Quantum theory introduction and principles

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

Pars, L.A. (1962). An Introduction to the Calculus of Variations (Wiley, New York). [436] Yourgrau, W. and Mandelstam, S. (1968). Variational Principles in Dynamics and Quantum Theory, 3rd edition (Dover, New York). 

The adaptation of these ideas to the principles of quantum theory and the introduction of the spin has been carried out to some extent, but still meets with great difficulties. 

H. E. Stanley Introduction to phase transitions and critical phenomena 32. A. Abragam Principles of nuclear magnetism 27. P. A. M. Dirac Principles of quantum mechanics 23. R. E. Peierls Quantum theory of solids... 

The synthesis of general relativity and quantum theory is embodied in the gauge principle that emerges as a natural feature of projective relativity and explains the unihcation of the electromagnetic and gravitational helds. A brief introduction to the concept of gauge invariance is provided in a second Appendix. 

The actual mode of variation of the specific heat is such as to suggest that in certain ranges of temperature some of the degrees of freedom pass entirely out of action. The principles so far introduced, then, need amplification by some quite fundamental new rules which provide reasons why sometimes degrees of freedom should be operative and sometimes not. These rules cannot be derived in any way except by the introduction of the quantum theory. In the meantime it appears that the equipartition principle is an incomplete statement. If the results it predicted were merely inaccurate in a numerical sense, the discrepancies could be attributed to causes within the framework of ordinary mechanics, for example, to the non-independence of rotations and vibrations, which would spoil the formulation of the energy as a sum of square terms, but the difficulty lies deeper. 

The conceptions of the kinetic theory and of chaotic molecular motion, even by themselves, provide interpretations of a surprising range of phenomena. They fail to give any account of absolute entropies, or to define the exact position of any equilibrium. They lack, as it were, an origin of reference. This failure is largely redeemed by the introduction of the quantum theory and of the statistical principles whereby the occupation of molecular states is defined. 

We shall try to show that not withstanding the uncertainty which the preceding conditions contain, it yet seems possible even for atoms with several electrons to characterize their motion in a rational manner by the introduction of quantum numbers. The demand for the presence of sharp, stable, stationary states can be referred to in the language of quantum theory as a general principle of the existence and permanence of quantum numbers." ... 

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