Quantum time of arrival distribution in a simple lattice model

Understand

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.