Download Nov 16 - BYU Physics and Astronomy

Physics 451
Quantum mechanics I
Fall 2012
Nov 16, 2012
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• HW #20 Thursday Nov 17 by 10pm
Pb 4.27, 4.28, 4.29, 4.33, 4.34, 4.35
Practice Test 3 review
Mon Nov 19 – Tu Nov 20
Test 3 :
Monday Nov 26 – Friday Nov 30
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
Sz 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
Quantum mechanics
State of the electron
a
  a    b    
b 
a  b 1
2
Normalization
Expectation values
2
Sx   Sx 
Pb 4.27, 4.28, 4.29
etc…
Quantum mechanics
Quiz 29a
When measuring Sx on an electron, what are the possible results?
A. 0
B.
C.

/2
D.  / 2
E.
 /3
Quantum mechanics
Adding spins
S  S1  S2
  1 2
S z   (m1  m2 )   m
: m  1
Possibilities
for two spins 1/2
: m  0
: m  0
: m  1
Quantum mechanics
Adding spins 1/2
Triplet
possibilities
: m  1
: m  0
: m  0
: m  1
11 
1
10 
   

2
1  1 
Singlet
1
00 
   

2
Quantum mechanics
Adding spins 1/2
Magnitude of the spin momentum ?
S 2  S12  S22  2S1S2
Triplet
Singulet
S 2 10  2
S 2 00  0
2
10
Quantum mechanics
Adding spins S
Possible values for S:
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
Pb 4.34, 4.35
Quantum mechanics
Clebsch- Gordan coefficients
Quantum mechanics
Quiz 29b
What is the value of the Clebsch-Gordan coefficient
A.
3 / 10
B.
8 / 15
C.
6 / 15
D.
1 / 15
E.
3/ 5
C
213
10 1?
Quantum mechanics
The spin and magnetic field
Quantum mechanics
The spin and magnetic field
S
B
 S
H   .B
Quantum mechanics
Larmor precession
B
z
S
Sz 

y
x
Pb 4.32
d
i
 H    B0 Sˆz 
dt
Sx 
2
2
cos 
sin  cos   B0t 
S y   sin  sin   B0t 
2
Larmor frequency
   B0
Quantum mechanics
Stern-Gerlach experiment
z

B
S

F    .B 
 (t )   ae
i tB0 /2
E 
  e
i  t /2  z
Moves up
 B0   z 

 ae
2
 i tB0 /2
  e
 i  t /2  z
Moves down