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Journal of Mathematical Physics

Dynamics of uncertainties for bound one-dimensional semiclassical wave packets

Predrag Punoševac, Sam L. Robinson

Published Sep 1, 2016• 5 citations
Publisher

Abstract

We study the time evolution of the uncertainties Δx and Δp in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as P1(t)+tP2(t), where P1 and P2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product ΔxΔp achieves its minimum of ħ/2 at arbitrarily large times.

DOI
10.1063/1.4962926

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We study the time evolution of the uncertainties Δx and Δp in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as P1(t)+tP2(t), where P1 and P2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product ΔxΔp achieves its minimum of ħ/2 at arbitrarily large times.
We complete our study on the uncertainties in position and momentum associated with the semiclassical Hagedorn wave packets by first filling in a technical gap in the dynamics of bound states for isochronous potentials. We then consider scattered states and show that, if the packet is reflected from the potential or transmitted through a symmetric potential, then a minimal uncertainty “initial” state cannot in general lead to a “final” state with minimal uncertainty, and we give an explicit relationship for the difference in terms of a characteristic time associated with classical trajectories. We also characterize the behavior of the uncertainty product in the case where the underlying classical dynamics lead to capture by the potential.
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.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.