Wave equation meaning

In this discussion we define the x direction to be the direction of propagation of the light waves. This means that the yz plane contains the oscillating electrical and magnetic fields which carry the energy of the radiation. Only the electric field concerns us in scattering. Since the oscillation is periodic in both time t and location x, the electric field can be represented by the equation... 

The study of shock-wave equations of state of porous materials provides a means to expand knowledge of the equation of state of condensed materials to higher temperatures at a given volume than can be achieved along the principal Hugoniot. Materials may be prepared in porous form via pressing... 

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... 

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

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... 

A wave equation governing the unsteady motions is then derived by decomposition of all dependent variables as sums of the mean and fluctuation parts. Thus... 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. Completeness means that any arbitrary function can be expressed exactly as a linear combination of these functions. Mathematically, completeness can be expressed as... 

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

The wave equation for the electric field traveling through a medium whose properties are expressible by means of a static dielectric tensor e reads... 

The H7+ molecule-ion, which consists of two protons and one electron, represents an even simpler case of a covalent bond, in which only one electron is shared between the two nuclei. Even so, it represents a quantum mechanical three-body problem, which means that solutions of the wave equation must be obtained by iterative methods. The molecular orbitals derived from the combination of two Is atomic orbitals serve to describe the electronic configurations of the four species H2+, H2, He2+ and He2. 

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. 

The second front in the pulse, moving with velocity c2, sees the conversion of B to C. Thus 0 must fall from 0+. to zero. There may also be a final decay in the reactant concentration from a+ to zero. However, a+ is already small, so we will make the approximation that a = a+ k2 throughout the whole of this front. Assuming constant a means that again the system is reduced to one governing equation. Substituting a = k2 into eqn (11.52) gives a quadratic wave equation... 

It has been assumed, necessarily, that the reader has some prior familiarity with the basic notions of quantum theory. He is expected to know in a general way what the wave equation is, the significance of the Hamiltonian operator, the physical meaning of a wave function, and so forth, but no detailed knowledge of mathematical intricacies is presumed. Even the contents of a rather qualitative book such as Coulson s Valence should be sufficient, although, of course, further background knowledge will not be amiss. 

The electromagnetic field is quantized as a set of harmonic oscillators. Maxwell s equations, and the resulting wave equations, are described by partial differential equations that formally have an infinite number of degrees of freedom. Physically this means that the electromagnetic held is described by an infinite number of harmonic oscillators, where one sits at every point in space. The modes of the electromagnetic held are then completely described by this ensemble of harmonic oscillators. 

Is resonance a real phenomenon The answer is quite definitely no. We cannot say that the molecule has either one or the other structure or even that it oscillates between them. .. Putting it in mathematical terms, there is just one full, complete and proper solution of the Schrodinger wave equation which describes the motion of the electrons. Resonance is merely a way of dissecting this solution or, indeed, since the full solution is too complicated to work out in detail, resonance is one way - and then not the only way - of describing the approximate solution. It is a calculus , if by calculus we mean a method of calculation but it has no physical reality. It has grown up because chemists have become used to the idea of localized electron pair bonds that they are loath to abandon it, and prefer to speak of a superposition of definite structures, each of which contains familiar single or double bonds and can be easily visualizable. [30]... 

Closer scrutiny of the general wave equation provides an answer to the dilemma. As a second-order differential equation in the time variable, it has solutions in positive and negative time5. The negative time (or advanced) solutions are routinely rejected as physically impossible. This decision is based on prejudice rather than insight. Without evidence to accept only retarded (positive time) solutions as physically real, there is the possibility of response from a prospective receptor by means of advanced waves to establish one-to-one contact between emitter and absorber before transmission occurs. 

It can also be shown that Schrodinger s wave equation is none other than a form of the classical differential equation for a wave phenomenon in which the new feature is to be found in the application of it to electrons by means of the experimentally verified De Broglie relation which, in turn follows, as was seen, from a combination of the fundamental relations of Planck and Einstein in the form Av = me2 (p. 107). 

Equation (2) is thus the characteristic time-independent wave equation which we shall now apply to an electron by means of De Broglie s relation ... 

Essentially only the total 9 has a meaning, because this describes the stationary state. The splitting 9 = 91 9n is artificial and this forms only a means of finding an approximate solution of the wave equation, thus a necessary consequence of our mathematical incapacity. Resonance is therefore also not a real physical phenomenon but a human mnemonic. It is the same as when we say of a cnild, just like his mother and later just like his father . We do not really mean that the person of the child alternates, resonates , between that of mother and father or that these latter are present as parts but only that to a first approximation we try to describe his personality as a sum of two other entities. An analysis of an entity such as a human being into intelligence, character etc, can be very useful for an opinion but is not based on the presence of distinct parts which are themselves the subject of investigation. 

The stationary solutions are eigenfunctions of the time-independent wave equation (7), characterized by constant Vq. For an atom in an s-state (or any V o-state) the wave function is real, which means that the electron is at rest. This result may seem surprising, because classically a dynamic equilibrium is advanced to explain why the potential does not cause the particle to fall... 

This equation is Schrodinger s wave equation, where h is Planck s constant and H is the Hamiltonian of the system to be investigated. The Schrodinger equation is a deterministic wave equation. This means that once ip t = 0) is given, ip t) can be calculated uniquely. Prom a conceptual point of view the situation is now completely analogous with classical mechanics, where chaos occurs in the deterministic equations of motion. If there is any deterministic quantum chaos, it must be found in the wave function ip. 

What does this equation mean We have simply specified that A and k are constants. What values can these constants have Note that if they could assume any values, this equation would lead to an infinite number of possible energies—that is, a continuous distribution of energy levels. However this is not correct. For reasons we will discuss presently, we find that only certain energies are allowed. That is, this system is quantized. In fact, the ability of wave mechanics to account for the observed (but initially unexpected) quantization of energy in nature is one of the most important factors in convincing us that it may be a correct description of the properties of matter. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

A simple example of the wave equation, the one-dimensional particle in a box, shows how these conditions are used. We will give an outline of the method details are available elsewhere.The box is shown in Figure 2-3. The potential energy V x) inside the box, between x = 0 and x = a, is defined to be zero. Outside the box, the potential energy is infinite. This means that the particle is completely trapped in the box and would require an infinite amount of energy to leave the box. However, there are no forces acting on it within the box. 

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