Physics 451
Quantum mechanics I
Fall 2012
Nov 12, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW #19 Tuesday Nov 13
Pb 4.14, 4.15, 4.16, 4.17
• HW #20 Thursday Nov 15
Pb 4.18, 4.19, 4.21, 4.22
Test 3 Review
Monday Nov 19- 20
Sign for practice test
Quantum mechanics
The angular momentum
L  Lx  iLy
Ladder operator
L2  L L  L2z
Top
Value
=+l
Lz


Eigenstates
f l m  Yl m
Lz f l m  mf l m
L2 fl m 
2
l (l  1) f l m

L
L
Bottom
Value = -l
Quantum mechanics
The angular momentum
LzYl m  mYl m
L2Yl m 
2
l (l  1)Yl m


L Yl m   lmYl m 1

Pb 4.18
L Yl m 
l (l  1)  m(m  1)Yl m1
L
L
Quantum mechanics
The angular momentum
In spherical coordinates
z


r
Lrp
r  

i
y
x



1
 
L  r  r  r   r 

r 
i
r
 sin 
 

 
1  
L  


i  
sin   




Lz 
i 
Quantum mechanics
z


The angular momentum
In spherical coordinates
r
y
L  Lx  iLy
x
L   e
 i
 
 
 i cot 


 
 
2


1


1



2
2
L  
 sin 
 2
2 
  sin   
 sin   
Pb 4.21, 4.22
Quantum mechanics
z

The angular momentum
eigenvectors
r
y

LzYl m 
x
and
L Yl  
2
m
2
 m
Yl  m Yl m
i 
 1  
 
1 2  m
Y 

 sin 
 2
2  l
  sin   
 sin   
2
l (l  1)Yl m
were the two angular equations for the spherical harmonics!
H nml  En nml
Spherical harmonics
are the
eigenfunctions
L2 nml 
2
l (l  1) nml
Lz nml  m  nml
Quantum mechanics
z


The angular momentum
and Schrödinger equation
r
y
x
1
2mr 2

  2   2

 r  r r   L   V  E




3 quantum numbers (n,l,m)
• Principal quantum number n: integer
• Azimutal and magnetic quantum numbers (l,m)
can also be half-integers.
Quantum mechanics
Quiz 26
For a given n value, how many eigenstates can we find for
the operator
L2
A. 0
B.
n
C. 2n  1
D.
n(n  1)
E.
n2
?
Quantum mechanics
The Spin
Types of angular momentum
Lrp
orbital
L  I
spin
Quantum mechanics
Agular moment in the atom
• Orbital moment (l)
• Spin moment (s)
Representation of
nlm r , ,  
Quantum mechanics
Spin in elementary particles
Each elementary particle is characterized
by an immutable spin S
• Fermions: (S half-integer)
S=1/2
Leptons: electrons,…
Quarks: u,b,c,s,t,b
Proton, neutron
• Bosons: (S integer)
Photon
Mesons
S=1
Quantum mechanics
The spin
L̂  Sˆ
 S x , S y   i S z
 S y , S z   i S x
Sz , Sx   i
Sy
 S 2 , S x    S 2 , S y    S 2 , S z   0
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
The spin 1/2
The “spinor”
3
S  
4
2
Sz  
2
2


Spin up
Spin down
1 
   
0
0
   
1 
3
S  
4
2
2

Sz    
2
3
S 
4
2
2
1 0


0 1
1 0 
Sz  

2  0 1
Quantum mechanics
Pauli matrices
3
S 
4
2
2
1 0


0 1
0 1
Sx  

2 1 0
x
 0 i 
Sy  

2i 0 
y
Pb 4.29
1 0 
Sz  

2  0 1
z