Physical wave solution

Expansions in terms of continuum wave functions As is well known, the time evolution of the decaying wave solution given by Eq. (47) may also be calculated by expanding the retarded Green function in terms of the complete set of continuum wave functions of the problem, the so-called physical wave solutions r) to obtain... 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves... 

One basic difficulty with the nonlinear equation arises from the following. Consider a physical situation where a source of particles is composed of many emitters, each emitting a particle at a time. If considered alone, each particle would be described by a localized wave /,- solution of the master equation. Now, what happens if, instead of emitting the particles one by one, the source emits many particles at the same time If the master equation were a linear equation, like the usual Schrodinger equation, the answer would be trivial. The general solution would be simply the sum of all particular solutions. 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

Because every matches an atomic orbital, there is no limit to the number of solutions of the Schrodinger equation for an atom. Each P describes the wave properties of a given electron in a particular orbital. The probability of finding an electron at a given point in space is proportional to A number of conditions are required for a physically realistic solution for P ... 

It can easily be seen that adding function P4(r, t) to any solution of the problem defined by (13.155) and (13.156) provides another solution of the same problem. Thus, it becomes clear that there exists a set of different mathematical solutions of (13.155) and (13.156). In other words, we cannot uniquely define the real physical process by assigning the sources of oscillations and the condition of field decay at infinity. At the same time, in the real world there is only one physical solution of this problem. It becomes clear now that we should impose an additional constraint to obtain a unique, physically meaningful, solution. This constraint should be able to reject convergent waves, acquiring and carrying energy firom infinity, which is physically impossible. 

However, we know, from high-energy physics experiments, that the interaction potential between a neutron (of typical energy ca lO MeV) and the nucleus is not weak but involves the strong nuclear force of ca 36 MeV. We can, however, avoid the nuclear force problem by searching for a new form for the interaction potential. The required potential must give S-wave solutions to the final neutron wavefunction when the Bom approximation is applied. 

The molecular dynamics constraint technique presented in the previous section is designed to simulate steady solutions of the Euler equations but there is no guarantee that all of the simulated solutions are physical. Some steady solutions are characterized by unboimded volume expansion, and others may not be the particular shock wave solutions desired. This section defines mechanical stability conditions that characterize shock waves and then shows that the molecular dynamics constraint technique naturally takes the system through states that satisfy these stability conditions. 

The equations of motion, Eqs. (36) and (37) in Section 5, contain many coefficients (a , bi, Ci, etc.) that are defined mathematically in that section. Similarly, in Section 6, when a particular solution is written, the coefficients (B, c, C, D ) are defined explicitly in terms of physical quantities. In Section 7, where the plane wave solution for the porosity diffusion wave is given, all the symbols and coefficients are defined immediately after they are written down, or have been previously defined. 

Nonlinear wave solutions of (14) were constructed by Shkadov (1973a) in the first time full theory by Demekhin et al. (1991) was developed. Popular weakly nonlinear equations (13) and (14) which follow from basic system (9) as > 0 don t contain physical... 

Sisoev, G. M., and Shkadov, V. Y. (1999). On two-parametric manifold of the waves solutions of equation for falling film of viscous liquid. Physics-Doklady 44(7) 454-459. 

The Ginzburg-Landau equation possesses a family of plane wave solutions. They are considered to be a special form of the plane waves whose existence was proved by Kopell and Howard (1973 a) for oscillatory reaction-diffusion systems in general. In view of the physical situation where the Ginzburg-Landau equation arises, the plane waves of Kopell and Howard are expected to reduce to this special form as the point of Hopf bifurcation (of the supercritical type) is approached from above. One of the important conclusions to be drawn below is that all the family of plane waves (including uniform oscillation as a special plane wave) can happen to be unstable, which is a property not shared by the A - co system with a diagonal diffusion matrix, see Sect. 2.4. 

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics... 

Problems in chemical physics which involve the collision of a particle with a surface do not have rotational synnnetry that leads to partial wave expansions. Instead they have two dimensional translational symmetry for motions parallel to the surface. This leads to expansion of solutions in terms of diffraction eigenfiinctions. 

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