Quantum time of arrival distribution in a simple lattice model

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.

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