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Equation 94 · Part 2 · The Atlas That Refuses to Close

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

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.