Scattering matrix definition

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

In terms of the definitions in Section 2, and noting in particular that the vibrational eigenfunctions av(ra) in equation (16) can be chosen to be purely real (so it is immaterial whether or not they are complex conjugated in bras), the reactive scattering version of the variational functional in equation (34) for a general scattering matrix element Sa /,aviE) becomes ... 

There are situations in which a definite wave function cannot be ascribed to a photon and hence cannot quantum-mechanically be described completely. One example is a photon that has previously been scattered by an electron. A wave function exists only for the combined electron-photon system whose expansion in terms of the free photon wave functions contains the electron wave functions. The simplest case is where the photon has a definite momentum, i.e. there exists a wave function, but the polarization state cannot be specified definitely, since the coefficients depend on parameters characterizing the other system. Such a photon state is referred to as a state of partial polarization. It can be described in terms of a density matrix... 

The single-site matrix t corresponds to a scattering operator t defined by t yi = iupL for any primitive basis orbital (pi in a given atomic cell. Here Xl = (Pl — fr GoV(pL = Jl Cl<,l From this definition it follows that tJi = v The t-matrix is... 

The 5-matrix is unitary and symmetric, while the T-matrix is symmetric. This particular definition of the T-matrix reduces for scattering by a central potential to the phase-shift factor in the scattering amplitude,... 

The basic concepts and definitions relating to sound propagation in a lossy material are reviewed. The material may be a viscoelastic polymer which converts the sound energy to heat by molecular relaxation, or the material may be a composite where sound is scattered by inhomogeneities (inclusions) in a host matrix material. 

The complex-valued energies singled out by the boundary conditions, Eq. (31), can also be described as poles of the 5-matrix. For very narrow resonances close to the real axis, the position e and the width T , defined in this way, agree with other definitions based only on real energies. The most popular alternative description of resonance is undoubtedly the step-like behavior of the scattering phaseshift 17(E), where the resonance is defined by the position of the maximum of the collisional time decay function (Le Roy and Liu, 1978)... 

Though Eq. (32) provides a rigorous quantum definition of the cumulative reaction probability, it is not helpful in a practical sense because a complete state-to-state reactive scattering calculation is required to obtain the S matrix. We seek a more direct (and thus presumably more efficient) route to N(E), but without approximation, to which approximations can be incorporated later as needed in specific applications. 

We do not intend to give here a profound description of scattering fheory that can be found in the textbooks, e.g.. Refs. [44 6] and review articles Refs. [47-50]. We shall recall fhe definition of fhe fransifion operafor and, in a second sfep, extend the model Hamiltonian of Section 2 fo derive exact expressions of fhe on-shell transition matrix [51]. 

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