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Equation 92 · Part 12 · The Entry a Relabeling Cannot Write

Numerator: hbaromega

ΔEΛ(t)=⟨Hf⟩−⟨HS⟩(t)=−ℏω2cos⁡(2gt).\Delta E_\Lambda(t) = \langle H_f\rangle - \langle H_S\rangle(t) = -\frac{\hbar\omega}{2}\cos(2gt).
ℏω\hbar\omega

What this part means

The complete quantity above the fraction bar.

Its job in the formula

hbaromega occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

…at that instant, matching the closed form by direct substitution rather than by coincidence. A system-only observer watching only ⟨\langle HSH_S⟩(t)\rangle(t) across this half-period would see the tracked energy swing by a full ℏ\hbarω\omega with no term in HSH_S alone to explain it; the ledger accounts for every joule of that swing at every instant, without approximation.

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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 article section

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