Stationary wave equation

The solutions to the stationary-wave equation (6.63) which satisfy the extrapolation boundary condition have been discussed in Chap. 5 in considerable detail. In this application, we have from (6.56a)... 

To integrate equation (10.149) we must know how dn is related to v. Debye assumed that a crystal is a continuous medium that supports standing (stationary) waves with frequencies varying continuously from v = 0 to v = t/m. The situation is similar to that for a black-body radiator, for which it can be shown that... 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

The energy values correponding to the various stationary states are found from the wave equation to be those deduced originally by Bohr with the old quantum theory namely,... 

The system to be considered consists of two nuclei and one electron. For generality let the nuclear charges be ZAe and ZBe. From Born and Oppenheimer s results it is seen that the first step in the determination of the stationary states of the system is the evaluation of the electronic energy with the nuclei fixed an arbitrary distance apart. The wave equation is... 

In a line of reasoning that many of the younger quantum physicists regarded as reactionary, Schrodinger built his treatment of the electron on the well-understood mathematical techniques of wave equations as partial differential equations involving second derivatives. Schrodinger s equation for stationary electron states, as written in the Annalen der Physik in 1926, took the form... 

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. 

In this section, a problem of nonlinear waveguide excitation by stationary light beam has been investigated. In the analysis, an approach traditional for nonlinear optics and based on solution to nonlinear paraxial wave equation has been used. The range of light beam powers that induce nonlinear variation of refractive index comparable with linear contrast of the step-index waveguide has been considered. 

Many phenomena such as dislocations, electronic structures of polyacetylenes and other solids, Josephson junctions, spin dynamics and charge density waves in low-dimensional solids, fast ion conduction and phase transitions are being explained by invoking the concept of solitons. Solitons are exact analytical solutions of non-linear wave equations corresponding to bell-shaped or step-like changes in the variable (Ogurtani, 1983). They can move through a material with constant amplitude and velocity or remain stationary when two of them collide they are unmodified. The soliton concept has been employed in solid state chemistry to explain diverse phenomena. 

Therefore, questioning the physical significance of potential is not relevant here. The new formulation of Maxwell s equations [20-23], where potentials are directly coupled to fields clearly indicates that potentials, play a key role in particle behavior. To make a long story short, the difference in nature between potentials and fields stems from the fact that potentials relate to a state of equilibrium of stationary waves in the medium usually nonaccessible to an observer (except when potentials are used in a measurement process of the interferometric kind, at a given instant in time). Conversely, fields illustrate a nonequilibrium state of the medium as an observable progressive electromagnetic wave, since this wave induces the motion of material particles. 

In the perturbative "transfer Hamiltonian approach developed by Bardeen 58), the tip and sample are treated as two non-interacting subsystems. Instead of trying to solve the problem of the combined system, each separate component is described by its wave function, i tip and i/zj, respectively. The tunneling current is then calculated by considering the overlap of these in the tunnel junction. This approach has the advantage that the solutions can be found, for many practical systems, at least approximately, by solution of the stationary Schrodinger equation. 

Coupled wave equations are not easy to handle. It will help greatly if we can somehow find some sort of approximate relationship between a and / , which will allow us to substitute for a, say, and reduce the system to a single equation. This relationship need only apply during the first wavefront. What should we take The behaviour of this model in a CSTR suggests one such form. By analogy with the stationary-state relationship, eqn (6.50), we try the form... 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

The galactic redshift could obviously be attributed to the damping of the electromagnetic waves emitted from various galaxies in random motion within a stationary universe. Now, comparison between Hubble relativistic linear law and the logarithmic law that comes out from Maxwell electromagnetic wave equation shows that, in any case, the logarithmic law fits experimental data very well and thus better than linear law. 

Note that wavefunction

time independent. The wave functions obtained from a time-independent equation are called standing (or stationary) waves. To obtain such an equation, we assume

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The first objective of quantum theory is indeed aimed at the electron. The wave-mechanical version of quantum theory, which is the most amenable for chemical applications, starts with solution of Schrodinger s wave equation for an electron in orbit about a stationary proton. There is no rigorous derivation of Schrodinger s equation from first principles, but it can be obtained by combining the quantum conditions of Planck and de Broglie with the general equation6 for a plane wave, in one dimension ... 

The most serious limitation of wave mechanics is the complexity of any wave equation that describes interacting particles and prevents application to atoms other than hydrogen. To separate the equation and solve for any situation of interest it is necessary that it be reduced to a one dimensional one-particle problem. In the case of the hydrogen atom this is done by assuming the proton to be of infinite mass and therefore stationary. The only molecular system that can be treated in the same way is H, if the two protons are clamped to remain at a fixed distance apart. 

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... 

In the stationary case the operators Qn(t) coincide with the (coordinate) quadrature components of the field mode operators. Putting (191) into the wave equation... 

We have thus found the formalism, according to which any mechanical problem can be treated. What we have to do is to find the one-valued and finite solutions of the wave equation for the problem. If in particular we wish to find the stationary solutions, i.e. those in which the wave function consists of an amplitude function independent of the time and a factor periodic in the time (standing vibrations), we make the assumption that ijj involves the time only in the form of the factor Schrodinger s equation, we find... 

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