One-particle propagator

Figure 4.4 SEM photographs of the compact tension fracture surface of 4 phr VTBN-modified VE resin showing a) transition from fast fracture to crack-initiation region and b) close-up of one particle. Propagation was from left to right. The specimen was stained with osmic acid. Reprinted with permission from J.S. Ullett, and R.P.Chartoff, Polymer Engineering and Science, 1995, 35, 13, 1086 1995, John Wiley and Sons Publishers...

This tight-binding two-band model simplifies the one-particle propagators in eq. (3.9) to... 

The propagator K can be interpreted as the Green s function for computing amplitudes for one-particle processes relative to the amplitude that the vacuum remain a vacuum under the influence of the external potential. To lowest order the amplitude that the vacuum remain a vacuum is given by... 

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

Note that Eq. (12) requires that the material system sustaining the quantum state went through the double slit the way it actually does is of no concern. Why is this so Answer QM works on all possibilities epitomized by quantum states. Remember the Feynman scheme where all possible paths must be in the calculation of the propagated quantum state [14]. Therefore, one particle path in real space does not make sense in Hilbert space. 

In concentrated systems, multiple scattering is important (I.e. a significant proportion of the energy scattered from one particle is incident upon neighbouring particles). If one considers multiple scattering in the derivation of the complex propagation constant [49], then Eq. 9.13 becomes ... 

In this latter case, one particle must be a neutral molecule and one a free radical. In both of these examples, a new radical is generated. This type of step is called propagation, since the newly formed radical can now react with another molecule and produce another radical, and so on, until two radicals do meet each other and terminate the sequence. The process just described is called a chain reaction, and there may be hundreds or thousands of propagation steps between an initiation and a termination. Two other types of propagation reactions do not involve a... 

The propagator [6-9] or Green s function method (GFM) [10-14] provides another approach to calculate the quasi-particle energy bands. The Dyson equation provides the exact E(N 1) energies in a formally one-particle picture, but the equation can only be solved approximately in real applications [57], With the irreducible self-energy part in the diagonal approximation being correct to second-order, the inverse Dyson equation can be written as [26]... 

Since the density in the disk of the galaxy is of order one particle per cm3 and the typical partial cross section for a light nucleus will be of order 100 mb, the scale for the characteristic time is 107 years. The full analysis requires self-consistent solution of the coupled equations for all species accounting for all loss terms, as described in John Wefel s lecture in this volume (Wefel, 2005). A nice overview of propagation models is given by Jones et al., 2001. 

The term zero-one designates that all latex particles contain either zero or one active free radical. The entry of a radical in a particle that already contains a free radical will instantaneously cause termination. Thus, the maximum value of the average number of radicals per particle, n, is 0.5. In a zero-one system, compartmentalization plays a crucial role in the kinetic events of emulsion polymerization processes. In fact, a radical in one particle will have no access to a radical in another particle without the intervention of a phase transfer event. Two radicals in proximity will terminate rapidly however, the rate of termination will be reduced in the process because of compartmentalization, as the radicals are isolated as separate particles. Consequently, the propagation rate is higher and the molecular weight of the polymer formed is larger than in the corresponding bulk systems. Which model is more appropriate depends primarily on the particle size. Small particles tend to satisfy the zero-one model, as termination is likely to be instantaneous. ... 

Creation of a propagating bound state is related to the fact that phonons can also propagate. However, the effective mass corresponding to the phonon motion is quite large so that the bands of one-particle states are narrow, and the respective Davydov splitting is small. 

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