Customer Reviews

Group Theory in Quantum Mechanics (Dover Books on Physics and Chemistry)

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This is my first book on group theory and its applications to physics. I did take a one semester course on algebra in mathematics department when I was an undergraduate, so that I knew the basic properties of groups before reading this book. In my opinion, this is a very good introduction to teach physics students or physicists how to use the group theory in QM. Most of books with a similar title strat with the basic theory of groups and then its applications to physical problems. That is, mathematics first, then physics. This book, however, takes a different philosophy. It tightly binds the mathematics of the group theory with QM at the begining and Keeps the style at later chapters, so that a person who is familiar with QM at the level like ``Principles of QM" by Shankar can absorb the materials which Dr. Heine tried to talk easily. Moreover, at the end of each section, there are a few problems to test your understanding. The proofs of the theorems are put at the appendix such that the main discussions will not be interrupted. The whole presentation is particluarly useful for a person who is not mathematically minded and has no ideas about this topic before. The prerequisites for understanding this book is the mathematics and physics contained in Shankar's QM. After finishing this book, you may consult Cornwell or Gilmore for more complete mathematics and Georgi for the applications of Lie algebra to particle physics. I cannot recommend this book to beginners majoring in physics too highly.

My quest to better understand group theory finally brought me to this book. I've tried several others, but this one has the best presentation by far. No, it's not elementary, but this book is very well written. Just the first chapter alone is worth the price. If you want to understand the relationship between group theory and quantum mechanics, I would start here.

Heine's book, although not as easy to understand as Tinkham's book Group Theory and Quantum Mechanics, addresses two topics that are not covered by Tinkham--in addition to covering the other topics covered by Tinkham (e.g. solid state physics; atomic physics). Like Tinkham, Heine begins his introduction to the basics of group theory, including the properties of character tables, with a simple example involving the symmetry of a figure from plane geometry. The two final chapters of Heine are "Nuclear Physics" and "Relativistic Quantum Mechanics". The nuclear physics chapter includes a discussion of isotopic spin. The chapter on relativistic QM would seem to provide an introduction for quantum field theory (QFT), including a discussion of parity conservation.

There are better books, however, that address topics that relate to QFT, especially Wu-Ki Tung's book Group Theory in Physics that addresses topics such as the Lorenz and Poincare transformations that are relevant for QTF. Indeed, Weinberg (in his book The Quantum Theory of Fields, Volume 1: Foundations)recommends Tung for QTF--and with good reason. Heine's book was published in 1965, while Tung's was published in 1990, so naturally the later book in more current. So if you need some insight into group theory for QTF, go to Tung. For other topics, I prefer Tinkham. I was also not able to understand Heine's proof of the Vector Addition Theorem for angular momentum. I found a version of the proof that I could understand, however, in Wigner's book Group Theory and It's Application to the Quantum Mechanics of Atomic Spectra, and I display this proof along with my review of Wigner's book.

Volker Heine's book on group theory and quantum mechanics is not for the fainthearted. To get the full benefit of the text, you'll need a college-level understanding of both mathematics and physics, possibly with a liberal sprinkling of chemistry. Nevertheless, within those bounds the presentation of the material is excellent. If you plan to work through this book on your own, take your time and enjoy (!) what he presents, probably in small pieces, and work through as many of the problems as you can to solidify your understanding before proceeding onward. No, it's not a walk in the park. Frankly, it's one of the toughest books I've ever read. But it also is one of the most rewarding because Mr. Heine can show you that with some effort you too can get a handle on this subject.