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Hc(t):=UHfU†+iℏU˙U†H_c(t) := UH_fU^\dagger + i\hbar\dot U U^\dagger

Why this formula appears here

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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U†U^\dagger

Symbol U^dagger

UdU^dagger is one of the signed contributions combined to compute the quantity on the left.

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U˙\dot U

Symbol dot U

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

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Published contexts (1)

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Hc(t):=UHfU†+iℏU˙U†H_c(t) := UH_fU^\dagger + i\hbar\dot U U^\dagger

Equation 115 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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