= Quantum state continuity at a finite Hamiltonian step
{title2=$|\psi(0^+)\rangle=|\psi(0^-)\rangle$}
For a fixed <Hilbert space>, a bounded <self-adjoint> <Hamiltonian operator> with a finite step $H_i\to H_f$ at $t=0$ does not deliver an instantaneous rotation of the <quantum state>. Integrating the <Schrödinger equation> across a short interval gives
$$
|\psi(\epsilon)\rangle-|\psi(-\epsilon)\rangle=-\frac{i}{\hbar}\int_{-\epsilon}^{\epsilon}H(t)|\psi(t)\rangle\,dt.
$$
For a normalized <quantum state> and $\|H(t)\|\le M$ in <operator norm>, the difference has <norm> at most $2\epsilon M/\hbar$, which tends to zero. The time <derivative> can jump even though the <quantum state> is continuous. Here $\hbar$ is the reduced <Planck constant>. An instantaneous replacement of the <eigenbasis> changes coordinates, not the vector being represented.
For a real switch occupying $0\le t\le\tau$, remove any real scalar contribution $a(t)I$ that produces only a <global phase>, and write $|\widetilde\psi(t)\rangle=e^{i\int_0^t a(s)ds/\hbar}|\psi(t)\rangle$. The same integral argument gives the sufficient bounded-model estimate
$$
\bigl\|\widetilde\psi(\tau)-\psi(0)\bigr\|\le\varepsilon,\qquad
\varepsilon=\frac1\hbar\int_0^\tau\|H(t)-a(t)I\|\,dt.
$$
Thus $\varepsilon\ll1$ controls the <sudden approximation> up to a <global phase>. For any normalized final <energy eigenstate>, its <probability amplitude> differs from the instantaneous-switch value by at most $\varepsilon$, and its <Born rule> <probability> differs by at most $2\varepsilon$. These are sufficient error bounds, not necessary criteria. For the fixed-gap <two-level sudden quench>, any switching path $H(t)=-\Delta\mathbf n(t)\cdot\boldsymbol\sigma/2$ with a unit vector $\mathbf n(t)$ has <operator norm> $\Delta/2$, giving $\varepsilon\le\Delta\tau/(2\hbar)$.
This estimate is deliberately for bounded <Hamiltonian operators>. An unbounded <energy> spectrum does not supply a finite global $M$; relevant <quantum states>, <operator domains> and excited <energy eigenvalues> require separate control. The <adiabatic theorem> also requires hypotheses about the path and its <spectral gaps>; there is no universal switch-duration criterion covering all spectra. The <cold-atom collapse and revival after a lattice quench> illustrates how a single switch can be sudden for one degree of freedom and slow for another.
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