Electromagnetic vector potential

This has been discussed above. Four real-valued fiinctions. 4 = ( ei, A) define the potential matrix 

If we perform a Poincare transformation of the electromagnetic potentials according to (84), then the corresponding new field strengths E and B will again satisfy Maxwell s equations. 

The Dirac operator with an electromagnetic field reads 

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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 identical transformation, equation (6), of the electromagnetic vector potentials was found before to leave the fields unaffected or gauge invariant. The fields Atl are not gauge invariant, but the fields described by the tensor, equation (33)... 

The laws of electromagnetism are based on the theory of gauge fields. The electromagnetic vector potential defines components of a gauge connection 1-form. This gauge connection defines a field strength 2-form ... 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

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... 

The effect of a magnetic field B = V x A, where A is an electromagnetic vector potential, can be included in the Hamiltonian in (1) by modifying transfer integrals as... 

It is assumed here that the electromagnetic vector potential, 4 ( is a (polar)... 

In order to obtain Poincare-covariance of the Dirac equation, Apy must behave as an electromagnetic vector potential, as far as proper Poincare transformations are concerned. The right behavior under a parity transformation would be... 

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... 

When a current source is located within a clad fiber of arbitrary profile, the determination of the radiation field is extremely complicated. However, if the fiber is weakly guiding the determination is greatly simplified [2]. It is intuitive that when the variation in the refractive-index profile is small, the source radiates as if it were located in a virtually uniform, infinite medium of refractive index equal to the cladding index n j. The problem is then analogous to the radiation from an antenna in free space. Consequently, we can borrow from standard antenna theory, and couch the solution to radiation from the weakly guiding fiber in terms of the electromagnetic vector potential A [3-5]. 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

The term a, therefore plays the role of a vector potential in electromagnetic theory, with a particularly close connection with the Aharonov-Bohra effect, associated with adiabatic motion of a charged quantal system around a magnetic... 

We have met the electrostatic potential 4> in earlier chapters. The vector potential A is a fundamental construct in electromagnetism (HinchUffe and Munn, 1985). 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

This may well appear not to produce anything new until the electron is examined in an external electromagnetic field, represented by a scalar potential V, and a vector potential A. The appropriate operators then become... 

It is important to note once again that <, and A in equation (75) are the scalar and vector potential resulting from the external electromagnetic radiation field. Also note that here, , = V(, - and S, is the spin of the ith particle. We can identify each term in equation (75) as corresponding to a certain type of physical interaction of moving charged particles. The list of physical interpretations of terms follows in the same order as the terms in equation (75). 

The interaction Hamiltonian contains the operator A, corresponding to the vector potential A of the electromagnetic field.2 Excluding magnetic scattering, the interaction Hamiltonian is given by... 

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)),... 

If there is no electromagnetic field present, the quantized vector potential fluctuates according to... 

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