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.