Group theory and quantum mechanics (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Bercksichtigung der Anwendungsgebiete)

Van der Waerden wrote this book in 1932, about the same epoch of Hermann Weyl's book on group theory, in German. After a staggering gap of 42 (!) years, he prepared this second edition, this time in English. It's probably the cleanest and most rigorous exposition of group theory ever made "for physicists". The quotes were needed because van der Waerden's slant is mainly towards math, as far as mathematical rigour is concerned. But what a clarity! And what a great choice of topics! After a review of basic quantum mechanics in a rather mathematically rigorous and clean style, it starts with group theory itself. It manages to proof the basic theorems about groups with or without operators, discuss representation theory in the context of observables and symmetries, expose thoroughly Lie groups (translations, rotations and the Lorentz group), spinors and the Dirac equation, and permutations, and introduce briefly molecule spectra, all that in about 200 pages!! No time-wasting like most other books, which seem to "over-wrap" the main core of the subject, making it hard to peel. There is a rather large amount of references to heavy Lie group theory and functional analysis' theorems, but here I have to agree with the author that inserting also their proofs wouldn't add anything profitable, because they would be beyond the book's scope and totally out of context. Even in these exclusions we notice the author's wisdom. I deeply regret that this masterpiece is out-of-print. I DO insist that Springer should reprint it as soon as possible! The German edition of this book appeared in 1932 under the title "Die gruppentheoretische Methode in der Quantenmechanik." Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory to the Quantum Mechanics of Atoms and Molecules. The book was mainly written for the benefit of physicists who were supposed to be familiar with Quantum Mechanics. However, it turned out that it was also used by. mathematicians who wanted to learn Quantum Mechanics from it. Naturally, the physical parts were too difficult for mathematicians, whereas the mathematical parts were sometimes too difficult for physicists. The German language created an additional difficulty for many readers. In order to make the book more readable for physicists and mathe maticians alike, I have rewritten the whole volume. The changes are most notable in Chapters 1 and 6. In Chapter t, I have tried to give a mathematically rigorous exposition of the principles of Quantum Mechanics. This was possible because recent investigations in the theory of self-adjoint linear operators have made the mathematical foundation of Quantum Mechanics much clearer than it was in t 932. Chapter 6, on Molecule Spectra, was too much condensed in the German edition. I hope it is now easier to understand. In Chapter 2-5 too, numerous changes were made in order to make the book more readable and more useful."