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∣ψ⟩=12(∣↑⟩+∣↓⟩)|\psi\rangle = \tfrac{1}{\sqrt2}(|{\uparrow}\rangle+|{\downarrow}\rangle)

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An exact two-level calculation shows how far this can depart from the intuitive picture. Preselect a spin-one-half system along x , |ψ\psi⟩\rangle = 12(∣↑⟩+∣↓⟩)\tfrac{1}{\sqrt2}(|{\uparrow}\rangle+|{\downarrow}\rangle) , and postselect along y , |f⟩\rangle = 12(∣↑⟩+i∣↓⟩)\tfrac{1}{\sqrt2}(|{\uparrow}\rangle+i|{\downarrow}\rangle) . 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

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∣ψ⟩=12(∣↑⟩+∣↓⟩)|\psi\rangle = \tfrac{1}{\sqrt2}(|{\uparrow}\rangle+|{\downarrow}\rangle)

Equation 94 · 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.

An exact two-level calculation shows how far this can depart from the intuitive picture. Preselect a spin-one-half system along x , |ψ\psi⟩\rangle = 12(∣↑⟩+∣↓⟩)\tfrac{1}{\sqrt2}(|{\uparrow}\rangle+|{\downarrow}\rangle) , and postselect along y , |f⟩\rangle = 12(∣↑⟩+i∣↓⟩)\tfrac{1}{\sqrt2}(|{\uparrow}\rangle+i|{\downarrow}\rangle) . 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

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