Physics 451
Quantum mechanics I
Fall 2012
Dec 5, 2012
Karine Chesnel
Homework
Quantum mechanics
Last assignment
• HW 24 Thursday Dec 65.15, 5.16, 5.18, 5.19. 5.21
Final exam
Wednesday Dec 12, 20127am – 10am
C 285
Class evaluation
Please fillthe class evaluation
survey online
Quiz 34: 5 points
Quantum mechanics
SolidsQuantum 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
Pb 5.15: Relation between Etot and EF
Pb 5.16: Case of Cu: calculate EF , vF, TF, and PF
Free electron gas
Quantum mechanics
Bravaisk-spacexk
yk
zk
Fk Fermi surface
2 2 2
2/3232 2
FF
kE
m m
Total energy contained inside the Fermi surface
2 52/3
20 0 10
F FE k
Ftot k k
k VE dE E n dk V
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 5totE
PV m
SolidsQuantum 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
Solids
Quantum mechanics
V(x)
( ) ( )V x a V x
Dirac comb
Bloch’s theorem
( ) ( )iKax a e x 2 2
( ) ( )x a x
1
0
( ) ( )N
j
V x x ja
Solids
Quantum mechanics
V(x)
( ) ( )x Na x
Circular periodic condition
1iNKae
2 nK
Na
x-axis “wrapped around”
Solids
Quantum mechanics
V(x)
( ) sin( ) cos( )x A kx B kx
Solving Schrödinger equation
0 a
2 2
22
dE
m dx
0 x a
Solids
Quantum mechanics
V(x)
( ) sin( ) cos( )x A kx B kx
Boundary conditions
0 a
0 x a
( ) ( )iKax a e x 0a x
( ) sin ( ) cos ( )iKax e A k x a B k x a or
Solids
Quantum mechanics
V(x)
( ) sin( ) cos( )right x A kx B kx
Boundary conditions at x = 0
0 a
• Continuity of
• Discontinuity of d
dx
sin( ) cos( )iKae A ka B ka B
2
2cos( ) sin( )iKa m
kA e k A ka B ka B
( ) sin ( ) cos ( )iKaleft x e A k x a B k x a
Solids
Quantum mechanics
2cos( ) cos( ) sin( )
mKa ka ka
k
Quantization of k:
sin( )( ) cos( ) cos( )
zf z z Ka
z
z ka
2
m a
Band structure
Pb 5.18Pb 5.19Pb 5.21
Quiz 33Quantum mechanics
A. 1
B. 2
C. q
D. Nq
E. 2N
In the 1D Dirac comb modelhow many electrons can be contained in each band?
Solids
Quantum mechanics
Quantization of k: Band structure
E
N states
N states
N states
Band
Gap
Gap
Band
Band
(2e in each state)
2N electrons
Conductor: bandpartially filled
Semi-conductor: doped insulator
Insulator: bandentirely filled
( even integer)q
Quiz 33Quantum mechanics
A. Conductor
B. Insulator
C. Semi-conductor
A material has q=3 valence electrons / atoms.In which category will it fall
according to the 1D dirac periodic potential model?
Final ReviewQuantum mechanics
What to remember?
Quantum mechanics
Wave function and expectation values
*x x dx
“Operator” x
*p i dxx
“Operator” p
* ,Q Q x i dxx
Quantum mechanics
Time-independent Schrödinger equation
2 2
22i V
t m x
Here ( )V xThe potential is independent of time
/( , ) ( ) ( ) ( ) iEtx t x t x e Stationary state
/
1
( , ) ( ) niE tn n
n
x t c x e
General state
Review IQuantum mechanics
Infinite square well
2 2 2
22n
nE
ma
Quantization of the energy
2sinn
nx
a a
x0 a
Ground state 1 1,E
Excited states
2 2,E
3 3,E
Quantum mechanics
Harmonic oscillator
x
V(x)
1
2a ip m x
m
1 1
2 2H a a a a
2 2 21 1( )
2 2V x kx m x
• Operator position 2
x a am
• Operator momentum 2
mp i a a
Review IQuantum mechanics
4. Harmonic oscillator
Ladder operators:
0
1
!
n
n an
nnE
Raising operator: 11n na n nE
a1n
Lowering operator: 1n na n nE
a1n
1
2nE n
Quiz 35Quantum mechanics
A.
