= Impulsive quantum kick
{title2=$U_{\rm kick}=e^{-iK/\hbar}$}
An <impulsive quantum kick> is the limit of a pulse with finite integrated <Hamiltonian operator>, rather than a bounded finite step. For a bounded <self-adjoint operator> $K$ with units of <action>, a rectangular pulse $H_\tau=K/\tau$ lasting $\tau$ has exact <unitary time evolution> $e^{-iK/\hbar}$ for every $\tau$. In the limiting notation $H(t)=K\delta(t)$, $\delta$ is the <Dirac delta function> and
$$
|\psi(0^+)\rangle=e^{-iK/\hbar}|\psi(0^-)\rangle.
$$
The integral estimate for <quantum state continuity at a finite Hamiltonian step> no longer tends to zero: the pulse height diverges. A bounded background <Hamiltonian operator> contributes vanishing evolution during this limit. A more general pulse with noncommuting generators can require <time ordering>; integrated area alone then need not specify the result.
For example, $K=\hbar\phi\sigma_y/2$, with a $y$ <Pauli matrix>, sends a <qubit> in $|0\rangle$ to $\cos(\phi/2)|0\rangle+\sin(\phi/2)|1\rangle$. At $\phi=\pi$, the <Born rule> predicts $|1\rangle$ with <probability> one, however short the regularized pulse is. Its finite rotation is precisely what a bounded finite step cannot accomplish at the switching instant.
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