Time-dependent wave packets, scattering states

Analysis of time-dependent wave packets in terms of scattering states - why and how... 

The states correspond to wave packet controlled in the far past and in the far future, respectively. Let us see what this means. In the absence of external time-dependent fields, the scattering component of the time-dependent wave function i/r(f) can be expanded in terms of either of the two sets of scattering states for example, those with incoming boundary conditions... 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

The aim is to establish the relation between the observable cross-sections and the collision dynamics. We denote the scattering state in the interaction region at t = 0 by x) and write the Hamiltonian in the form Hc.m. + Hre, i.e., the Hamiltonians associated with the center-of-mass motion and the relative motion. The propagator can be written in the form U(t) = exp(—iHc.mt/h)exp(—iHre t/h), and x(t)) = [/(f) x) describes the time-dependent scattering state at any time, i.e. (il x(f)) is the associated wave packet. 

The most straightforward application of the TD approach to a scattering problem is to launch an initial wave packet in a specific internal state from the asymptotic region with a positive momentum toward the interaction region. There are different ways to extract the scattering information once the time-dependent wavefunction R(r) is obtained. One approach is to directly project out the specific product states from the final R(r) using the relation (1)... 

During this lecture I hope I impressed you with the fact that an electron in a fluid like Ar is bound, its wave function is extended, and, in a "frozen" liquid (without thermal motion of the atoms), the electron in the conduction band would not scatter, i.e., it would be in a stationary state. Scattering corresponds to a transition from one stationary state to another. It is not the result of the interaction with a single atom but instead with a change of potential brought about by the displacement of the atoms of the fluid. This can be described by phonons, if we consider their time dependence (usually in the GHz range), or "static" if we consider a much slower time dependence so that the electron wave packet (whose dimensions are of the order of the thermal wave length of the electron, A = moved far from the... 

Classical Dynamics of Nonequilibrium Processes in Fluids Integrating the Classical Equations of Motion Control of Microworld Chemical and Physical Processes Mixed Quantum-Classical Methods Multiphoton Excitation Non-adiabatic Derivative Couplings Photochemistry Rates of Chemical Reactions Reactive Scattering of Polyatomic Molecules Spectroscopy Computational Methods State to State Reactive Scattering Statistical Adiabatic Channel Models Time-dependent Multiconfigurational Hartree Method Trajectory Simulations of Molecular Collisions Classical Treatment Transition State Theory Unimolecular Reaction Dynamics Valence Bond Curve Crossing Models Vibrational Energy Level Calculations Vibronic Dynamics in Polyatomic Molecules Wave Packets. 

The understanding, interpretation and practical tools to approach the problem of resonances states in quantum chemistry and molecular physics are basically very well studied. Generally one has either (i) concentrated on the properties of the stationary time-independent scattering solution (ii) attempted to extract the Gamow wave by analytic continuation and/or (iii) considered the time-dependent problem via a suitably prepared reference function or wave-packet. In each case the analysis prompts different explanations, numerical techniques and understanding, see e.g. Ref. [15] for a review and more details. 

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