  We consider an exact population transition, defined as the probability of
finding a state at a final time being exactly equal to the probability of
another state at the initial time. We prove that, given a Hamiltonian, there
always exists a complete set of orthogonal states that can be employed as
time-zero states for which this exact population transition occurs. The result
is general: it holds for arbitrary systems, arbitrary pairs of initial and
final states, and for any time interval. The proposition is illustrated with
several analytic models. In particular we demonstrate that in some cases, by
tuning the control parameters a \textit{complete} transition might occur, where
a target state, vacant at $t=0$, is fully populated at time $\tau$.
