View
217
Download
1
Category
Tags:
Preview:
Citation preview
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
Recommended