The quantum adiabatic theorem governs the evolution of a

wavefunction under a slowly time-varying Hamiltonian. I will

consider the opposite limit of a Hamiltonian that is varied

impulsively: a strong perturbation U(x,t) is applied over a time

interval of infinitesimal duration e->0. When the strength of the

perturbation scales like 1/eˆ2, there emerges an interesting

dynamical behavior characterized by an abrupt displacement of

the wave function in coordinate space. I will solve for the

evolution of the wavefunction in this situation. Remarkably, the

solution involves a purely classical construction, yet describes

the quantum evolution exactly, rather than approximately. I will

use these results to show how appropriately tailored impulses

can be used to control the behavior of a quantum wavefunction.

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