Vector potential electromagnetic wave

Hence, the method of Mead and Truhlar [6] yields a single-valued nuclear wave function by adding a vector potential A to the kinetic energy operator. Different values of odd (or even) I yield physically equivalent results, since they yield (< )) that are identical to within an integer number of factors of exp(/< )). By analogy with electromagnetic vector potentials, one can say that different odd (or even) I are related by a gauge transformation [6, 7]. 

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. 

XVIII. Beltrami Vector Potential Associated with TEM (Transverse Electromagnetic) Standing Waves with E//B... 

XVIII. BELTRAMI VECTOR POTENTIAL ASSOCIATED WITH TEM (TRANSVERSE ELECTROMAGNETIC) STANDING WAVES WITH E//B... 

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 ... 

It is essential that plasmons are longitudinal electric waves, that is, that the electric field strength vector is parallel to the wave vector q. For a plane wave, the vector potential of electromagnetic field is... 

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]... 

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

The potentials which will be considered here are stepwise constant. In each region of these potentials the time-dependent wavefunction is a linear combination of solutions of the wave equation for a particle interacting with an electromagnetic field of vector potential with A(t) [10] ... 

Further difficulties arise when the electromagnetic field is described in terms of the quantum theory, This is accomplished by treating the potentials of the electromagnetic field as operators, subject to the quantum commutation rules. It is convenient first to expand the vector potential A (r, t) into harmonic waves of all frequencies, oj 2rrv. The attempt to express the energy H of the field in terms of the number of photons nw of angular frequency a> (see, for example, [117]) then results in the expression... 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation... 

In terms of multipoles only the electric dipole is coupled to the electromagnetic field and so this approximation is termed the electric dipole approximation. It may appear strange that the electromagnetic field, which is transversal, in this approximation is given solely by the scalar potential. It must, however, be remembered that the scalar and vector potentials of (197) do not describe the electromagnetic wave as such. Rather, it models the interaction of the electromagnetic wave with the molecule [65]. 

In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as follows ... 

The general solution of these equations of the electromagnetic wave yields a rather complicated mathematical expression. So, by using the second equation of Eq. (1.60) and remembering the formula, V(Vx ) = 0 given in Table 1.3, we introduce the vector potential, A(x,y,z,t), which satisfies the following equations ... 

Putting it in a different way, if the wave function undergoes a phase change, then the particle moves in a vector potential of an electromagnetic field. 

The scalar potential in the rapidly varying electromagnetic field is supposed to be zero. The vector potential A satisfies the wave equation in free space ... 

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. 

The parentheses surrounding the quantity V A in Eq. 1.76 indicate that V operates only on the vector potential A immediately following it. The choice of Coulomb gauge (V A = 0) for electromagnetic waves propagating in free space (< = 0) reduces the Hamiltonian to... 

As a concrete example, the vector potential for a linearly polarized monochromatic electromagnetic plane wave with wave vector k may be written... 

We may associate the perturbations lV(tt), lV(t2), and W Ctj) with vector potentials for electromagnetic waves with frequencies coj, 0)2, and CO3 respectively ... 

We now investigate the effect of a light wave entering an atom, which is in the type of short-lived state we investigated in the previous section. So we introduce, except for the interaction potential of the two electrons A(/(xX), simultaneously the influence of a light wave with perio c, electromagnetic vector potential... 

Solving Maxwell s equations [see Exercise 7.1] for the vector potential of a plane or linear polarized electromagnetic wave oscillating with angular frequency w gives... 

---