Scalar waves theory

The fundamental problem in scalar diffraction theory is to determine the value of a scalar wave xp at a point P given the value of xp and Vxp on a closed surface S surrounding P. A concise and lucid treatment of this theory is given by Wangsness (1963), who shows that... 

Green s functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product,... 

The divergence of the gradient of a scalar field occurs in several fundamental equations of electromagnetism, wave theory, and quantum mechanics. In Cartesian coordinates. 

Wavemaker theories play several important roles in coastal and ocean engineering. The most important role is the application to laboratory wavemakers for both wavemaker designs and wave experiments. A second role for wavemaker theories is to compute a scalar radiated wave potential to compute the wave-induced loads on large solid bodies appl5dng potential wave theory. The displacements and rotations of a semi-immersed six degrees-of-freedom large Lagrangian solid body are related to the displacements and rotations of wavemakers. The boundary between a planar wavemaker and an ideal fluid requires special care because the fluid motion... 

The combination of solutions of the scalar wave equation for the transverse fields of weakly guiding fibers of circular cross-section are given in Table 13-1, page 288. As we showed in the previous section, these combinations can be derived using perturbation theory. In this section we show how the combinations can be deduced using only symmetry arguments [2]. We start with the four vector solutions constructed from the solutions of the scalar wave equation with the common propagation constant P, and denote them by... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

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. 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

In 1904 Whittaker [28] showed that any EM field or wave consists of two scalar potential functions, initiating what is known as superpotential theory [77]. By Whittaker s [8] 1903 paper, each of the scalar potential functions is derived from internally structured scalar potentials. Hence all EM fields, potentials, and waves may be expressed in terms of sets of more primary interior or infolded longitudinal EM waves and their impressed dynamics.35 This is indeed a far more fundamental electrodynamics than is presently utilized, and one that provides for a vast set of new phenomenology presently unknown to conventional theorists. 

Further, in 1904 Whittaker [56] (see also Section V.C.2) showed that any electromagnetic field, wave, etc. can be replaced by two scalar potential functions, thus initiating that branch of electrodynamics called superpotential theory [58]. Whittaker s two scalar potentials were then extended by electrodynamicists such as Bromwich [59], Debye [60], Nisbet [61], and McCrea [62] and shown to be part of vector superpotentials [58], and hence connected with A. 

Another possible description is given by the 3D electron density pel(re, qnuc) which is a scalar function of re and contains qnuc as parameters. These two representations of the electron subsystem form the basis for the development of either conventional quantum chemistry methods or electron Density Functional Theory (DFT). The electron subsystem generates an effective potential, U(qnuc), acting on the classical nuclei, which can be expressed as an average of the full potential V over the electron wave function IP, and written as ... 

Equation (2) is recognized as the four equations of electromagnetism modified by a wave-like scalar field. Equation (1) represents the 10 Einstein equations of general relativity, equated to energy and momentum derived from the fifth dimension. In short, KK theory is a unified account of gravity, electromagnetism and a scalar field. Kaluza s case, 744 = — 2 = — 1, together with the identification 1... 

Group Theory for Non-Rigid Molecule (NRG), permits us to classify the torsional wave-functions according to the irreducible representations of the symmetry group of the molecule. As it is well known, the scalar product of (119) does not vanish when the direct products of the irreducible representations, under which 4, and / transform, contain at least one of the components of the dipole moments variation. Thus, when symmetry properties of these components are known, Selection Rules may be established. 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

For mixtures with 5c 1 the classical turbulence theory for scalars fields predicts that the scalar spectrum has a shape similar to for k < kx- For kx < k < kx, the so-called Batchelor spectrum [4] is found for which Ec k) oc k, and for k > kx, the scalar spectrum falls rapidly towards zero due to the effect of scalar dissipation by molecular diffusion, as illustrated in Fig 7.9. The Batchelor [4] wave number is of the order kx 27r(e/j/D ) /. Molecular... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

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