The time distribution of quantum events

We develop a general theory of the time distribution of quantum events, applicable to a large class of problems such as arrival time, dwell time and tunneling time. A stopwatch ticks until an awaited event is detected, at which time the stopwatch stops. The awaited event is represented by a projection operator $\pi$, while the ideal stopwatch is modeled as a series of projective measurements at which the quantum state gets projected with either $\bar{\pi }=1-\pi$ (when the awaited event does not happen) or $\pi$ (when the awaited event eventually happens). In the approximation in which the time $\delta t$ between the subsequent measurements is sufficiently small (but not zero!), we find a fairly simple general formula for the time distribution $\mathcal{P}(t)$, representing the probability density that the awaited event will be detected at time $t$.