= Sudden approximation
The <sudden approximation> treats a rapid, finite change of a <Hamiltonian operator> as leaving the <quantum state> approximately unchanged during the switch. The <energy> measurement can nevertheless change immediately: the same <quantum state> has different <probability amplitudes> in the new <eigenbasis>. Compute their <inner products> with the new <energy eigenstates>, then apply the <Born rule>. The <two-level sudden quench> below makes this distinction quantitative without assuming the change is small.
An ideal finite step has exact <quantum state continuity at a finite Hamiltonian step>; a finite-duration implementation has a small evolution error under the bounded-model conditions below. An <impulsive quantum kick> is different: its <Hamiltonian operator> grows as its duration shrinks, retaining a finite integrated <action> and a finite change of <quantum state>. A <sudden approximation> therefore concerns short evolution during switching, not <conservation of energy> during externally driven change. The subsequent <unitary time evolution> can be substantial.<Fluctuation theorems: Work is not an observable by Peter Talkner and others>[Talkner, Lutz and Hänggi, p. 2, paragraph before equation (7): sudden switch gives the identity time-evolution operator]{ref}
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