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∣ψ′(t)⟩:=U(t)∣ψ(t)⟩|\psi'(t)\rangle := U(t)|\psi(t)\rangle

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A third regime sits between the first two and needs its own treatment rather than being folded into either. Nothing here is discarded — Hc\mathcal H_c = Hf\mathcal H_f still — but the relabeling itself now carries explicit time dependence, because the “coarse” observer describes the same system from a frame in motion relative to the frame that defines HfH_f : a rotating platform, a driven interaction picture, a magnet ramping in time. Let U(t) be the corresponding time-dependent unitary and |ψ\psi'(t)⟩\rangle := U(t)|ψ(t)\psi(t)⟩\rangle , with |ψ(t)\psi(t)⟩\rangle solving iℏ\hbar\, d|ψ\psi⟩\rangle/dt = HfH_f|ψ\psi⟩\rangle . Differentiating the product and using U†U^\dagger U = I gives

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∣ψ′(t)⟩:=U(t)∣ψ(t)⟩|\psi'(t)\rangle := U(t)|\psi(t)\rangle

Equation 109 · 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.

A third regime sits between the first two and needs its own treatment rather than being folded into either. Nothing here is discarded — Hc\mathcal H_c = Hf\mathcal H_f still — but the relabeling itself now carries explicit time dependence, because the “coarse” observer describes the same system from a frame in motion relative to the frame that defines HfH_f : a rotating platform, a driven interaction picture, a magnet ramping in time. Let U(t) be the corresponding time-dependent unitary and |ψ\psi'(t)⟩\rangle := U(t)|ψ(t)\psi(t)⟩\rangle , with |ψ(t)\psi(t)⟩\rangle solving iℏ\hbar\, d|ψ\psi⟩\rangle/dt = HfH_f|ψ\psi⟩\rangle . Differentiating the product and using U†U^\dagger U = I gives

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