Affiliations: Laboratoire de Physique Théorique et Hautes Energies, Bâtiment 211, Université de Paris-Sud, 91405 Orsay cedex, France. E-mail: [email protected] | Département de Mathématiques, Université de Nantes, 2 rue de la Houssinière, 44072 Nantes cedex 03, France. E-mail: [email protected]
Note: [] Unités de Recherche Associées au Centre National de la Recherche Scientifique.
Note: [] Unités de Recherche Associées au Centre National de la Recherche Scientifique.
Abstract: Precise semiclassical estimates for the spreading of quantum wave packets are derived, when the initial wave packet is in a coherent state. We find asymptotics for the quantum evolution of coherent states, at any order in the Planck constant ħ, with a control in large time of the remainder term depending explicitely on ħ and on the stability matrix. Our results extend Hagedorn's work on the propagation of Gaussian coherent states. We present here a proof simplifying Hagedorn's arguments, and extending it to general, possibly time-dependent Hamiltonians, not necessarily in the form of kinetic energy plus potential energy (p2 + V(x)). Our proof also works for more general coherent states and extends recent results by Paul and Uribe. As a first application of our semiclassical estimates we show that, if the initial quantum state is a coherent state located around an unstable fixed point α of the classical flow, the spreading of the quantum wave packet at time t grows like e2λt for times not larger than (γ/λ)log (1/ħ), where λ is the classical instability exponent associated to the fixed point α and γ is a numerical constant.
