An experiment such as that in Figure 2.6 represents a measurement out-
come: the average value of the z-component of spin of the input beam is
+1/2, obtained by the weighted probabilities of spin 1/2 at the up port and
spin −1/2 at the down port, which in quantum notation is:
hS
z
i
|ii
=
1
2
|h↑
z
|↑
z
i|
2
+ (−
1
2
)|h↓
z
|↑
z
i|
2
=
1
2
.
Similarly, the experiment in Figure 2.4 gives us the average value of the x-
component of spin the input beam |↑
z
i, which is zero:
hS
z
i
|ii
= +
1
2
|h↑
x
|↑
z
i|
2
+ (−
1
2
)|h↓
x
|↑
z
i|
2
= 0.
Consider a beam of spin-up electrons from the filter SG
z
↑. We now set up
a second Stern–Gerlach machine, but rotated by an angle θ to z. Let us label
this machine “SG
θ
”. Into this machine we pass the beam of |↑
z
i electrons.
What would be the measured output? The schematic setup is in Figure (2.7)
and the outputs are |↑
θ
i state at the up port and |↓
θ
i state at the down port.
We wish to predict the probabilities of each state.
FIGURE 2.7: An experiment with the SG along an arbitrary direction θ.
Now the “classical” projection cos θ×1/2 of the spin of the incoming beam
along the θ direction, gives us an average value for the measured spin, weighing
in both the output ports. The intensity of the up beam gives an average
spin of +1/2 with probability P(↑
θ
) = |h↑
θ
|↑
z
i|
2
and −1/2 with probability
P(↓
θ
) = |h↓
θ
|↑
z
i|
2
. The total probability of this happening is cos θ. So we have
1
2
cos θ =
1
2
P(↑
θ
) −
1
2
P(↓
θ
). (2.8)
24 Introduction to Quantum Physics and Information Processing
We also have an equation for conservation of probability:
1 = P(↑
θ
) + P(↓
θ
). (2.9)
From these two, we get the probabilities of the up and down spin states in the
output:
P(↑
θ
) = cos
2
θ
2
, P(↓
θ
) = sin
2
θ
2
. (2.10)
Let’s check this result by comparing with the special cases:
• θ = 0: SG
θ
= SG
z
,
P(↑) = 1, P(↓) = 0.
• θ = π: SG
θ
= SG
−z
or an SG
z
turned upside down,
P(↑) = 0, P(↓) = 1.
• θ = π/2: SG
x
(or SG
y
)
P(↑) =
1
2
= P(↓)
Mathematically we can write the state of the input electron as a superpo-
sition of the output states of SG
θ
:
|↑
z
i = α|↑
θ
i + β|↓
θ
i,
where
|α|
2
= P(↑
θ
) = cos
2
θ
2
, |β|
2
= P(↓
θ
) = sin
2
θ
2
.
The experiment only tells us the magnitudes of the complex amplitudes α and
β.
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