A beam of spin one-half particles can also be separated by a Stern-Gerlach apparatus (See section
18.5) which uses a large gradient in the magnetic field to exert a forceon particles proprtional
to the component of spin along the field gradient. Thus, we can measure the component of spin
along a direction we choose. A field gradient will separate a beam of spin one-half particles into two
beams. The particles in each of those beams will be in a definite spin state, the eigenstate with the
component of spin along the field gradient direction either up or down, depending on which beam
the particle is in.
We may represent a Stern-Gerlach appartatus which blocks the lower beam by the symbol below.
{
- −|
}
z→
This apparatus is equivalent to the operator that projects out the + ̄h 2 eigenstate.
|+〉〈+|=
(
1
0
)
( 1∗ 0 ∗) =
(
1 0
0 0
)
We can perform several thought experiments. The appartus below starts with an unpolarized beam.
In such a beam we don’t know the state of any of the particles. For areally unpolarized beam, half
of the particles will go into each of the separated beams. (Note that an unpolarized beam cannot
be simply represented by a state vector.) In the apparatus below,we block the upper beam so that
only half of the particles come out of the first part of the apparatus and all of those particles are in
the definite state having spin down along the z axis. The second partof the apparatus blocks the
lower separated beam. All of the particles are in the lower beam so nothing is left coming out of the
apparatus.
Unpolarized Beam (N particles)→{
+|
−
}
z→
N
2
(
0
1
)
→
{
+
−|
}
z→ 0
The result is unaffected if we insert an additional apparatus that separates in the x direction in the
middle of the apparatus above. While the apparatus separates, neither beam is blocked (and we
assume we cannot observe which particles go into which beam). This apparatus does not change
the state of the beam!
(N particles)→{
+|
−
}
z→
N
2
(
0
1
)
→
{
+
−
}
x→
N
2
(
0
1
)
→
{
+
−|
}
z→ 0
Now if we block one of the beams separated according to the x direction, particles can get through
the whole apparatus. The middle part of the apparatus projects the state onto the positive eigenstate
ofSx. This state has equal amplitudes to have spin up and spin down along the z direction. So
now, 1/8 of the particles come out of the apparatus. By blocking one beam, the number of particles
coming out increased from 0 toN/8. This seems a bit strange but the simple explanation is that the
upper and lower beams of the middle part of the apparatus were interfering to give zero particles.
With one beam blocked, the inteference is gone.
(N)→
{
+|
−
}
z→
N
2
(
0
1
)
→
{
+
−|
}
x→
N
4
( 1
√
2
√^1
2)
→
{
+
−|
}
z→
N
8
(
1
0
)
→
Note that we can compute the number of particles coming out of thesecond (and third) part by
squaring the amplitude to go from the input state to the output state
N
2∣
∣
∣
∣
( 1
√
2
√^1
2)
(
0
1
)∣
∣
∣
∣
2
=