Customer Reviews

Group Theory and Quantum Mechanics (Dover Books on Chemistry)

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Even after taking 3 semesters of quantum mechanics, I felt like I had a pretty shaky grasp on topics such as selection rules and the addition of angular momenta. I had heard about the important role that group theory plays in quantum mechanics, so I took a mathematics class in abstract algebra. Though this covered a lot of interesting topics in group structure and ring theory, I was left with almost no idea how the material applied to quantum mechanics. Tinkham's book is invaluable in that it develops the parts of group theory that are extremely relevant to physics and chemistry such as the theory of representations (topics that mathematicians seem bored by) and then shows beautifully how it applies to quantum mechanics. Not only did I understand the selection rules, angular momentum, etc... I had a much better understanding of quantum mechanics overall. Group theory makes much more evident what is meant by "good quantum numbers", where degeneracies come from, and other basic issues in quantum mechanics. Particularly clever was the discussion of the Bloch wavefunction ansatz as a consequence of the Abelian symmetry group of a periodic crystal lattice. Invaluable for quantum chemistry, a subject which is touched on, but which was not nearly as developed when the book was written as it is today. Tinkham knows his math, but he knows his physics even better. If you have any interest in quantum mechanics, get this book!

I began reading this book having just finished a course on Abstract Algebra through my school's math department, and the semester before I took a graduate course on the exact subject.

After taking the math course, I was presented with group theory as if it were some muddled mix of facts, and the course came across as a poorly taught class on number theory. After reading just the first chapter of Tinkham's book, I developed a new, deeper understanding of group theory as a whole. For example, the way that Tinkham presents normal subgroups makes vastly more intuitive sense than the presentation I received in my math course.

The first two chapters alone are probably worth 80% of the book's sale price. The rest is made up entirely of the fact that the book does not piddle around with trivial examples, but genuinely frames quantum mechanics in the language of group theory, and the most important part is that Tinkham does it well.

This book, along with his book on superconductivity, are must-haves for any serious condensed matter person, and this book should be at least read (if not owned) by any physics grad student.

My background is that of theoretically inclined inorganic chemist and this review is intended for those with interests in inorganic and physical chemistry or solid-state chemistry/physics.

Tinkham's text is the first textbook one should go to for a reasonably rigorous introduction to the theory and use of group representations in physics and theoretical chemistry. Modern theoretical chemists should become familiar with all of this book, with the possible exception of the some of the material in Chapter 5 that will be applicable only to physicists (and not a lot of that, actually). The pervasiveness of band theory, even in general inorganic chemistry journals now, should convince chemists who teach this subject to include a lot of Chapter 8 (Solid-State Theory) and chemical theorists will even have to go beyond the symmorphic groups treated here.

The purely mathematical aspects of the subject are treated briefly, but much more completely, than "chemical group theory books" like Cotton's, for example. Naturally, this comes at a price of more mathematical abstractness, but that is unavoidable. These sections, like the rest of the book, are very well written.

Chapter 7, on applications to molecular quantum mechanics, is now quite dated. It was quite incomplete even when written, since it did not include any discussion of ligand-field theory. The effects of antisymmetric wavefunctions for electrons are touched on briefly in Chapter 5 (atoms), but are not adequately accounted for in discussion of molecules. (Incidentally, the failure to use Mulliken notation in molecular QM is an unfortunate annoyance.)

These objections aside, this book is an excellent buy for the price of a Dover edition. Indeed, if I'd included price in my rating, it would be 5 stars - easily!

This book has the advantage of applying group theory directly to solvable physical problems. In most areas of applied physics it is
very important to know the basics concepts of group theory, but
there is no need to have a deep knowledge as well as to know how to
proof all the main theorems. As an introductory course for undergrad
students this book is well recommended.

I have attempted to read other books on group theory, especially those intended for physicists, including Weyl's book The Theory of Groups and Quantum Mechanics. Tinkham's book, however, is the only one that I have been able to understand relatively well. Tinkham gently takes you by the hand and starts you out on a tutorial that addresses the symmetry of a simple example from plane geometry, and then gradually builds up to more sophisticated problems. Character tables and the various orthogonality and normalization relations that make them useful are developed and used for both simple (e.g. plane geometry) and more sophisticated problems. Lie Groups, Schur's Lemma, angular momentum, crystal symmetry, and nature's inability to conserve parity are among the topics addressed.

The treatment of Lorentz and Poincare groups required for a more sophisticated understanding of quantum field theory, however, is not included in this book--for those topics Weinberg's (The Quantum Theory of Fields, Volume 1: Foundations) suggestion of Tung's Group Theory in Physics would seem to be reasonable. I was also not able to understand Tinkham'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.

This book is an excellent reference for group theory. It gives you the detailed math behind group theory (which is great for me). It also gives you a brief introduction so you can work with molecular group theory. This was the recommended text in my chemical group theory class. It serves as a good mathematical reference. Also, see Cotton's group theory book, and Carters group theory book.

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