Stationary scattering states

The system of integral equations [Eq. (66)] is eventually discretized and solved with numerical linear algebra procedures. At each energy, the system (66) must be solved for each of the open channels. A complete set of linearly independent degenerate real (i.e., stationary) continuum solutions if"E is thus obtained. The stationary scattering states xjr E are not orthogonal it can be shown that their superposition is given by... 

Alternatively, it turns out that these probabilities can be extracted from stationary scattering states, i.e., eigenfunctions of the Hamiltonian with energy E0 = pl/(2m), and the asymptotic behavior... 

Stationary scattering states were used in the derivation of Eq. (5.98). Quantum mechanical traces are, however, independent of the representation in which they are carried out, so that there is no longer any explicit reference to these states, and any other orthonormal set of functions can be used in the trace. The quantum mechanical traces can then, e.g., be evaluated in a coordinate basis. 

The trace in Eq. (5.98) can be rewritten in various alternative and more convenient forms [2]. Time evolution can be introduced in the expression using that Eq. (4.132) is also valid for stationary scattering states [7]. Thus,... 

Crucial to all types of computation is the possibility of defining and computing consistently matrix elements. According to standard QM, for the stationary scattering state of real energy, = the norm is... 

It has to be emphasized that only an ultrashort laserpulse can create a localized wave packet as displayed in the figure. The longer the pulse, the more the prepared state will be delocalized in coordinate space and thus resemble a single stationary scattering state of the molecule. The time evolution of such a state is given by a phase factor and thus the whole idea of pump/probe spectroscopy is lost. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

The first convincing description of stationary quantum states was provided by the assumed wave nature of matter proposed by Louis de Broglie. The proposal had its roots in Einstein s explanation of the photoelectric effect and Compton s analysis of X-ray scattering. 

The novelty in the present approach is the absence of potential energy surfaces. It is sufficient to identify the stationary states involved in a given mechanism. An example of this is given in our paper [22]. Only state correlation diagrams are retained. The chemical interconversion is seen from a perspective of the stationary scattering approach. The asymptotic states also include the transition structures. Since we are interested in molecular mechanisms, these intermediates states are depicted to emphasize the chemical change. They actually define path histories. 

Third, the choice of the form of the atom-field interaction operator must be correct and appropriate for the TDMEPs under investigation. This is crucial not only for the accuracy of the computation of h (f) and (f) but also for ensuring that (f) and bE t) dE indeed represent the time-dependent occupation probabilities that correspond to the stationary states of interest labeled by (discrete states) or by (scattering states). 

One aspect of the mathematical treatment of the quantum mechanical theory is of particular interest. The wavefunction of the perturbed molecule (i.e. the molecule after the radiation is switched on ) involves a summation over all the stationary states of the unperturbed molecule (i.e. the molecule before the radiation is switched on ). The expression for intensity of the line arising from the transition k —> n involves a product of transition moments, MkrMrn, where r is any one of the stationary states and is often referred to as the third common level in the scattering act. 

The rather long lifetimes of many triplet states often allow an appreciable buildup of the triplet-state population under constant illumination. Already in 1954 Craig and Ross 33) measured reliable T—T spectra on a single-beam recording spectrophotometer. The observation of new absorption bands in the visible part of the spectrum is not difficult and there is satisfactory agreement between the peak positions reported from different laboratories. The determination of triplet extinction coefficients requires additionally a knowledge of the stationary triplet concentration, which is much harder to obtain. The corresponding literature values are widely scattered. 

A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

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