Time of arrival in the semiclassical regime for Gaussian wave packets
Abstract
We study the time-of-arrival (TOA) distribution of a Gaussian wave packet in the semiclassical regime for a general one-dimensional quadratic Hamiltonian, namely a forced harmonic oscillator with time-dependent frequency. Expanding the Gaussian probability density as a series in derivatives of the Dirac delta and using the standard rules for the composition of distributions with the classical trajectory, we obtain compact closed-form expressions for the leading semiclassical mean arrival time and its standard deviation. The mean acquires a quantum shift of order σ2 set by the classical velocity and acceleration and by the time derivative of the Ermakov width, while the spread reduces to the simple form Δ𝒯x≃σ(tx)/vx. The latter implies a time--position uncertainty relation that depends only on the fundamental solutions of the classical equation of motion. We apply the framework to free fall and to two models of a time-dependent opening trap under gravity, and we validate every expression against exact numerical integration of the TOA distribution. These results extend earlier free-fall predictions to realistic settings in which the trapping potential cannot be neglected.
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BibTeX
@misc{beau2026time,
title = {{Time of arrival in the semiclassical regime for Gaussian wave packets}},
author = {Mathieu Beau and Maximilien Barbier and Vincent Bucourt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19027},
eprint = {2609.19027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19027},
}
RIS
TY - UNPB
TI - Time of arrival in the semiclassical regime for Gaussian wave packets
AU - Mathieu Beau
AU - Maximilien Barbier
AU - Vincent Bucourt
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19027
UR - https://arxiv.org/abs/2609.19027
AB - We study the time-of-arrival (TOA) distribution of a Gaussian wave packet in the semiclassical regime for a general one-dimensional quadratic Hamiltonian, namely a forced harmonic oscillator with time-dependent frequency. Expanding the Gaussian probability density as a series in derivatives of the Dirac delta and using the standard rules for the composition of distributions with the classical trajectory, we obtain compact closed-form expressions for the leading semiclassical mean arrival time and its standard deviation. The mean acquires a quantum shift of order σ^2 set by the classical velocity and acceleration and by the time derivative of the Ermakov width, while the spread reduces to the simple form Δ𝒯_x≃σ(t_x)/v_x. The latter implies a time--position uncertainty relation that depends only on the fundamental solutions of the classical equation of motion. We apply the framework to free fall and to two models of a time-dependent opening trap under gravity, and we validate every expression against exact numerical integration of the TOA distribution. These results extend earlier free-fall predictions to realistic settings in which the trapping potential cannot be neglected.
ER -