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Equation 113 · Part 5 · The Entry a Relabeling Cannot Write

Symbol U

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,
UU

What this part means

U is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

U is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

…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 operator that actually generates…

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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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