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Equation 90 · Part 8 · The Atlas That Refuses to Close

Numerator: langle f|hat A|psirangle

Aw=⟨f∣A^∣ψ⟩⟨f∣ψ⟩,A_w = \frac{\langle f|\hat A|\psi\rangle}{\langle f|\psi\rangle},
⟨f∣A^∣ψ⟩\langle f|\hat A|\psi\rangle

What this part means

The complete quantity above the fraction bar.

Its job in the formula

langle f|hat A|psirangle occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

There is a fifth kind of answer to “where,” and it is worth stating precisely how it differs from all four above before using it. Prepare a system in a state |ψ\psi⟩\rangle , couple it weakly to a pointer, and then postselect on a final state |f⟩\rangle measured after the coupling. The weak value of any observable A^\hat A , position among them, conditioned on that postselection is Aw=⟨f∣A^∣ψ⟩⟨f∣ψ⟩A_w = \frac{\langle f|\hat A|\psi\rangle}{\langle f|\psi\rangle}. introduced by Aharonov, Albert, and Vaidman as the quantity a weakly coupled pointer’s mean shift actually reports, to leading order in the coupling strength, once the postselection has been performed [ 11 ] . AwA_w is not an eigenvalue of A^\hat A , not an expectation value in the ordinary sense, and…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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