This book requires a rudimentary knowledge of quantum mechanics. The purpose of this Appendix is to make the reader somewhat familiar with the basic structure/principles of quantum mechanics. How we proceed is quite similar to our approach to statistical thermodynamics; mathematics and empirical facts are synthesized. For statistical thermodynamics the reader has only to understand the time-independent Schrödinger equation and energy eigenvalue problem A.24 (the harmonic oscillator is discussed in A.29). Those who are not familiar with the bra-ket notation should read Appendix 17A first. The following exposition includes an outline of Fourier analysis in the Dirac notation.

How We Proceed

Everyone reading Dirac's The Principles of Quantum Mechanics should realize that respecting both the salient empirical facts and the mathematical aesthetics leads to the most natural exposition of an empirical theory. Therefore, the following exposition of mechanics is constructed around the empirical facts that suggest/justify the mathematical principles.

Remarks for Those Who are Familiar with Introductory Mechanics

For the reader who knows an outline of (analytical and quantum) mechanics, an outline of this Appendix is given here.

The first part A.3–A.8 is a preparation of our language – Fourier analysis. This part explains the Fourier transformation in Dirac's notation.

Based on experimental facts and the thought experiments they inspire, Schrödinger's equation of motion is derived (A.14). At the same time the fundamental postulates of quantum mechanics are summarized (Postulate 1–6). The perturbation method (A.31) and the minimax principle for eigenvalues (A.32) as well as the uncertainty relation (A.28) are explained. The harmonic oscillator eigenenergies are obtained (A.29). For convenience 1/2 spins are also outlined (A.30).

We then proceed to the classical approximation. The reader should be familiar with the rudiments of analytical mechanics. First, the Feynman path integral expression of the system evolution is derived (A.33), from which classical mechanics is deduced as a macroscopic approximation (A.34). The correspondence between Heisenberg's and Hamilton's equations of motion introduces the Poisson bracket and the canonical quantization condition (A.37).

Remark on the notation. In the first half of this Appendix, although the notation x is used for coordinates, this can be a vector of any dimension; xy should be interpreted as the scalar product, dx is a volume element.