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Equation 120 · Part 5 · The Entry a Relabeling Cannot Write

Symbol dot U^dagger

(iℏU˙U†)†=−iℏUU˙†=iℏU˙U†(i\hbar\dot U U^\dagger)^\dagger = -i\hbar U\dot U^\dagger = i\hbar \dot U U^\dagger
U˙†\dot U^\dagger

What this part means

dot UdU^dagger has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

Its job in the formula

dot UdU^dagger has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

The passage around this formula

so the operator that actually generates |ψ\psi'(t)⟩\rangle ’s evolution is Hc(t)H_c(t) := UHfH_fU†U^\dagger + iℏ\hbarU˙\dot U U†U^\dagger , not the bare conjugate UHfH_fU†U^\dagger alone. The extra piece, iℏ\hbarU˙\dot U U†U^\dagger , is Hermitian — differentiating UU†U^\dagger = I gives U˙\dot U U†U^\dagger = -UU˙†\dot U^\dagger , so (iℏ\hbarU˙\dot U U†U^\dagger)^†\dagger = -iℏ\hbar UU˙†\dot U^\dagger = iℏ\hbar U˙\dot U U†U^\dagger — and it carries units of energy, ℏ\hbar times a rate. Call its expectation value the frame’s inertial term, Δframe(t)\Delta_{\rm frame}(t) := iℏ\hbar⟨\langleU˙(t)\dot U(t) U(t)^†\dagger⟩\rangle .

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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