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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

Why this formula appears here

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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tt

Symbol t

t occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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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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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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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,

Equation 113 · 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 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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