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⟨σz⟩w=12(1+i)12(1−i)=i\langle \sigma_z \rangle_w = \frac{\tfrac12(1+i)}{\tfrac12(1-i)} = i

Why this formula appears here

​ 1 ​ ( ∣ ↑ ⟩ + i ∣ ↓ ⟩) . Then ⟨\langle f|ψ\psi⟩\rangle = 12(1−i)\tfrac12(1-i) and ⟨\langle f|σz\sigma_z|ψ\psi⟩\rangle = 12(1+i)\tfrac12(1+i) , so the weak value of σz\sigma_z is ⟨σz⟩w=12(1+i)12(1−i)=i\langle \sigma_z \rangle_w = \frac{\tfrac12(1+i)}{\tfrac12(1-i)} = i. σz\sigma_z has eigenvalues ±\pm1 only, yet its weak value here is purely imaginary, not merely outside the eigenvalue range but off the real line entirely. This is an exact, closed-form evaluation of the defining ratio, not a measured or simulated number, stated here only to make concrete what “anomalous weak value” means before the same word is applied to a spatial coordinate rather than a spin component.

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12(1−i)\tfrac12(1-i)

Denominator: tfrac12(1-i)

The complete quantity below the fraction bar; it must be nonzero for this division.

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With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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⟨σz⟩w=12(1+i)12(1−i)=i.\langle \sigma_z \rangle_w = \frac{\tfrac12(1+i)}{\tfrac12(1-i)} = i.

Equation 100 · Evolutionary Physics

The Atlas That Refuses to Close

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

​ 1 ​ ( ∣ ↑ ⟩ + i ∣ ↓ ⟩) . Then ⟨\langle f|ψ\psi⟩\rangle = 12(1−i)\tfrac12(1-i) and ⟨\langle f|σz\sigma_z|ψ\psi⟩\rangle = 12(1+i)\tfrac12(1+i) , so the weak value of σz\sigma_z is ⟨σz⟩w=12(1+i)12(1−i)=i\langle \sigma_z \rangle_w = \frac{\tfrac12(1+i)}{\tfrac12(1-i)} = i. σz\sigma_z has eigenvalues ±\pm1 only, yet its weak value here is purely imaginary, not merely outside the eigenvalue range but off the real line entirely. This is an exact, closed-form evaluation of the defining ratio, not a measured or simulated number, stated here only to make concrete what “anomalous weak value” means before the same word is applied to a spatial coordinate rather than a spin component.

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