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 π, while the ideal stopwatch is modeled as a series of projective measurements at which the quantum state gets projected with either $\overline{\pi }=1-\pi$ (when the awaited event does not happen) or π (when the awaited event eventually happens). In the approximation in which the time δt 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.

我们开发了量子事件时间分布的一般理论，适用于诸如到达时间，驻留时间和隧穿时间之类的一大类问题。秒表会滴答作响，直到检测到等待事件为止，这时秒表将停止计时。等待的事件由一个投影算子π表示，而理想的秒表被建模为一系列的投影测量，在该测量下，量子态将被其中一个投影$\overline{\pi }=\mathrm{1个}-\pi$（当等待事件没有发生时）或π（当等待事件最终发生时）。在后续测量之间的时间δt足够小（但不为零！）的近似值中，我们找到了一个非常简单的时间分布通用公式$\mathcal{P}（Ť）$，表示在时间t处将检测到等待事件的概率密度。