Physics 451
Quantum mechanics I
Fall 2012
Dec 5, 2012
Karine Chesnel
Quantum mechanics
Homework
Last assignment
• HW 24 Thursday Dec 6
5.15, 5.16, 5.18, 5.19. 5.21
Final exam
Wednesday Dec 12, 2012
7am – 10am
C 285
Quantum mechanics
Class evaluation
Please fill
the class evaluation
survey online
Quiz 34: 5 points
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
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
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)
or
 ( x)  eiKa  A sin k ( x  a)  B cos k ( x  a)
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 Y
• 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)
Quantum mechanics
Quiz 33
A material has q=3 valence electrons / atoms.
In which category will it fall
according to the 1D dirac periodic potential model?
A. Conductor
B. Insulator
C. Semi-conductor
Quantum mechanics
Final Review
What to
remember?
Quantum mechanics
Wave function and expectation values

 

p   Y  i
 Ydx
x 



x 
 Y xYdx
*
*

“Operator” x
“Operator” p

 

Q   Y Q  x, i
 Ydx
x 


*
Quantum mechanics
Time-independent Schrödinger equation
2
Y
2Y
i

V Y
2
t
2m x
Here
V ( x)
The potential is independent of time
Stationary state
Y( x, t )   ( x) (t )   ( x)eiEt /

General state
Y ( x, t )   cn n ( x)e  iEnt /
n 1
Review I
Quantum mechanics
Infinite square well
 3 , E3
n 
2
 n
sin 
a
 a

x

Excited states
 2 , E2
Quantization of the energy
Ground state
0
 1 , E1
a
x
n2 2 2
En 
2
2ma
Quantum mechanics
Harmonic oscillator
V(x)
V ( x) 
1 2 1
kx  m 2 x 2
2
2
a 
1
2m 

ip  m x 
• Operator position
• Operator momentum
x
x
pi
2m
m
2
 a  a 
 a  a 
1
1


H    a a      a a  
2
2


Review I
Quantum mechanics
4. Harmonic oscillator
Ladder operators:
a 
n 1
Raising operator:
a n  n  1 n1
Lowering operator:
a n  n n1
n 
1
n
 a   0
n!
1

En   n   
2

a
n
 n1
En  
En
En  
Quantum mechanics
Quiz 35
What is the result of the operation  a   3 ?
4
A.
7 7
B.
3 2
C.
 4! 0
D.
 3! 0
E. 0
Quantum mechanics
Square wells and delta potentials
V(x)
Physical considerations
 incident  x   Aeikx
 reflected  x   Beikx
Scattering
States E > 0
 transmitted  x   Feikx
x
Symmetry considerations
Bound states
E<0
 even   x    even  x 
 odd   x    odd  x 
Ch 2.6
Quantum mechanics
Square wells and delta potentials
Continuity at boundaries
Delta functions

d
dx
is continuous
is continuous except where V is infinite
 d
D
 dx

2 m


 0 

2
h

Square well, steps, cliffs…

d
dx
is continuous
is continuous
Quantum mechanics
The delta function well/ barrier
V  x     x 
E 0
Scattering state
 right  x   Feikx
 left  x   Aeikx  Beikx
A
F
B
x
0
Reflection coefficient
1
R
1  2 2 E / m 2


Transmission coefficient
T

1
1  m 2 / 2 2 E

“Tunneling”
Quantum mechanics
Formalism
Wave function
Operators
Y Y
Vector
Hˆ   H ij 
Q Q
†
Observables are Hermitian operators
Q̂ a   a
Linear transformation
(matrix)
a
is an eigenvector of Q
 is an eigenvalue of Q
Quantum mechanics
Eigenvectors & eigenvalues
To find the eigenvalues:
T   I  a
 0
det T   I   0
We get a Nth polynomial in : characteristic equation
Find the N roots
 1, 2 ,...N 
Find the eigenvectors
 e1, e2 ,...eN 
Spectrum
Quantum mechanics
The uncertainty principle
Finding a relationship between standard deviations
for a pair of observables
 A B 
 A, B
2i
Uncertainty applies only for incompatible observables
Position - momentum
Dx Dp 
2
Quantum mechanics
The uncertainty principle
Energy - time
DE Dt 
Derived from the
Heisenberg’s equation
of motion
d Q
dt

i
Special meaning of Dt
2
Q
 H , Q 
t
Dt 
Q
d Q
dt
Quantum mechanics
Quiz 33
Which one of these commutation relationships
is not correct?
A.
 x, p  i
B.
V ( x), x  o
C.
 H , x  o
D.  Ly , Lz   i Lx
E.
 Lx , L2   0
Quantum mechanics
Schrödinger equation in
spherical coordinates
z


r
y
x
H  
2
   V (r )  E
2
2m
 nlm  r, ,   Rnl  r  Yl m  , 
The radial equation
1 d  2 dR  2mr 2
r
  2 V (r )  E   l (l  1)
R dr  dr 
1 1  
Y 
1  2Y 
 l (l  1)
 sin 
 2
The angular equation 
2 
Y  sin   
  sin   
Quantum mechanics
The hydrogen atom
R(r ) 
1 l 1  
 e v(  )
r
jmax
v(  )   c j  j
j 0
c j 1 
  kr
2( j  l  1  n)
cj
( j  1)( j  2l  2)
Quantization of the energy
2
 m  e2 2  1
1

En    2 


 2  4 0   n2
2ma 2 n2


Bohr radius
4 0 2
10
a

0.529

10
m
2
me
Quantum mechanics
The hydrogen atom
Spectroscopy
Energies levels
E1
En  2
n
0
E
E4
Energy transition
Paschen
E3
E2
 1

hc
1
DE 
 E1  2  2 
n


n
i
f


Balmer
 1
1 

R 2  2
n


 f ni 
1
Rydberg constant
E1
Lyman
Quantum mechanics
z


The angular momentum
eigenvectors
r
y
H nlm  En nlm
x
Spherical harmonics
are the
eigenfunctions
1
2mr 2
L2 nlm 
2
l (l  1) nlm
Lz nlm  m  nlm

  2   2
  r  r r   L   V  E




Quantum mechanics
The spin
S 2 sm 
2
s(s  1) sm
S z sm  m sm
S sm 
s ( s  1)  m(m  1) s  m  1
Quantum mechanics
Adding spins S
Possible values for S when adding spins S1 and S2:
S   S1  S2  ,  S1  S2  1 ,  S1  S2  2  ,...  S1  S2 
sm 

s1s2 s
m1m2 m
C
m1  m2  m
s1m1 s2 m2
Clebsch- Gordan coefficients
Quantum mechanics
Periodic table
Filling the shells
2
2
6
1s  2s  2 p 3s 3 p  4s ...
Quantum mechanics
Periodic table
1s  2s  2 p 3s 3 p  4s ...
2 S 1
LJ
Quantum mechanics
Solids
• Crystal Bloch’s theory
•Free electron
gas theory
eEnx 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
2
kx
2
kF 2
EF 

3 2
2m
2m


2/3
Quantum mechanics
Thank you for your participation!
Good luck for finals
And Merry Christmas!