T-matrix formalism

Usually, it is not just a calculation for an individual stracture, but an averaged result over the statistical ensemble and orientation that is required. Formally, such an averaging can be carried out by simple summation of the results calculated for various orientations and polarizations of the incident light, but this is inefficient. It is more convenient to perform orientation averaging for clusters using T-matrix formalism [65], even for arbitrary nonspherical particles [66]. If we formally invert Eq. (3.15), we get the T-matrix of an individual cluster-particle [62]... 

The null-field method is used to compute the T matrix of each individual particle and the T-matrix formalism is employed to analyze systems of particles. For homogeneous, composite and layered, axisymmetric particles, the null-field method with discrete sources is applied to improve the munerical stability of the conventional method. Evanescent wave scattering and scattering by a half-space with randomly distributed particles are also discussed. To extend the domain of applicability of the method, plane waves and Gaussian laser beams are considered as external excitations. 

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

We shall use the T-/ -isomorphism that allows us to consider the orbital triplet T2 as a state possessing the fictitious orbital angular momentum L = 1, keeping in mind that the matrix elements of the angular momentum operator L within T2 and P bases are of the opposite signs, L(T2) = —L(P) [2]. As it was shown in our recent paper [10] this approach provides both an efficient computational tool and a clear insight on the magnetic anisotropy of the system that appears due to the orbital contributions. Within T-P formalism the spin-orbital and Zeeman terms can be represented as ... 

Since it is the dynamics of the system that is of interest, it would be convenient to preaverage over the environment variables and obtain an equation of motion for ps(t), the system component of the density matrix. Formal work of this kind [161,. .. 162] yields the so-called generalized master equation. Deriving the generalized f master equation, and extracting the various approximations utilized, goes well "astray of the central focus of this book. For this reason we just sketch the models id direct the reader to suitable review articles [161, 162] that provide an appropriate pview. 

ITTFA starts calculating a PCA model of the original data matrix, D. There is a formal analogy between the PCA decomposition, i.e., D = TPT, and the CR decomposition, i.e., D = CST, of a data matrix. The scores matrix, T, and the loadings matrix, PT, span the same data space as the C and the ST matrices thus, their profiles can be described as abstract concentration profiles and abstract spectra, respectively. This means that any real concentration profile of C belongs to the score space and can be described as a linear combination of the abstract concentration profiles in the T matrix. 

The limit e —> 0-1- is kept in the formalism. We introduce a notation for it below. The important quantity in this limit is the T-matrix element... 

In the limit L — oo the index i has become a convenient discrete notation including the projectile continuum for channel i, defined by (6.7), or including the projectile—target continuum when the notation is defined by (6.8). We will retain this notation for formal convenience, but use the more-explicit forms (6.7,6.8) when it is necessary to specify electron momenta. The more-explicit form for the T-matrix element is... 

By using the density-matrix formalism for calculation of the detected signal, the evolution of the double-quantum part p2Q of the density matrix during the double-quantum space-encoding period t is needed as an intermediate result [Gun2, Gotl],... 

Two-state quantum beats (discussed in Section 6.5.3 in a simpler but less powerful notation, and in Sections 9.2.1, 9.2.2 and 9.3.2), provide the simplest illustration of the density matrix formalism. The key ideas are the creation by E of a coherent superposition state, which produces off-diagonal elements of p(t) (known as coherences ), and the simultaneous selection of the desired coherence and destruction of unwanted coherences by D. It is instructive to consider the essential features of E and D by which specific coherences axe selectively created and detected. 

The analysis of multiple-layer systems considers partially reflected and transmitted light beams at the interface of two or more phases (media). Figure 2.20 shows a simple three-phase vertical structure where the partial wave interference yields the reflectance Rp (and transmittance T). Mathematically, the expressions for R can be simplified by a matrix formalism [146]. In the three-layer system considered, consisting of ambient, intermediate film, and substrate, for nonmagnetic media a complex index of refraction can be defined for the ith layer ... 

In the case of the three-nucleon system, the formalism is different, but the mechnisms at work are very similar. A two-nucleon J -matrix is used as input for the Faddeev equations. The energy parameter of this T-matrix runs from — 8.5 MeV to — 00, causing a large dispersive effect (there is no Pauli effect). 

Within the density-matrix formalism (Vol. 1, Sect. 2.9) the coherent techniques measure the off-diagonal elements pab of the density matrix, called the coherences, while incoherent spectroscopy only yields information about the diagonal elements, representing the time-dependent population densities. The off-diagonal elements describe the atomic dipoles induced by the radiation field, which oscillate at the field frequency radiation sources with the field amplitude Ak(r, t). Under coherent excitation the dipoles oscillate with definite phase relations, and the phase-sensitive superposition of the radiation amplitudes Ak results in measurable interference phenomena (quantum beats, photon echoes, free induction decay, etc.). 

To explore this connection on a more formal basis, we can replace the expression based on transmission coefficients Tby an equivalent expression based on scattering amplitudes, or T matrix elements, between zero order states localized on the electrodes. This can be derived directly from Eqs. (29 or 31) by using the identity... 

The t/-matrix propagates the wave function forward in time. A formal solution is given by chopping the interval ( l, -tt) "to small intervals of width T. Then in the limit as T tends to zero,... 

Isobar excitation in the two-body NN system is incorporated in the NN r-matrix, whether it is empirical or calculated from a model. Direct three-body and higher order intermediate isobar excitation effects are formally included in the first portion of the second-order optical potential. For example, the elastic channel to isobar channel transition t-matrix can be obtained from the nucleon-isobar coupled channels model. Estimates indicate that this effect is significant for p + He scattering predictions but that it quickly diminishes in importance as target mass increases and is negligible for the heavier nuclei under consideration here [Wa77, Wa81]. 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

The term decoherence describes the process by which the off-diagonal elements of the reduced density matrix tend to zero when evolving with time. Our objective is to reach an understanding of the molecular mechanisms governing decoherence with an atomic resolution. In addition we wish to be in a position to treat systems consisting of tens to thousands of atoms since the brute force simulation of the time evolution of p t) by the Liouville-von Neuman equation (p (i) = ih [H, p ]), the equivalent of the TDSE in the density matrix formalism, is out of question for such molecular systems. 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

The polarization results from the dipole moments, t r,t), of the gas atoms induced by the electric field (7.7). It can be calculated in the framework of the atomic density matrix formalism. The main features of the phenomena, however, are also reproduced by the classical model of a damped oscillator. We are interested in the case where the frequency of light, m, is close to the atomic transition frequency, coq- The time evolution of the atomic dipole moment can be described, therefore, by the equation for an oscillator driven by the external force eEf (r, f) (Landau and Lifshitz 1978)... 

Consider the multilevel system depicted in Figure 10.1 the optical transition between any pair of levels (i and j) is mediated by an electric dipole. In the density matrix formalism, the expectation value of the dipole moment at any time t... 

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