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Dynamics of uncertainties for bound one-dimensional semiclassical wave packets
G. Gallavotti, The Elements of Mechanics (Springer, 1983).
C. Chicone, “Ordinary differential equations with applications,” in Texts in Applied Mathematics (Springer, 2006).
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V. Maslov, Théorie des Perturbations et Méthodes Asymptotiques (Dunod, 1972).
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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 , where P
1 and P
2 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.
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