Quantum II (PHYS 4410)
Lecture 2
Spin ½
Homework session: Scheduled for
Thursdays, 4-6PM in Duane D-142.
Prof. Chuck Rogers: [email protected]
303-492-4476
Office hours: MF 4-4:45P after class, start here and then migrate to
Duane F-631
Homework sessions: Thursdays 4-6PM Duane D-142
Website: http://www.colorado.edu/physics/phys4410
Physics Colloquium 4PM Today:
Duane G1B20
Prof. Dmitry Reznik.
Looking inside quantum materials with
the Spallation Neutron Source
In spin space, the basis states (eigenstates of S2,
Sz ) are orthogonal:
  0.
Are the following matrix elements zero or
non-zero?
S 
2
 Sz 
A) Both are zero
B) Neither are zero
C) The first is zero; second is non-zero
D) The first is non-zero; second is zero
149
Phys3220, U.Colorado at Boulder
TRUE (A) or FALSE (B):
0
Sy  
2  i
i 

0
… is a hermitian operator.
Phys3220, U.Colorado at Boulder
The 2x2 matrix representation of the s =1/2
operator for Sy is:
0
Sy  
2  i
i 

0
Sy is a hermitian operator. The possible
results of a measurement of Sy are:
A) +/- 1
B) +/- i
C) +/- 2
D) Impossible to know due to the
Uncertainty Principle
Phys3220, U.Colorado at Boulder
A spin ½ particle in the spin state
a
  a  b   
b
A measurement of Sz is made. What is the
probability that the value of Sz will be ħ/2?

A)
D)
150
 Sz 
2
 Sz 

2
B)

2
C)
 Sz 
2
E) None of these

Phys3220, U.Colorado at Boulder
A spin ½ particle is in a spin state (a “spinor”)
a
  a  b   
b
A measurement of Sz is made. What is the
probability that the value of Sz will be -ħ/2?

A)
D)
 Sz 

2

150
2
B)
 Sz 
2
C)
 Sz 
2
E) None of these

Phys3220, U.Colorado at Boulder
A large number of spin ½ particles are in:
a
  a  b   
b
Many measurements of Sz are made. What is
the average outcome of those measurements?

A)
D)
 Sz 

2
2
B)
 
2

 Sz 
2
C)
 Sz 
2
2
E) None of these

Phys3220, U.Colorado at Boulder
A spin ½ particle is in the +ħ/2 eigenstate of Sˆ x
(i.e, it has a definite value for the x-component
of spin, +ħ/2)
Suppose we immediately measure S
z.
What is the probability that this measurement
will yield Sz= +ħ/2?
154
A) Zero
B) 25%
C) 50%
D) 100%
E) other/Impossible to say
Phys3220, U.Colorado at Boulder
Suppose a spin ½ particle is in the spin state
1
       , the  /2 eigenstate of Sˆ z .
0
Suppose we measure Sx and then immediately
measure Sz. What is the probability that the
second measurement (Sz) will leave the particle
0
in the Sz = down state        ?
1
A) Zero
B) Non-zero
154
Phys3220, U.Colorado at Boulder
Consider two possible states for a spin ½ particle:
1 1
1 1 
 I   , and  II   
2 1
2 1
Is there any physical (measurable) difference
between these two states?
A) No, they are indistinguishable
(phases, like -1, don’t matter in QM)
B) Yes, they are easily distinguishable
Phys3220, U.Colorado at Boulder
A classical particle of charge, q, and mass,
m, follows a circular path at speed, v. The
electrical current due to this motion is on
average equal to what around the path?
qv
A)
,
2
r
qv
C)
,
2 r
q
E)
2mv
q
B)
vr
r
D)
qv
The raising operator operating on the up and
down spin states:
S     ,
S   0
What is the matrix form of the operator S+ ?
A)   0
1

1

0
0
1


D)  
 1 1


151
B)
0 1

 
 0 0
C)
 0 0

 
1 0
E) None of these
Phys3220, U.Colorado at Boulder
The raising operator is
0 1

S+ =  
 0 0
Is the raising operator S+ hermitian?
A)Yes, always
B) No, never
C) Sometimes
152
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