Measurement of Time-of-Arrival in Quantum Mechanics

TL;DR

The paper investigates how time-of-arrival can be defined and measured for quantum particles, arguing that direct, arbitrarily precise measurements are thwarted by quantum back-action on clocks. Through several toy models (clock coupling, two-level detectors, energy boosters, and gradual triggering), it derives a dynamical limit Δt_A > 1/E_k and shows that higher precision leads to reflection or distortions in the arrival-time distribution. It contrasts operator-based time-of-arrival formulations with continuous measurement approaches, proving that an exact time-of-arrival operator cannot generally represent physical measurements and that continuous schemes provide a more faithful accounting of arrival times. The results highlight a fundamental distinction between mathematical time operators and physically realizable measurements, with implications for how arrival times should be defined in quantum theory.

Abstract

It is argued that the time-of-arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $Δt_A \sim 1/E_k$, where $E_k$ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here.