Basic simplifications of the quantum model

The quantum mechanical description of a material system is obtained as solution of the pertinent Schrodinger equations. 

The first Schrodinger equation is the famous equation everybody knows  

The first step in the sequence of operations necessary to reach a description of properties of the system is to give an explicit formulation of the Hamiltonian. This is not a difficult task often its formulation is immediate. The problems of applied mathematics are related to the solution of the equation, not to the formulation of H. 

The second Schrodinger equation adds more details, ft reads  

Let us come back to the expression of H(x). The number of parameters within (x) is exceedingly large if the system is a liquid. Fortunately, things may be simplified by using factorization techniques. 

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We shall start with the introduction of some basic simplifications in the quantum model. 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

Chemistry and the molecular sciences start with the many-particle theories of physics part III of the book deals with these many-electron extensions of the theoretical framwork, which have their foundations in the one-electron framework presented in part II. The first chapter in part III is on the most general many-electron theory known in physics quantum electrodynamics (QED). From the point of view of physics this is the fundamental theory of chemistry, although far too complicated to be used for calculations on systems with more than a few electrons. Standard chemistry does not require all features covered by QED (such as pair creation), and so neither does a basic and at the same time practical theory of chemistry. Three subsequent chapters describe the suitable approximations, which provide a first-quantized theory for many-electron systems with a, basically, fixed number of particles. A major result from this discussion is the fact that this successful model is still plagued by practical as well as by conceptual difficulties. As a consequence further simplifications are introduced, which eliminate the conceptual difficulties these simplifications are discussed in part IV. 

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