Vector potential classical electromagnetic field

An arbitrary free classical electromagnetic field is described by the vector potential which obeys the wave equation [14,24,25]... 

An extension of this representation to more general form so as to include the vector potential arising from classical electromagnetic field will be made later in this book. 

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. 

As a next step we also need to specify the magnetic and retardation interactions experienced by an electron i and generated by all other electrons. In a first approximation retardation is neglected and we assume that electron i experiences the electromagnetic field immediately. For the scalar potential j,unret, and the vector potential A/ Unret created by electron j and felt by electron i the classical expression reads ... 

So far, we considered only the unretarded electromagnetic field. However, for the correct expression, we have to include the retardation of the vector potential due to the finite speed of light. We may obtain from Darwin s classical electromagnetic interaction energy expression (21) (correct up to 0(c 2)),... 

As is well known in classical electromagnetics, the fields described by the Maxwell equations can be derived from a vector potential and a scalar potential. However, there are various forms that are possible, all giving the same fields. This is referred to as gauge invariance. In making measurements at some point... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

Besides its appearance in the FFMF equation in plasma physics, as well as associated with time-harmonic fields in chiral media, the chiral Beltrami vector field reveals itself in theoretical models for classical transverse electromagnetic (TEM) waves. Specifically, the existence of a general class of TEM waves has been advanced in which the electric and magnetic field vectors are parallel [59]. Interestingly, it was found that for one representation of this wave type, the magnetic vector potential (A) satisfies a Beltrami equation ... 

Just by considering equation (4) one may speculate that the NACTs might be similar to the electromagnetic vector potential, S. It is known from classical mechanics that the momentum p of a charged particle in an electromagnetic field changes to p — p + eS - a substitution termed as the minimal principle [1]. Due to the correspondence principle the quantum mechanical minimal principle becomes V—>- V+ i(e/fi)S. However, the NACTs in equation (4), when considering each element separately, do not combine with V (because the... 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... 

Px, Py and pz are the components of the linear momentum of the particle, in a cartesian coordinate system which we need not define further at this stage. The electromagnetic field is described by the vector potential A and scalar potential 4>. Using the operator form for the linear momentum gives us the semi-classical expression,... 

The electric field generated by the classical vector potential of the electromagnetic field of a pulse laser to be used in Eq. (5.23) is of the standard form... 

The electromagnetic field pulse is described classically on the basis of Maxwell s equations. We are considering the dielectric and magnetic properties of CNTs [10] and two-dimensionality of the problem the Maxwell equations for the vector potential A in the gage e = will be ... 

Let us consider that the molecule is subjected, not to the infiuence of an electromagnetic field similar to that associated with light, but to a constant electric field in the z direction, of strength Ez, For this case, the vector potential A may be taken equal to zero the scalar potential is —zFa. The classical Hamiltonian function for a system of charged particles will then be... 

In its broadest sense, spectroscopy is concerned with interactions between light and matter. Since light consists of electromagnetic waves, this chapter begins with classical and quantum mechanical treatments of molecules subjected to static (time-independent) electric fields. Our discussion identifies the molecular properties that control interactions with electric fields the electric multipole moments and the electric polarizability. Time-dependent electromagnetic waves are then described classically using vector and scalar potentials for the associated electric and magnetic fields E and B, and the classical Hamiltonian is obtained for a molecule in the presence of these potentials. Quantum mechanical time-dependent perturbation theory is finally used to extract probabilities of transitions between molecular states. This powerful formalism not only covers the full array of multipole interactions that can cause spectroscopic transitions, but also reveals the hierarchies of multiphoton transitions that can occur. This chapter thus establishes a framework for multiphoton spectroscopies (e.g., Raman spectroscopy and coherent anti-Stokes Raman spectroscopy, which are discussed in Chapters 10 and 11) as well as for the one-photon spectroscopies that are described in most of this book. 

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