Abstract
Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass m falling in a uniform gravitational field g, we use this approach to obtain an exact explicit expression for the probability density of the time-of-arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by δ = q²/2 when the factor q≡ ℏ/2mσ √2gx ≪ 1 (semi-classical regime), and by δ = √2/πq when q≫ 1 (quantum regime), where σ is the width of the initial Gaussian wavepacket and x is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.