Vector potential quantum electrodynamics

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

In quantum electrodynamics (QED) the potentials asume a more important role in the formulation, as they are related to a phase shift in the wavefunction. This is still an integral effect over the path of interest. This manifests itself in the phase shift of an electron around a closed path enclosing a magnetic field, even though there are no fields (approximately) on the path itself (static conditions). As can be shown the result of such an experiment is gauge-invariant, allowing the use of various choices of the vector potential (all giving the same result). 

E. Baum, Vector and Scalar Potentials away from Sources, and Gauge Invariance in Quantum Electrodynamics, Physics Note 3 (1991). 

It is apparent that for A3, = 0, the electric field component does not contain a product of potential terms. In general the vanishing of this term occurs if there are no longitudinal electric field components. Within the framework of most quantum electrodynamic, or quantum optical, calculations this is often the case. The B(3) field then is a Fourier sum over modes with operators a qaq. The B(3 ) field is then directed orthogonal to the plane defined by A1 and A2. The fourdimensional dual to this term is defined on a time-like surface that has the interpretation, under dyad-vector duality in three dimensions as, as an electric... 

A negative imaginary potential in the time-independent Schrodinger equation absorbs the particle flux, thus violating the law of conservation of flux, which is satisfied for real potentials [12,13]. Then, the quantum electrodynamical phenomenon of pair annihilation can be represented by particle loss due to an effective absorption potential H = —zVabs since the exact mechanism of positron loss is totally irrelevant to the study of the atomic processes in consideration [9,10,14-16]. The only important aspect of pair annihilation for the present purpose is the correct description of the loss rate. The absorption potential H is proportional to the delta function 5 (r) of the e+-e distance vector r (Section 4.2). 

The following picture emerges The radiation field, represented classically by a vector potential—that is, by a superposition of plane waves, as before, with transverse electrical and magnetic fields—is now, in quantum electrodynamics, a system with quantized energies. The general eigenfunction is then a product eigenfunction of the type... 

The potential energy surface used in solution, G (R), is related to an effective Hamiltonian containing a solute-solvent interaction term, Vint- In the implementation of the EH-CSD model, that will be examined in Section 6, use is made of the equilibrium solute-solvent potential. There are good reasons to do so however, when the attention is shifted to a dynamical problem, we have to be careful in the definition of Vint - This operator may be formally related to a response function TZ which depends on time. For simplicity s sake, we may replace here TZ with the polarization vector P, which actually is the most important component of TZ (another important contribution is related to Gdis) For the calculation of Gei (see eq.7), we resort to a static value, while for dynamic calculations we have to use a P(t) function quantum electrodynamics offers the theoretical framework for the calculation of P as well as of TZ. The strict quantum electrodynamical approach is not practical, hence one usually resorts to simple naive models. 

Having left the framework of field theory outlined in chapter 7 and thus having avoided any need for subsequent renormalization procedures, the mass and charge of the electron are now the physically observable quantities, and therefore do not bear a tilde on top. In contrast to quantum electrodynamics, the radiation field is no longer a dynamical degree of freedom in a many-electron theory which closely follows nonrelativistic quantum mechanics. Vector potentials may only be incorporated as external perturbations in the many-electron Hamiltonian of Eq. (8.62). From the QED Eqs. (7.13), (7.19), and (7.20), the Hamiltonian of a system of N electrons and M nuclei is thus described by the many-particle Hamiltonian of Eq. (8.66). In addition, we refer to a common absolute time frame, although this will not matter in the following as we consider only the stationary case. 

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