FOUNDATIONS OF QUANTUM MECHANICS

III. 6. SPIN 1/2 PARTICLES 73
The triplet states, with s = 1 and m = 1, 0, −1 are
|1, 1⟩ = |z ↑⟩ ⊗ |z ↑⟩
√
|1, 0⟩ = 1 ( )
2 2 |z ↑⟩ ⊗ |z ↓⟩ + |z ↓⟩ ⊗ |z ↑⟩
|1, − 1⟩ = |z ↓⟩ ⊗ |z ↓⟩. (III. 166)
III. 6. 3. 2
CORRELATIONS
In chapter VII we will use the spin correlation function of the singlet,
E QM (⃗a, ⃗ b) := ⟨0, 0|⃗a · ⃗σ 1 ⊗ ⃗ b · ⃗σ 2 |0, 0⟩, (III. 167)
where ⃗a, ⃗ b ∈ R 3 are unit vectors. E QM (⃗a, ⃗ b) is the expectation value to find both for particle 1 spin
up along ⃗a and for particle 2 spin up along ⃗ b. To find E QM (⃗a, ⃗ b), first choose the z - axis along ⃗a
as in diagram III. 3, next choose the x - axis in such a way that ⃗ b is in the xz - plane. The spherical
symmetry of the singlet state allows such a choice.
z
⃗a
θ ⃗a, ⃗ b
⃗ b
Figure III. 3: Spin up for particle 1 along ⃗a, for particle 2 along ⃗ b
x
With ⃗a = ⃗e z , ⃗ b similar to ⃗n in (III. 137), and θ ⃗a, ⃗ b
the angle between ⃗a and ⃗ b, we have
E QM (⃗a, ⃗ b) = ⟨0, 0| σ 1z ⊗ (sin θ ⃗a, ⃗ b
σ 2x + cos θ ⃗a, ⃗ b
σ 2z ) |0, 0⟩. (III. 168)
Now σ z |z ↑⟩ = |z ↑⟩, σ x |z ↑⟩ = |z ↓⟩ etc., so that we have, using (II. 100), (III. 165) and (III. 166),
√
(σ 1z ⊗ σ 2x ) |0, 0⟩ = 1 ( )
2 2 |1, 1⟩ + |1, −1⟩ (III. 169)
which is perpendicular to |0, 0⟩, and
(σ 1z ⊗ σ 2z ) |0, 0⟩ = − |0, 0⟩, (III. 170)
from which we see that
E QM (⃗a, ⃗ b) = − cos θ ⃗a, ⃗ b
. (III. 171)