Wave equation determinant

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... 

The dynamical elastic and inelastic scattering ofhigh-energy electrons by solids may be described by three fundamental equations [5]. The first equation determines the wave amplitude G ( r, r, E), or the Green function, at point r due to a point source of electrons at r in the averaged potential (V (r)) ... 

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

One must adopt the mental attitude and procedure of an optimist.. . . The optimist... is satisfied with approximate solutions of the wave equation. If they favor, say, tetrahedral and plane hexagonal models of methane and benzene, respectively, or a certain sequence among activation energies, or a paramagnetic molecule, he is content that these properties will be possessed by more accurate solutions. He appeals freely to experiment to determine constants, the direct calculation of which would be too difficult. 149... 

Belles prediction of the limits of detonability takes the following course. He deals with the hydrogen-oxygen case. Initially, the chemical kinetic conditions for branched-chain explosion in this system are defined in terms of the temperature, pressure, and mixture composition. The standard shock wave equations are used to express, for a given mixture, the temperature and pressure of the shocked gas before reaction is established (condition 1 ). The shock Mach number (M) is determined from the detonation velocity. These results are then combined with the explosion condition in terms of M and the mixture composition in order to specify the critical shock strengths for explosion. The mixtures are then examined to determine whether they can support the shock strength necessary for explosion. Some cannot, and these define the limit. 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

D. A. Mazziotti, Contracted Schrodinger equation determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57, 4219 (1998). 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

In this example the relative order of the orbitals is somewhat arbitrary. Only by solving an appropriate wave equation could the actual order be determined in a particular case, but the result would have to correspond with this diagram in regard to the types and the number of each type of wave functions obtained. 

The possible energy levels are determined by Schrodinger s wave equation [Reif, 1965], For translational motion of a particle, the wave equation takes the form... 

In alloy theory, this equation determines the variational wave function for an atomic cell I/, embedded in a statistical medium defined by the vector of coefficients ft for all other cells [157, 281]. 

What is of further interest here, as a model of the hydrogen atom and its angular momentum, is the vibration of a three-dimensional fluid sphere in a central field. As in 2D the wave equation separates into radial and angular parts, the latter of which determines the angular momentum and is identical with the angular part of Laplace s equation. 

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