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Equation 113 · Part 12 · The Entry a Relabeling Cannot Write

subscript

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,
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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