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UHfU†UH_fU^\dagger

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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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UU

Symbol U

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HfH_f

Symbol H_f

HfH_f is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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U†U^\dagger

Symbol U^dagger

UdU^dagger is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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Published contexts (2)

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UHfU†UH_fU^\dagger

Equation 116 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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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UHfU†UH_fU^\dagger

Equation 138 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The crank-magnet error this section exists to name lives exactly at the boundary between these two Hamiltonians: an observer who correctly computes UHfH_fU†U^\dagger — a legitimate operator identity, unchanged from the passive-relabeling case above — and then uses that operator, rather than Hc(t)H_c(t) , to predict or track the rotating frame’s own dynamics will find a discrepancy with no term in their own equations to explain it, on a clock the lab frame does not keep. The discrepancy is not missing energy. It is a dropped term whose physical origin is whatever classical control is enacting U(t) in the first place — the field ramp, the driven rotation — already doing ordinary, accountable work on…

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