Physics 451
Quantum mechanics I
Fall 2012
Nov 30, 2012
Karine Chesnel
Quantum mechanics
Test 3
Last day Today!
Homework
• HW 23 Tuesday Dec 4
• HW 24 Thursday Dec 6
Quantum mechanics
Periodic table
Filling the shells
s: “sharp”
p: “principal”
d: “diffuse”
f: “fundamental”
2
2
6
1s  2s  2 p 3s 3 p  4s ...
Quantum mechanics
Periodic table
Filling the shells
Quantum mechanics
Periodic table
Spectroscopic symbol
1s  2s  2 p 3s 3 p  4s ...
2 S 1
LJ
J  LS
J   L  S , L  S  1,...., L  S 
Quantum mechanics
2 S 1
Periodic table
LJ
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=IL-SI
has lowest energy
If shell more than half filled, the state with J=L+S
has lowest energy
Quantum mechanics
Periodic table
1s  2s  2 p 3s 3 p  4s ...
2 S 1
LJ
Quantum mechanics
Quiz 31
What is the spectroscopic symbol for Silicon ?
Si: (Ne)(3s)2(3p)2
A.
2
S1
3
P2
3
P0
4
S2
B.
C.
D.
E.
4
D2
Quantum mechanics
Periodic table
Filling the shells
s: “sharp”
p: “principal”
d: “diffuse”
f: “fundamental”
2
2
6
1s  2s  2 p 3s 3 p  4s ...
Quantum mechanics
Periodic table
Filling the shells
Quantum mechanics
Quantum mechanics
Periodic table
Spectroscopic symbol
1s  2s  2 p 3s 3 p  4s ...
2 S 1
LJ
J  LS
J   L  S , L  S  1,...., L  S 
Quantum mechanics
Periodic table
1s  2s  2 p 3s 3 p  4s ...
2 S 1
LJ
Quantum mechanics
2 S 1
Periodic table
LJ
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
Quantum mechanics
Quiz 33
What is the spectroscopic symbol for Silicon ?
Si: (Ne)(3s)2(3p)2
A.
2
S1
3
P2
3
P0
4
S2
B.
C.
D.
E.
4
D2
Quantum mechanics
Quiz 33
What is the spectroscopic symbol for Chlorine ?
Cl: (Ne)(3s)2(3p)5
A.
B.
C.
2
S1
4
D2
3
P0
4
S2
D.
E.
2
P3/ 2
Quantum mechanics
Solids
eWhat is
the wave function
of a valence electron
in the solid?
Quantum mechanics
Solids
e-
Basic Models:
• Free electron gas theory
• Crystal Bloch’s theory
Quantum mechanics
Free electron gas
lz
e-
e-
ly
lx
Volume
V  lxl y lz
Number of electrons:
N .q
Quantum mechanics
Free electron gas
e-
 (r , t )
H 
2
2m
3D infinite
square well
 2  V (r )
V  x, y, z  

2
2m
0

 2  E
inside the cube
outside
Quantum mechanics
Free electron gas

2
2m
e-
 2  E
Separation of variables
 (r , t )   x ( x) y ( y ) z ( z )

2
2m
i 2 i  Ei i
 n x x   n y  y   n z  z 
8
 (r , t ) 
sin 
 sin 
 sin 

lxl y lz
l
l
l
 x   y   z 
Enx ny nz 
 2  nx 2
2
ny 2
 2  2
2m  l x
ly
2 2
nz 2 
k
 2 
lz  2m
Quantum mechanics
Free electron gas
Enx ny nz 
Bravais
k-space
kz
 2  nx 2
2
ny 2
 2  2
2m  l x
ly
2 2
nz 2 
k
 2 
lz  2m
ky
Volume of
unit cell
3
lxl y lz
kx

3
V
Quantum mechanics
Quiz 33c
How many electrons can be contained in one
unit cell in the Bravais k- lattice ?
A. 1
B. 2
C. q
D. Nq
E. An infinity
Quantum mechanics
Free electron gas
kz
kF
Fermi surface
Bravais
k-space
ky
Free electron density
kx

k F  3
2

1/3
Nq

V
Quantum mechanics
Free electron gas
kz
kF
Fermi surface
ky
2
kx
2
kF 2
EF 

3 2
2m
2m
Bravais
k-space

Total energy contained inside the Fermi surface
EF
2 5
F
2
kV
Etot   dE 
10 m
0

2/3
Quantum mechanics
Free electron gas
kz
kF
Fermi surface
ky
kx
Bravais
k-space
Solid Quantum pressure
arises from
Pauli exclusion principle
Solid Quantum pressure
2
dV
dEtot   Etot
3
V
2 Etot
P

3 V

3
2

2/3
5m
2
 5/3
Quantum mechanics
Solids
e-
Enx ny nz 
 2  nx 2
2
ny 2


2m  lx 2 l y 2
2 2
nz 2 
k
 2 
lz  2m
kz
kF
Fermi surface
ky
kz
Bravais
k-space
kx
ky
kx
Number of unit cells
N A  6.02  1023