A systematic theory of sudden perturbations is derived for quantum systems whose states are described both by wave functions (a pure ensemble) and by a quantum density operator (a mixed ensemble). A perturbation series is written in powers of the parameter $\omega\tau$, which is small when the perturbation is ``sudden''; $\hbar\omega$ is the typical eigenvalue of the unperturbed system; and $\tau$ is the characteristic collision time. When the perturbation $V(t)$, taken at different times, commutes with itself, the theory yields a compact analytic expression for the probabilities for stimulated transitions for any value of $V(t)/\hbar$. The results of many cross-section calculations for atomic collision processes are discussed from a common standpoint: the processes are treated as ``jarring'' processes which stimulate transitions in the quantum system. If a momentum $\delta_p$ is rapidly transferred to the system in a collision, regardless of the physical nature of the ``jarring'', the probabilities for the stimulated transitions are governed by the parameter $N\sim\delta_p\cdot\delta R/\hbar$ where $\delta R$ is a measure of the uncertainty in the coordinates which is due to the relatively slow motions in the unperturbed system.