Physics 451
Quantum mechanics I
Fall 2012
Dec 3, 2012
Karine Chesnel
Quantum mechanics
Homework
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
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 32a
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 32b
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:
Nq
Quantum mechanics
Free electron gas
e-
 (r , t )
H 
2
2m
3D infinite
square well
 2  V (r )
V  x, y, z  

2
2m
 2  E
0

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
kx
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
kF
2 5
F
2
kV
Etot   dE   Ek nk dk 
 V 2/3
10 m
0
0

2/3
Quantum mechanics
Free electron gas
kz
kF
Fermi surface
ky
kx
Bravais
k-space
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
Quantum mechanics
Solids
e-
Enx ny nz 
 2  nx 2
2
ny 2


2m  lx 2 l y 2
Pb 5.15:
Relation between Etot and EF
Pb 5.16:
Case of Cu:
calculate EF , vF, TF, and PF
2 2
nz 2 
k
 2 
lz  2m
kz
kF
Fermi surface
ky
kz
Bravais
k-space
kx
ky
kx
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
Quantum mechanics
Solids
Dirac comb
V(x)
N 1
V ( x)     ( x  ja)
Bloch’s theorem
j 0
V ( x  a )  V ( x)
 ( x  a)  eiKa ( x)
 ( x  a )   ( x)
2
2
Quantum mechanics
Solids
Circular periodic condition
V(x)
x-axis “wrapped around”
 ( x  Na)   ( x)
eiNKa  1
2 n
K
Na
Quantum mechanics
Solids
Solving Schrödinger equation
V(x)
0
a
d 2

 E
2
2m dx
2
 ( x)  A sin(kx)  B cos(kx)
0 xa
Quantum mechanics
Solids
Boundary conditions
V(x)
0
a
0 xa
 ( x)  A sin(kx)  B cos(kx)
a  x  0
 ( x  a)  eiKa ( x)
 ( x)  eiKa  A sin(kx)  B cos(kx)
Quantum mechanics
Solids
Boundary conditions at x = 0
V(x)
a
0
 left ( x)  eiKa  A sin k ( x  a)  B cos k ( x  a)
eiKa  A sin(ka)  B cos(ka)  B
• Continuity of 
• Discontinuity of
 right ( x)  A sin(kx)  B cos(kx)
d
dx
kA  e
 iKa
k  A cos(ka )  B sin(ka )  
2m
2
B
Quantum mechanics
Solids
Quantization of k:
m
cos( Ka )  cos(ka )  2 sin(ka)
k
z  ka

m a
2
sin( z )
f ( z )  cos( z )  
 cos( Ka )
z
Band structure
Pb 5.18
Pb 5.19
Pb 5.21
Quantum mechanics
Quiz 33
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
Quantum mechanics
Solids
Quantization of k:
Band structure
E
Band
Conductor: band
partially filled
N states
( q even integer)
Gap
Band
Insulator: band
entirely filled
N states
Semi-conductor:
doped insulator
Gap
Band
N states
2N electrons
(2e in each state)