Time-dependent wave packets, scattering

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

Time-dependent wave packets (interpretation of scattering results)... 

Sun Z, Yang W, Zhang DH (2012) Higher-order split operator schemes for solving the Schrodinger equation in the time-dependent wave packet method applications to triatomic reactive scattering calculations. Phys Chem Chem Phys 14 1827... 

Jackson, B. (1995). Time-dependent wave packet approach to Quantum Reactive Scattering. nn. Rev. Phys. Chem. 46, 251. 

Jackson [15,16] proposes a quantum-mechanical theory for nonelastic scattering of particles as a result of interaction with surface phonons. Translational degrees of freedom of the gas particle are presented as a time-dependent wave packet. The wave functions describing the scattering of the particles satisfy a Schrodinger-type equation with a potential of interaction between flie gas particle and the solid surface, which depends both on time and temperature. In this model the Hamiltonian of the system is... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

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. 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

Althorpe, S.C. (2002) Time-dependent plane wave packet formulation of quantum scattering with application to H + D2 HD + D, J. Chem. Phys. 117, 4623-4627. 

D. E. Weeks and D. J. Tannor, A time-dependent formulation of the scattering matrix using Moller operators, Chem. Phys. Lett. 207 301 (1993) D. J. Tannor and D. E. Weeks, Wave packet correlation function formulation of scattering theory the quantum analog of classical 5-matrix theory, /. Chem. Phys. 98 3884 (1993). 

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)... 

---