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Spin one-half along an axis ( $n \cdot S$ )

For

n = (sin θ, 0, cos θ)

, the normalized eigenstates of

n \cdot S

are

∣ ↑_{θ} ⟩ = cos \frac{θ}{2} ∣ ↑ ⟩ + sin \frac{θ}{2} ∣ ↓ ⟩, ∣ ↓_{θ} ⟩ = - sin \frac{θ}{2} ∣ ↑ ⟩ + cos \frac{θ}{2} ∣ ↓ ⟩,

with eigenvalues

+ ℏ/2

and

- ℏ/2

.

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Solution