subscript

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 — $\mathcal H_c$ = $\mathcal H_f$ 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 $H_f$ : a rotating platform, a driven interaction picture, a magnet ramping in time. Let U(t) be the corresponding time-dependent unitary and |$\psi$'(t)$\rangle$ := U(t)|$\psi(t)$$\rangle$ , with |$\psi(t)$$\rangle$ solving i$\hbar$\, d|$\psi$$\rangle$/dt = $H_f$|$\psi$$\rangle$ . Differentiating the product and using $U^\dagger$ U = I gives

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