Electrodynamics, classical

Equation (3.5) implies that different modes are distinguished by their time and space dependence, characterized by two vectors A wave vector k that points to the direction of spatial modulation of (X, t) and a polarization unit-vector a k, that specifies the direction of A itself. The transversality of A expresses the fact that k and A are perpendicular to each other, that is. 

for every wave vector there are two possible polarization directions perpendicular to it. 

Given A, that is given Uk.ak for every (k, Ok), the electric and magnetic fields can be found as sums over modes using Eqs (3.4) and (3.3). For example, this leads to 

The second form of this result is written in anticipation of the analogous quantum result. 

The standard approach of quantum mechanics considers molecules as a collection of positively charged nuclei fixed in space and negatively charged electrons, the latter with spin. Most molecular properties then arise from the introduction of external electric and magnetic fields as well as nuclear spins, possibly coupled to displacements of the nuclei. These properties therefore fall 

The theoretical content of classical electrodynamics can be summarized by the Lorentz force law and Maxwell s equations. The Lorentz force law describes the force on a charge q moving with velocity v in the presence of an electric field E and a magnetic field B  

The second inhomogeneous equation (80) splits into three, one for each component of the current density j. Maxwell s equations accordingly constitute a system of six coupled first-order differential equations for the components of E and B. A more compact formulation is obtained with the introduction of potentials. We invoke Helmholtz theorem [37] stating that the fields, in view of the boundary conditions stated above, can be written as V = -I-V-, where the 

An even more compact formulation is obtained in covariant notation. We will follow the advice of Sakurai [38, p.6] and not introduce the Minkowski space metric g p, since the distinction between covariant and contravariant 4-vectors is not needed at the level of special relativity. We shall, however, employ the Einstein summation convention in which repeated Greek indices implies summation over the components a = 1,2,3,4 of a 4-vector. From the 4-gradient 

It was slowly realized [39] that whereas the electromagnetic potentials uniquely defines the electromagnetic fields, they are themselves unique only to within gauge transformations 

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Jackson, J. D., Classical Electrodynamics, John Wiley Sons, Inc., New York, 1967. 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

Abstract The statistical properties of the electromagnetic field find their origin in its quantum nature. While most experiments can be interpreted relying on classical electrodynamics, in the past thirty years, many experiments need a quantum description of the electromagnetic field. This gives rises to distinct statistical properties. 

There was thus the need for optical experiments showing the flaws of classical electrodynamics. An important difference between a wave and a particle is with respect to a beam splitter a wave can be split in two while a photon can not. An intensity correlation measurement between the two output ports of the beamsplitter is a good test as a wave would give a non zero correlation while a particle would show no correlation, the particle going either in one arm or the other. However, when one takes an attenuated source, such as the one used by Taylor, it contains single photon pulses but also a (small) fraction of two... 

From the classical electrodynamics, the Dirac Hamiltonian of a hydrogen molecule moving in a constant magnetic field B is [102]... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

MSN.27.1. Prigogine and B. Leaf, On the field-matter interaction in classical electrodynamics, 1., Physica 25, 1067-1079 (1959). 

DIV.4. I. Prigogine and F. Henin, Radiation damping and the equation of motion in classical electrodynamics, Physica 27, 982-984 (1961). 

D1V.9. 1. Prigogine and F. Henin, On the structure of elementary particles in classical electrodynamics, in Nucleon Structure, R. Hofstadter and Schiff eds., Stanford University Press, Palo Alto, CA, 1964, pp. 334-336. 

In the leading nonrelativistic approximation the denominator of the photon propagator cancels the exchanged momentum squared in the numerator, and we immediately obtain the Hamiltonian for the interaction of two magnetic moments, reproducing the above result of classical electrodynamics. 

The sign of this contribution may easily be understood from purely classical considerations, if one thinks about the magnetic dipoles in the context of the Ampere hypothesis about small loops of current. According to classical electrodynamics parallel currents attract each other and antiparallel ones repel. Hence, it is clear that the state with antiparallel magnetic moments (parallel spins) should have a higher energy than the state with antiparallel spins and parallel magnetic moments. 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

In the 1990s, however, there have been several attempts to extend the received view of classical electrodynamics, for example, the work of Barrett... 

Therefore, this is a statement of our fundamental hypothesis, specifically, that the topology of the vacuum defines the field equations through group and gauge field theory. Prior to the inference and empirical verification of the Aharonov-Bohm effect, there was no such concept in classical electrodynamics, the ether having been denied by Lorentz, Poincare, Einstein, and others. Our development of 0(3) electrodynamics in this chapter, therefore, has a well-defined basis in fundamental topology and empirical data. In the course of the development of... 

A simple example in classical electrodynamics of what is now known as gauge invariance was introduced by Heaviside [3,4], who reduced the original electrodynamical equations of Maxwell to their present form. Therefore, these equations are more properly known as the Maxwell-Heaviside equations and, in the terminology of contemporary gauge field theory, are identifiable as U(l) Yang-Mills equations [15]. The subj ect of this chapter is 0(3) Yang-Mills gauge theory applied to electrodynamics and electroweak theory. 

Barrett has argued convincingly that there are several effects in classical electrodynamics [3,4] where the potential must be physical, and Ref. 3 lists... 

In electromagnetic theory, we replace W 1 by GM the relativistic helicity of the field. Therefore, Eq. (770) forms a fundamental Lie algebra of classical electrodynamics within the Poincare group. From first principles of the Lie algebra of the Poincare group, the field B is nonzero. 

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