FOUNDATIONS OF QUANTUM MECHANICS

III. 6. SPIN 1/2 PARTICLES 71
EXERCISE 25. Prove the following statements.
(a) ⟨⃗σ⟩ W = ⃗w,
(b) det W = 1 4 (1 − ⃗w 2 ),
(c) the eigenvalues of W are 1 2 ± 1 2 ∥ ⃗w∥.
EXAMPLES
In the following two examples, consider vectors ⃗w with ∥ ⃗w∥ = 1, thus corresponding to pure
states.
(a) Since in this case ⃗w equals the unit vector ⃗n, for ⃗w = (0, 0, 1) ∈ R 3 we have
( )
W = 1 (11 1 0
2 + σ z) = , (III. 155)
0 0
which is a 1 - dimensional projector, it is the matrix representation of W = |z ↑⟩ ⟨z ↑|.
Likewise we have
⃗w = (1, 0, 0) =⇒ W = 1 2 (11 + σ x) = |x ↑⟩ ⟨x ↑|, (III. 156)
⃗w = (0, 1, 0) =⇒ W = 1 2 (11 + σ y) = |y ↑⟩ ⟨y ↑|,
and we see that generally W = 1 2
(11 + ⃗n · ⃗σ) corresponds to the pure state |⃗n, +⟩, as was
already shown in (III. 149).
In the same way, for |⃗n, −⟩ we have
etc.
⃗w = (0, 0, − 1) =⇒ W = 1 2 (11 − σ z) = |z ↓⟩ ⟨z ↓|, (III. 157)
(b) For the probability to find spin up in the direction ⃗n ′ in the state |⃗n, +⟩, with (III. 45)
and (III. 127) we find
µ W ⃗n
(W ⃗n ′) = Tr W ⃗n ′ W ⃗n = Tr ( 1
2 + ⃗n ′ · ⃗σ) · 1
2
(11 + ⃗n · ⃗σ))
= 1 4 Tr ( 11 + ⃗n ′ · ⃗σ + ⃗n · ⃗σ + (⃗n ′ · ⃗n)11 + i⃗σ · (⃗n ′ × ⃗n) )
= 1 2 (1 + ⃗n ′ · ⃗n) = 1 2 (1 + cos θ) = cos2 1 2θ, (III. 158)
with θ the angle between ⃗n and ⃗n ′ . This is in accordance with (III. 148).
The following examples concern mixed state operators W , for which ⃗w has its endpoint somewhere
inside the sphere, ⃗w 2 < 1.
(c) Choosing ⃗w to be 1 2
(0, 1, 0) yields
( 1
W = 1 (11 2 + 1 2 σ 2
y) =
− 1 4 i
This can, for instance, be factorized as
1
4 i 1
2
)
. (III. 159)
W = 1 4 |z ↑⟩ ⟨z ↑| + 1 4 |z ↓⟩ ⟨z ↓| + 1 2
|y ↑⟩ ⟨y ↑|, (III. 160)
which clearly is a mixture.