Quantum Mechanics and the Wave Function

As the twentieth century progressed, it became increasingly clear that quantization was not only a characteristic of light, but also of the fundamental particles from which matter is constructed. Bound electrons in atoms, in particular, are clearly limited to discrete energies (levels) as indicated by their ultraviolet and visible line spectra. This phenomenon has no classical correspondence - in a classical system, obeying Newtonian mechanics, energy can vary continuously. 

In order to describe microscopic systems, then, a different mechanics was required. One promising candidate was wave mechanics, since standing waves are also a quantized phenomenon. Interestingly, as first proposed by de Broglie, matter can indeed be shown to have wavelike properties. However, it also has particle-Uke properties, and to properly account for this dichotomy a new mechanics, quanmm mechanics, was developed. This chapter provides an overview of the fundamental features of quantum mechanics, and describes in a formal way the fundamental equations that are used in the construction of computational models. In some sense, this chapter is historical. However, in order to appreciate the differences between modem computational models, and the range over which they may be expected to be applicable, it is important to understand the foundation on which all of them are built. Following this exposition. Chapter 5 overviews the approximations inherent 

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer 

We begin with a brief recapitulation of some of the key features of quantum mechanics. The fundamental postulate of quantum mechanics is that a so-called wave function, P, exists for any (chemical) system, and that appropriate operators (functions) which act upon h return the observable properties of the system. In mathematical notation. 

These postulates place certain constraints on what constitutes an acceptable wave function. For a bound particle, the normalized integral of q - over all space must be unity (i.e., the probability of finding it somewhere is one) which requires that F be quadratically integrable. In addition, q must be continuous and single-valued. 

From this very formal presentation, the nature of 4 can hardly be called anything but mysterious. Indeed, perhaps the best description of 4 at this point is that it is an oracle - when queried with questions by an operator, it returns answers. By the end of this chapter, it will be clear the precise way in which 4 is expressed, and we should have a more intuitive notion of what 4 represents. However, the view that 4 is an oracle is by no means a bad one, and will be returned to again at various points. 

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