B.
C.
D.
E. 0
What is the result of the operation ? 4 3a
77
23
04!
03!
Quantum mechanics
Square wells and delta potentials
V(x)
x
Bound statesE < 0
ScatteringStates E > 0
Symmetry considerations
even evenx x
odd oddx x
Physical considerations
ikxreflected x Be
ikxincident x Ae
ikxtransmitted x Fe
Quantum mechanics
Square wells and delta potentials
Ch 2.6
Continuity at boundaries
Delta functions
Square well, steps, cliffs…
dx
d
is continuous
is continuous except where V is infinite
022
m
dx
d
dx
d
is continuous
is continuous
Quantum mechanics
Scattering state 0E
0
ikx ikxleft x Ae Be ikx
right x Fe
A F
Bx
Reflection coefficient Transmission coefficient
2 2
1
1 2 /R
E m 2 2
1
1 / 2T
m E
The delta function well/ barrier
V x x
“Tunneling”
Formalism
Quantum mechanics
ˆijH H Linear transformation
(matrix)Operators
Wave function Vector
Observables are Hermitian operators †Q Q
Q̂ a a a is an eigenvector of Q
is an eigenvalue of Q
Quantum mechanics
Eigenvectors & eigenvalues
0T I a
det 0T I
To find the eigenvalues:
We get a Nth polynomial in : characteristic equation
Find the N roots 1 2, ,... N Spectrum
Find the eigenvectors 1 2, ,... Ne e e
Quantum mechanics
The uncertainty principle
,2A B
A B
i
Finding a relationship between standard deviations for a pair of observables
Uncertainty applies only for incompatible observables
Position - momentum 2x p
Quantum mechanics
The uncertainty principle
Energy - time
2E t
Special meaning of t
Qtd Q
dt
,d Q i QH Q
dt t
Derived from the Heisenberg’s equation
of motion
Quiz 33Quantum mechanics
A.
B.
C.
D.
E.
Which one of these commutation relationships is not correct?
,x p i
,y z xL L i L
2, 0xL L
( ),V x x o
,H x o
Quantum mechanicsSchrödinger equation in
spherical coordinates
2
2 2
1 1 1sin ( 1)
sin sin
Y Yl l
Y
The angular equation
, , ,mnlm nl lr R r Y
The radial equation 2
22
1 2( ) ( 1)
d dR mrr V r E l l
R dr dr
x
y
z
r2
2 ( )2
H V r Em
Quantum mechanics
The hydrogen atom
11( ) ( )lR r e v
r
Quantization of the energy
22 2
2 2 2 20
1 1
2 4 2n
m eE
n ma n
max
0
( )j
jj
j
v c
1
2( 1 )
( 1)( 2 2)j j
j l nc c
j j l
Bohr radius2
1002
40.529 10a m
me
kr
Quantum mechanics
The hydrogen atom
21
n
EEn Energies levels
Spectroscopy
221
11
fi nnE
hcE
Energy transition
22
111
if nnR
Rydberg constant
E0
E1
E2
E3
E4
Lyman
Balmer
Paschen
Quantum mechanics
The angular momentumeigenvectors
x
y
z
r
Spherical harmonicsare the
eigenfunctions
nlm n nlmH E
2 2 ( 1)nlm nlmL l l
z nlm nlmL m
2 22
1
2r L V E
mr r r
Quantum mechanics
The spin
2 2 ( 1)S sm s s sm
zS sm m sm
( 1) ( 1) 1S sm s s m m s m
Quantum mechanics
Adding spins S
Possible values for S when adding spins S1 and S2:
1 2 1 2 1 2 1 2, 1 , 2 ,...S S S S S S S S S
1 2
1 2
1 2
1 1 2 2s s sm m m
m m m
sm C s m s m
Clebsch- Gordan coefficients
Periodic table
Quantum mechanics
Filling the shells
1 2 2 3 3 4 ...s s p s p s2 2 6
Periodic table
Quantum mechanics
1 2 2 3 3 4 ...s s p s p s
2 1SJL
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
yk
zk
Bravaisk-space
xk
xk
yk
zk
FkFermi surface
e-
•Free electron gas theory
• Crystal Bloch’s theory
2 2 2
2/3232 2
FF
kE
m m
Quantum mechanics
Thank you for your participation!
And Merry Christmas!
Good luck for finals