Free Particles Propagator

There are many ways for evaluation and equivalent forms of free Green function. Here, we will start with evaluating its Fourier transformation 

From the need the two expressions be equivalent we find the momentum-energy retarded free Green function as  

With this the space-time free Green function becomes  

Remarkably, this expression is in accordance with the spectral expansion of the general Green function, however here adapted to the free motion case. This is another confirmation for that the momentum-energy free Green function takes indeed the above ( , ) expression. However, going further with the free Green function evaluation in terms of plane wave, one yields the following transformations  

Quantum Nanochemistty-Volume I Quantum Theory and Observability 

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Suppose that we have no potentials in our system, but just free electrons. Then the Hamiltonian is simply Ho = —V2, and the Schrodinger equation can be solved exactly in terms of the free-particle propagator, or Green s function, Go(r, t r, t ), which satisfies the equation ... 

To obtain the SDVR of the free particle propagator, one takes the infinite N limit of Eq. (3.23) keeping Ax fixed, giving [97]... 

With the variable transformation p = firry/Ax, the free particle propagator in Eq. (3.24) takes on the more familiar form... 

There are several avenues for future study suggested by the present results. In this Section we consider the following first, possible improvements to the iterative calculation of reaction probabilities by an ABC Green s function on a grid and second, a generalization of Makri s effective free particle propagator. 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

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... 

A new kernel corresponding to the free two-particle propagator in (1.8) may be defined via this amputated two-particle Green function... 

A not-trivial ratchet effect can be observed when the injected charge density is voltage-independent, EL/R = Ep eV/2. Symmetry considerations require an asymmetric U (x) for a non-vanishing ratchet current in this case. Also an electron interaction must be present. Indeed, for free particles the reflection coefficient R(E) is independent of the electron propagation direction [14] and hence I(V) = —/(—V). 

A more accurate, but also more complex form, will be discussed later in the section on Path Integral Monte Carlo. Note that we have S3mimetrized the primitive form in order to reduce the systematic error of the factorization [19]. The explicit form of the kinetic propagator is the Green s function of the Bloch equation of a system of free particles [19,21], i.e. a diffusion equation in configurational space... 

It follows from the above discussion that considering excitons and intramolecular phonons as independent particles is approximate. It is clear, in particular, that a sufficiently strong exciton-phonon interaction can create propagating bound states, when electronic and vibronic excitations are centered on the same molecule. These states correspond to the previously discussed weak resonant interaction case. But the existence of such states in the vibronic spectrum does not exclude the existence of free excitons and intramolecular phonons states. Both types of states can usually coexist in the vibronic spectrum, in analogy to the case of two interacting particles, where continuum states, corresponding to free particles, coexist with bound states. 

Bulk semiconductors and powders have been used as initiators for radical polymerization reactions [140-144], Recently the study has been extended to semiconductor nanoclusters [145-147]. It was found that polymerization of methyl methacrylate occurs readily using ZnO nanoclusters. Under the same experimental conditions, no polymerization occurred with bulk ZnO particles as photoinitiators [145], In a survey study, several semiconductor nanoclusters such as CdS and Ti02, in addition to ZnO, were found to be effective photoinitiators for a wide variety of polymers [146], In all cases nanoclusters are more effective than bulk semiconductor particles. A comparison of the quantum yields for polymerization of methyl methacrylate for different nanoclusters revealed that Ti02 < ZnO < CdS [146]. This trend is parallel with the reduction potential of the conduction band electron. The mechanism of polymerization is believed to be via anionic initiation, followed by a free-radical propagation step. 

This form of the effective potential incorporates some degree of quantum effects as well as the anharmonicity of the potential. In the case of the free-particle reference system, the centroid-constrained propagator is... 

Again, any quadratic reference propagator a(r) could be used here instead of the free-particle one. 

This circular motion, being along a simple connected line (the circle) can be projected on the free-particle (line) motion (on real space 91) over which specific constraints are imposed to regain the circular path and motion towards the present path integral model of the Bohr s atom. Therefore, one re-considers the free-motion propagator of Eq. (4.102)... 

We consider the generalization of Makri s effective free particle kernel. We will not pursue these ideas in this dissertation, but rather, offer them for contemplation. For simplicity we consider a one dimensional system with meiss m. The success of Makri s propagator is based on the filtering of high frequency momenta which are unimportant and difficult to integrate. To this end, we denote a general momentum filter by w p) with the property that it goes to zero as p oo. This... 

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