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Equation 113 · Part 3 · The Entry a Relabeling Cannot Write

Symbol psi

iℏ d∣ψ′⟩dt=(UHfU†+iℏ U˙U†)∣ψ′⟩,i\hbar\,\frac{d|\psi'\rangle}{dt} = \Big(U H_f U^\dagger + i\hbar\,\dot U U^\dagger\Big)|\psi'\rangle,
ψ\psi

What this part means

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

Its job in the formula

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

The passage around this formula

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 iℏ d∣ψ′⟩dt=(UHfU†+iℏ U˙U†)∣ψ′⟩i\hbar\,\frac{d|\psi'\rangle}{dt} = \Big(U H_f U^\dagger + i\hbar\,\dot U U^\dagger\Big)|\psi'\rangle. so the…

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