Mathematical and physical solutions

It IS wonn notmg mat not an solutions ot me j cnroamger equation are pnys-ically acceptable. 

For example, for bound states, all solutions other than those of class Q (see p. 895) must be rejected. In addition, these solutions if/, which do not exhibit the proper symmetry, even if i/rp does, have also to be rejected. They are called mathematical (non-physical) solutions to the Schrodinger equation. Sometimes such mathematical solutions correspond to a lower energy than any physically acceptable energy (known as underground states). In particular, such illegal, non-acceptable functions are asymmetric with respect to the label exchange for electrons (e.g., symmetric for some pairs and antisymmetric for others). Also, a fully symmetric function would also be such a non-physical (purely mathematical) solution. 

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Mathematical and Physical Solutions The Time-Dependent SdirS i er Eiquation (A) p. 84... 

Klein R (1991) In Chen S-H, Huang JS, Tartaglia P (eds) Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution, NATO ASI Series C Mathematical and Physical Sciences, vol 369. Kluver, Dordrecht, p 39... 

Harris, R. K. (1983). Solution-state NMR studies of Group IV elements (other than carbon). NATO ASI Series, Series C Mathematical and Physical Sciences, 103 (Multinucl. Approach NMR Spectrosc.), 343-359. 

At the same time, when we impose the dimensions of the plant, we can focus on obtaining one or more of the optimal solutions (maximum degree of species transformation, minimum chemical consumption, maximum degree of species transformation with minimum chemical consumption etc.) For this purpose, it is recommended to use both mathematical and physical simulations. 

In this chapter, I shall describe the basic methodology of the laser-induced proton pulse. Starting with the inidal event of a synchronous proton dissociation, going through the reaction of a proton with other solutes in a true solution, and ending with the complex multiphasic system of protons, macromolecules, and interfaces associated with the real life of biochemical reaction. In each level of complexity, I shall point out the pertinent information available for interpretation and the mode of mathematical and physical analysis. In some cases, I shall also reflect the conclusions on current hypotheses of biochemical proton transfer. 

With mathematical and physical considerations, Gross-man reaches the conclusion that the real physical space-time is a Riemannian space and that in vacuum it must obey the equation R/uy = 0. This is the simplest covariant solution for a space-time with gravitation. (Publication in June 1913 see references. 

Thus the key point is to obtain the connections of the asymptotic solutions of Eq. (60) at - oo. The underlying mathematics to carry this out analytically is the Stokes phenomenon of asymptotic solutions of differential equations and is explained briefly in Sec. V. This is very important mathematics for the general semiclassical theory and various physical phenomena. It is interesting that the apparent small differences in q(%) of Eqs. (61) between the LZ and NT cases make a big difference mathematically and physically. The mathematical differences are not detailed here, but the physical differences are obvious, as pointed out in the Introduction. 

From the end of sixties, the principal studies in the theory of chemical technology were based on mathematical and physical modelling of the total set of superimposed processes. Vigorous development of computers and numerical methods of analysis promoted a fast development of investigations and continuous complication of models, which enable us to percive new details of the processes. At present, the fundamental physical principles and phenomena are understood in principle, mathematical models of processes have been developed in the main types of reactors, the fields of their application have been determined and computational methods for solution and analysis have been defined. Since the mid 1970s, the main attention of the researchers has been attracted to the study of the peculiarities of processes. 

ABSTRACT With the increase of mine exploitation depth and appliance widely of large-scale full-mechanized equipment, coal block gas emission has been one of the most gas effusion source. Base on unsteady diffusion theory and mass transmission fundamental, the mathematical and physical model of gas diffusion through coal particles with third type boundary condition was founded and its analytical solution was obtained by separate variableness method. The characteristics of gas through coal particles was analyzed according as mass transmission theory of porous material. The results show that the Biot s criterion of mass transmission can reflect the resistance characteristic of gas diffusion and the Fourier s criterion of mass transmission can represent the dynamic feature of diffusion field varying with time. 

Compton RG, Unwin PR. The dissolution of calcite in aqueous solution at pH < 4 Kinetics and mechanism. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 1990 330 1-45. 

The solutions of Eqs.(2.1.34), (2.1.35) with appropriate boundary conditions such as Eqs. (2.4.3), (2.4.4) will be called the steady states of the system. Various properties of the steady states, such as the invariant manifolds and a priori bounds, the existence and uniqueness of solutions, the asymptotic behavior, and the stability will be treated in the sections that follow. There is a strong similarity in the properties of the uniform open systems investigated in Sections 1.8,1.9 and the distributed systems to be studied now. In both types of systems the interplay between reaction and transport rates (or flow rates) creates the possibility of multiple steady states for certain types of reaction kinetics. Furthermore, the conditions for uniqueness and stability of the steady state have a common mathematical and physical basis. 

Analogously, calculus provides access to tractable mathematics and analytical solutions previously inaccessible to the human brain. Augmentation can then be considered as a qualitative shift in abilities. With results attainable only with calculus, the foundation can be solidly laid for theories that capture and explain physical phenomena. The development of the gravitational theory, the electromagnetic theory, or the quantum mechanical theory, is now possible, resulting, in turn, in tectonic changes in the human mindset. 

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

Molecular modeling has evolved as a synthesis of techniques from a number of disciplines—organic chemistry, medicinal chemistry, physical chemistry, chemical physics, computer science, mathematics, and statistics. With the development of quantum mechanics (1,2) ia the early 1900s, the laws of physics necessary to relate molecular electronic stmcture to observable properties were defined. In a confluence of related developments, engineering and the national defense both played roles ia the development of computing machinery itself ia the United States (3). This evolution had a direct impact on computing ia chemistry, as the newly developed devices could be appHed to problems ia chemistry, permitting solutions to problems previously considered intractable. 

While orbitals may be useful for qualitative understanding of some molecules, it is important to remember that they are merely mathematical functions that represent solutions to the Hartree-Fock equations for a given molecule. Other orbitals exist which will produce the same energy and properties and which may look quite different. There is ultimately no physical reality which can be associated with these images. In short, individual orbitals are mathematical not physical constructs. 

A long path has been followed to overcome these two physical problems raised by the laser guide star, namely the cone effect and the tilt determination. And there is still a lot of work to be carried out before everything is solved. But I would say that the mathematics and the physics involved in the solutions we have today in hands are mastered, it is now clear that ... 

It may turn out that the mathematical model is too rough, meaning numerical results of computations are not consistent with physical experiments, or the model is extremely cumbersome for everyday use and its solution can be obtained with a prescribed accuracy on the basis of simpler models. Then the same work should be started all over again and the remaining stages should be repeated once again. 

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