x y z x y z

n yn x n zr t

l l l l l l

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

Free electron gas

Quantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

xk

yk

zk

Bravaisk-space

Volume of unit cell

3 3

x y zl l l V

Quiz 33cQuantum mechanics

A. 1

B. 2

C. q

D. Nq

E. An infinity

How many electrons can be contained in one unit cell in the Bravais k- lattice ?

Free electron gas

Quantum mechanics

xk

yk

zk

Bravaisk-space

Fk Fermi surface

Free electron densityNq

V

1/323Fk

Free electron gas

Quantum mechanics

Bravaisk-spacexk

yk

zk

Fk Fermi surface

2 2 2

2/3232 2F

F

kE

m m

Total energy contained inside the Fermi surface

2 5

20 10

FE

Ftot

k VE dE

m

Free electron gas

Quantum mechanics

Bravaisk-spacexk

yk

zk

Fk Fermi surface

Solid Quantum pressure

2

3tot tot

dVdE E

V

2/32 2

5/332

3 5totEPV m

Solid Quantum pressure

arises from Pauli exclusion principle

Solids

Quantum mechanics

22 22 2 2 2

2 2 22 2x y z

yx zn n n

x y z

nn n kE

m l l l m

e-

yk

zk

Bravaisk-space

xk

xk

yk

zk

FkFermi surface

Number of unit cells 236.02 10AN