Electromagnetic field momentum

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

To express the angular momentum M of the electromagnetic field corresponding to one photon in terms of the wave function for the photon, M is identified with the expectation value of the angular momentum in the state... 

Finally - and equally important - Jens contribution to the formal treatment of GOS based on the polarization propagator method and Bethe sum rules has been shown to provide a correct quantum description of the excitation spectra and momentum transfer in the study of the stopping cross section within the Bethe-Bloch theory. Of particular interest is the correct description of the mean excitation energy within the polarization propagator for atomic and molecular compounds. This motivated the study of the GOS in the RPA approximation and in the presence of a static electromagnetic field to ensure the validity of the sum rules. 

We now turn to the momentum and energy balance of the electromagnetic field. In analogy with conventional deductions, Eq. (1) is multiplied vectorially by and Eq. (2), by eoE. The sum of the resulting equations is then rearranged into the local momentum balance equation... 

Similarly, the magnitude of the linear momentum of the electromagnetic field can be obtained by using the proportionality g = e/h in either Eqs. (443) or (444), giving... 

In the system of atom plus electromagnetic field, there must be valid energy and angular momentum conservation laws. A free atom, being in... 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

Definitions for electromagnetic field in Eqs. (79) and (80) are similar to Hofer s [99]. There is a difference we start from an equation of motion for a 4D ether, while Hofer starts from a wave equation for 3D momentum density (his eq. 16). Our B is also similar to Marmanis [100], but his E is quite different. 

Potential energy in the electromagnetic field is a result of linear momentum transport. 

The second and third terms are the interaction terms that couple the atom, here modeled as a two-state system with Pauli matrices, to the electromagnetic field. We consider the momentum to be the operator p - - V and consider this operator as not only operating on the vector potential but on the wavefunction. Hence we find that... 

The total Lagrangian X = JS G + JS D + JS , then involves the interaction between fermions and the gauge field. The Dirac field will be generically considered to be the electron and the gauge theory will be considered to be the non-Abelian electromagnetic field. The theory then describes the interaction between electrons and photons. A gauge theory involves the conveyance of momentum form one particle (electron) to another by the virtual creation and destruction of a vector boson (photon) that couples to the two electrons. The process can be diagrammatically represented as... 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

At present the density effect has been quite thoroughly studied both theoretically and experimentally. There are different ways of obtaining the calculation formulas for Se. In particular, we can make allowance for the effect the surrounding medium has on the electromagnetic field of a particle by making the substitution c2—c2/e(a>) in the formula for the relativistic differential cross section of energy and momentum transfer... 

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 vector of the electromagnetic field defines a well specified direction in the laboratory frame relative to which all other vectors relevant in photodissociation can be measured. This includes the transition dipole moment, fi, the recoil velocity of the fragments, v, and the angular momentum vector of the products, j. Vector correlations in photodissociation contain a wealth of information about the symmetry of the excited electronic state as well as the dynamics of the fragmentation. Section 11.4 gives a short introduction. Finally, we elucidate in Section 11.5 the correlation between the rotational excitation of the products if the parent molecule breaks up into two diatomic fragments. 

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

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