Field wave equations

If here e = const and n = const, field equations (8.15) and (8.17) go over into the wave equations  

For example, in free space this equation gives us the velocity of the light c  

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Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

The necessary boundary conditions required for E and //to satisfy Maxwell s equations give rise to tire well known wave equation for tire electromagnetic field ... 

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] ... 

Schrodinger (IV) has tentatively advanced a form of the wave equation in which magnetic fields are considered. 

Molecular mechanics (also known diS force-field calculations) is a method for the calculation of conformational geometries. It is used to calculate bond angles and distances, as well as total potential energies, for each conformation of a molecule. Steric enthalpy can be calculated as well. Molecular orbital calculations (p. 34) can also give such information, but molecular mechanics is generally easier, cheaper (requires less computer time), and/or more accurate. In MO calculations, positions of the nuclei of the atoms are assumed, and the wave equations take account only of... 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

In many problems for which no direct solution can be obtained, there is a wave equation which differs but slightly from one that can be solved analytically. As an example, consider die hydrogen atom, a problem that was resolved in Section 6.6. Suppose now that an electric field is applied to the atom. The energy levels of the atom are affected by the field, an example of the Stark effect. If die field (due to the potential difference between two electrodes, for example) is gradually reduced, the system approaches that of the unperturbed hydrogen atom. 

Yasui et al. [29] have used solution of wave equation based on finite element method for characterization of the acoustic field distribution. A unique feature of the work is that it also considers contribution of the vibrations occurring due to the reactor wall and have evaluated the effect of different types of the reactor walls or in other words the effect of material of construction of the sonochemical reactor. The work has also contributed to the understanding of the dependence of the attenuation coefficient due to the liquid medium on the contribution of the vibrations from the wall. It has been shown that as the attenuation coefficient increases, the influence of the acoustic emission from the vibrating wall becomes smaller and for very low values of the attenuation coefficient, the acoustic field in the reactor is very complex due to the strong acoustic emission from the wall. 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Invariance of the fields with respect to changes in potential is known as gauge invariance. It is used to simplify Maxwell s equations in regions where there is no free charge. In this case ip itself is a solution of the wave equation, so that it can be adjusted to cancel and eliminate the scalar potential. This means that in (13) V A= 0 and, as before... 

In surface sensing, the adsorbed biomolecules onto the inner wall of a microtube will change the wave vector of a supported mode from A 0 to k and the electric field from E0 to ). These unperturbed (L0) and perturbed (E ) electric fields still satisfied wave equation ... 

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

In cylindrical resonant cavities there exist Electric (E) and Magnetic (B) fields orthogonal to each other. Eigenvalue solutions of the wave equation subjected to proper boundary conditions are called the modes of resonance and are labeled as either transverse electric (TEfom) or transverse magnetic (TM/mn). The subscripts l,m,n define the patterns of the fields along the circumference and the axis of the cylinder. Formally, these l,m,n values are the number of full-period variations of A... 

Propagation of non-stationary light beam in a nonlinear medium with material dispersion is described by the scalar wave equation for the linearly-polarized y-component of electrical field E x,z,t) ... 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

Theoretical approach based on solution to the (2D-I-T) parabolic wave equation traditional for methods of nonlinear optics has been used. This approach is feasible if linear refractive index contrast in the waveguide is small and comparable with the nonlinear part of the refractive index. Then the back-reflected field can be ignored. 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

The basic idea of the EIM is to use a separation ansatz for the x- and the y-dependency of the field, x,y)- x)-constant refractive indices the wave equation thus reads as... 

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