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arXiv:2609.19027

Time of arrival in the semiclassical regime for Gaussian wave packets

Mathieu Beau, Maximilien Barbier, Vincent Bucourt

quant-phSubmitted Jul 19, 2026• Announced Jul 21, 2026• 0 citations
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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 $σ^2$ set by the classical velocity and acceleration and by the time derivative of the Ermakov width, while the spread reduces to the simple form $Δ\mathcal{T}_x\simeqσ(t_x)/v_x$. 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.

Comments
7 pages (+ 6 pages in appendix), 4 figures

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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 σ2σ^2σ2 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)/vxΔ\mathcal{T}_x\simeqσ(t_x)/v_xΔTx​≃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.
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 mmm falling in a uniform gravitational field ggg, 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 $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and xxx is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.
Abstract The probability distribution of a time measurement Tx at position x can be inferred from the probability distribution of a position measurement Xt at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g, the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.
Abstract The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in (1/c21/c^{2}1/c2). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach 0.15%−1.5%0.15\% - 1.5\%0.15%−1.5% for an electron wave packet confined within the 1−10keV1-10 \mathrm{keV}1−10keV energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.