  Imagine an experiment where a quantum particle inside a box is released at
some time in some initial state. A detector is placed at a fixed location
inside the box and its clicking signifies arrival of the particle at the
detector. What is the \emph{time of arrival} (TOA) of the particle at the
detector ? Within the paradigm of the measurement postulate of quantum
mechanics, one can use the idea of projective measurements to define the TOA.
We consider the setup where a detector keeps making instantaneous measurements
at regular finite time intervals {\emph{till}} it detects the particle at some
time $t$, which is defined as the TOA. This is a stochastic variable and, for a
simple lattice model of a free particle in a one-dimensional box, we find
interesting features such as power-law tails in its distribution and in the
probability of survival (non-detection). We propose a perturbative
calculational approach which yields results that compare very well with exact
numerics.
