Postulates and Theorems of Quantum Mechanics

The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to understand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates. From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest. 

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The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles. Expressed somewhat loosely, the fundamental postulates of quantum mechanics assert dial microscopic systems are described by wave functions diat completely characterize all of die physical properties of the system. In particular, there aie quantum mechanical operators corresponding to each physical observable that, when applied to the wave function, allow one to predict the probability of finding the system to exhibit a particular value or range of values (scalar, vector. 

The measurement process is one of the most controversial areas in quantum mechanics. Just how and at what stage in the measurement process reduction occurs is unclear. Some physicists take the reduction of as an additional quantum-mechanical postulate, while others claim it is a theorem derivable from the other postulates. Some physicists reject the idea of reduction [see M. Jammer, The Philosophy of Quantum Mechanics, ley, 1974, Section 11.4 L. E. Ballentine, Am. J. Phys, 55, 785 (1987)]. Ballentine advocates Einstein s statistical-ensemble interpretation of quantum mechanics, in which the wave function does not describe the state of a single system (as in the orthodox interpretation) but gives a statistical description of a collection of a large number of systems each prepared in the same way (an ensemble) in this interpretation, the need for reduction of the wave function does not occur. [See L. E. Ballentine, Am. J. Phys, 40,1763 (1972) Rev. Mod. Phys, 42,358 (1970).] There are many serious problems with the statistical-ensemble interpretation [see Whitaker, pp. 213-217 D. Home and M. A. B. Whitaker, Phys Rep., 210,223 (1992) Problem 10.3], and this interpretation has been largely rejected. 

Chapter 3 Quantum Roots of Crystals and Solids) The quantum mechanics postulates are shortly reviewed for their application in providing the basic crystal Bloch theorem, further specialization to the quantum modeling of crystals in reciprocal space, on various levels of electronic behavior from free to quasi-free, to quasi-binding, to tight-binding models. 

The third law of thermodynamics postulates that the entropy of chemically pure, crystalline substances at absolute zero 0 K is zero. Originally, this postulate was put forward as the Nernst heat theorem (1906) and later proved by quantum mechanics (Planck 1912). From this absolute zero, the standard entropy S T) of substances can formally be determined at an arbitrary temperature T, by adding the entropy increases AS from heating and phase transformations to 5(0). 

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