Collision wave packet

For 1, de Broglie s wavelength is small enough compared to the classical collision radius b so that a wave packet can be constructed which, approximately, follows the classical Coulomb trajectory [3]. The opposite limit, where the Sommerfeld parameter Zie hv<, denotes the case of weak Coulomb interaction where the Born approximation may be expected to be valid. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross sections. Using the method of SQ, it is possible to extract a Siegert state wavefunction from time-dependent quantum wave packets using the Fourier relation Eq. (28). The state obtained in this way for / = 0 is shown in Figure 3.7 this state is localized in the collinear F-H-D arrangement with three quanta of excitations in the asymmetric stretch... 

Kouri, D.J. and Mowrey, R.C. (1987). Close coupling-wave packet formalism for gas phase nonreactive atom-diatom collisions, J. Chem. Phys. 86, 2087-2094. 

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. 

Initial collision energy for this calculation was 0.01 a.u. At this energy the non-I radiative reaction probability is negligible. Here the effect of the laser pulse is to ["induce a near-complete (> 99%) population transfer from the wave packet of... 

For 4-atom systems such as formaldehyde the photodissociation dynamics have yet to be established. A fully quantal description of the photodissociation is still very expensive computationally. So far 6-dimensional wave packet studies have been applied only to one electronic surface, for example for scattering of a molecule or atoms on surfaces or collisions of molecules. Currently, for molecular systems with more than four atoms, a reduced dimensionality model must be used. This means that certain degrees of freedom are fixed throughout the dynamical simulation. 

The experimental situation is that times characteristic of an experiment are of the order 10 s, while the time characteristic of an atomic collision is, for example, the time it takes an electron projectile to traverse an atom. This is of the order 10 s. It is therefore physically reasonable to consider the limit t —> oo or e —> 0+. Experiments involving time resolution have been devised with resonant states whose lifetimes are greater than 10 s. Such experiments must be described by explicit wave packets rather than the formalism of the present section. An example is given in section 4.6. 

We use (6.22) to obtain from (6.21) the box-normalised wave-packet collision state at t = 0. 

The key quantity in the calculation of experimental observables is the collision amplitude (6.13). The box-normalised collision amplitude for the wave packet is given for t = 0 by using (6.26). 

Next we turn to the normalisation No of the collision amplitude. Introducing the unit operator for the channel space into the definition (6.15) in the wave-packet case we have... 

In WP analysis the time evolution of an initial wavefunction (or wave-packet) is obtained by the solution of the appropriate time-dependent Schrbdinger equation. The initial wavefunction is determined by the conditions of the collision. The Schrbdinger equation is then integrated, which given the complexity of the potentials usually has to be performed numerically. Information about the crossections can be obtained from this technique and again canonical rate coefficients obtained by the above averaging procedure. 

Sun, Y., Judson, R.S., and Kouri, D.J. (1989) Body frame close coupling wave packet approach to gas phase atom-rigid rotor inelastic collisions, J. Chern. Phys. 

Neuhauser, D., Baer, M., Judson, R.S. and Kotiri, D.J. (1990) A time-dependent wave packet approach to atom diatom reactive collision probabilities - theory and application to H -P H2 (J=0) system,, 7. Chem. Phys. 93, 312-322. 

Now can estimate the influence of the finite BEC temperature T on the nonexponential decay of impurity wave packets. Let us consider [i/ki> < T Boltzmann constant, T = 3.31 nf/ /(m2 k/j), [Isihara 1971], and rit = n I v is the total number density of the degenerate atomic sample. The number density of the above-condensate (thermal) fraction is na = nt (T/Tc)3/2, [Isihara 1971], The characteristic lime scale Ty/, on which an impurity atom experiences a collision with the atom belonging to the abovecondensate fraction, is given by... 

To begin with, Schrbdinger attempted to interpret corpus(il( .s, and particularly electrons, as wave packets. Although his formuhn are entirely correct, his interpretation cannot be maintained, since on the one hand, as we have already explained above, the wave packiits must in course of time become dissipated, and on the other hand the description of the interaction of two electrons as a collision of two wave packets in ordinary three-dimensional space lands us in grave difficulties. 

In gas-phase dynamics, the discussion is focused on the TD quantum wave packet treatment for tetraatomic systems. This is further divided into two different but closed related areas molecular photofragmentation or half-collision dynamics and bimolecular reactive collision dynamics. Specific methods and examples for treating the dynamics of direct photodissociation of tetraatomic molecules and of vibrational predissociation of weakly bound dimers are given based on different dynamical characters of these two processes. TD methods such as the direct projection method for direct photodissociation, TD golden rule method and the flux method for predissociation are presented. For bimolecular reactive scattering, the use of nondirect product basis and the computation of the initial state-selected total reaction probabilities by flux calculation are discussed. The descriptions of these methods are supported by concrete numerical examples and results of their applications. 

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