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Equation 94 · The Atlas That Refuses to Close

What does this equation mean?

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

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Inputs and operationstfrac1sqrt2(|uparrowrangle+|downarrowrangle)
Result or condition|psirangle
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ψ\psi

Symbol psi

psi is part of the quantity the equation computes from the expression on the right.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

addition

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How to interpret it

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What the article says around this equation

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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