The publication of Steven Weinberg's *Lectures on Quantum Mechanics* is a cause for celebration. Young readers learning quantum mechanics are fortunate to have this volume. So, for that matter, are those of us who teach it.

The book is intended to accompany a one-year introductory graduate course in quantum mechanics. At 375 pages it has brevity not achieved since Dirac's *Principles of Quantum Mechanics* (314 pages, 4th edition, Oxford University Press). Dirac's is one of the great books in the history of human civilization. But after four score and more years it is beginning to age. The appearance of a new book by a scientist of Weinberg's stature is thus well timed. The explicitly pedagogical intent of Weinberg's book sets it apart from books written by other great masters. Although concise it is written with the exceptional clarity and logic that readers would expect of Weinberg.

Any textbook on graduate quantum mechanics is an exercise in selectivity. Although there are no universal selection rules, the choices Weinberg has made are quite conventional. What is distinctive about this book is not the choice of topics, but the treatment of those topics and the illustrative examples.

The first three chapters introduce quantum mechanics. Chapter 1 is on history, Chapter 2 on simple applications of the Schrödinger equation, and Chapter 3 on the general principles.

Reading Weinberg's historical introduction may be the next best thing to reading the original papers. Points that are normally glossed over, such as Bohr's use of the Correspondence Principle to deduce that the quantum of angular momentum is $h/2\pi $, are elucidated. The account of Heisenberg's exploits, while recovering from hay fever on the North Sea island of Helgoland, is absolutely riveting. But the description of Schrödinger's work falls comparatively flat as Weinberg succumbs to the temptation of giving a schematic account of how it could have been done instead of telling us how it was actually done.

In Chapter 3, Weinberg sets forth the general principles of quantum mechanics. A distinctive feature of his presentation is the importance given to symmetry principles, which allows him to downplay and postpone discussion of the canonical formalism. When he does return to the canonical formalism in Chapter 9, he gives an excellent account of it, including both Feynman path integrals and Dirac's theory of constraints (a subject not normally covered in books at this level, although it should be). Perhaps the most unusual feature of Chapter 3 is the discussion of the measurement principle. More space is devoted to alternatives to the Copenhagen interpretation than to the Copenhagen interpretation itself, and Weinberg concludes that the problems with interpretation might be telling us to look for a more satisfactory theory that supersedes quantum mechanics itself.

The remainder of the book applies quantum mechanics. Chapter 4 is concerned with angular momentum, spin, statistics, and further development of symmetry. Chapters 5 and 6 are concerned with approximation methods (perturbation theory, variational principle, WKB approximation, adiabatic approximation). The two short chapters on scattering theory are particularly masterful. Chapter 7 is the scattering theory I normally teach in such a course; Chapter 8 is the scattering theory I wish I taught. Chapter 10 deals with particles in electromagnetic fields including discussion of gauge invariance, Landau levels, and the Aharonov-Bohm effect. Chapter 11 is on Quantum Electrodynamics; field quantization is dealt with here with unusual sophistication. The final chapter is on entanglement and an appropriately brief introduction to quantum computing.

Weinberg is interested in explaining natural phenomena. Marvels abound in every chapter. In less than one page he explains the shell model of nuclei and how to calculate the first few magic numbers (Chapter 4). Also in Chapter 4 is an enthralling six-page introduction to the symmetries of elementary particles, and the hidden SO(4) symmetry of hydrogen. Time-dependent perturbation theory is applied to calculate the ionization of atoms (Chapter 6). The Low equation and shallow bound states are introduced in Chapter 8 and the formalism is applied to proton-neutron scattering. Also in Chapter 8 is a brief intuitive explanation of the Froissart bound and its latest tests at the Large Hadron Collider and the Pierre Auger observatory. Deep ideas like broken symmetry and effective theories appear in Chapter 5. If you teach quantum mechanics you have probably struggled for years to incorporate similar material in your course. And in this book you will find many of the things you have only attempted, all beautifully and fully realized. One of the strengths of the book is its bibliography that allows the student to see quantum mechanics as a living, growing subject.

Knowledgeable readers sometimes carp about the non-standard notation that pervades all of Weinberg's books. Generally, this criticism seems misplaced because Weinberg always explains his notation meticulously. A rose by any other name is still a rose. But in this book Weinberg ups the ante by eschewing Dirac's bra-ket notation because “for some purposes it is awkward.” Weinberg's notation is in fact well adapted to discuss concepts that Dirac notation obfuscates such as operator adjoints. In Weinberg's notation, which is also used in more mathematically-oriented books, $\u27e8\phi |\psi \u27e9$ is written as $(\phi ,\psi )$. Thus, $\u27e8\phi |A|\psi \u27e9$ must now be unambiguously written as either $(\phi ,A\psi )$ or $(A\u2020\phi ,\psi )$. However, Dirac notation is better at conveying the operator character of outer products, at constructing resolutions of the identity, for formulating path integrals, and much else. Dirac notation is also typographically beautiful and emblematic of quantum mechanics. More pragmatically, it is the notation used in the literature including Weinberg's field theory books. Instructors using this book will likely adopt a middle ground using Weinberg's notation where it is evidently superior while making sure their students remain fluent in Dirac's notation.

One topic that Weinberg skips without regret is the relativistic Dirac theory. He explains his distaste for the subject in the preface to this book and at greater length in volume 1 of his field theory book (*The Quantum Theory of Fields,* vol. 1, Cambridge University Press, 2005). Essentially, his point is that the idea of holes and the Dirac sea does not apply to bosons and that some of the original motivation for the work was based on misunderstandings. It has become the norm to omit this subject in quantum field theory books in favor of alternative formulations. But its absence in a quantum mechanics text is regrettable; the hole theory does apply to fermions even if not to bosons. Historically it not only provided the first explanation of anti-matter, it also led directly to the understanding of p-type carriers in semiconductors, a discovery that has transformed the way we now live. The Dirac equation and hole theory remain the basis of some of the most interesting work in condensed matter physics today on materials like graphene and topological insulators. In the condensed matter analogs the Dirac sea has an undeniable reality. But readers of this book will have to look elsewhere to learn the Dirac equation.

The twelve chapters of the book are divided into a total of 78 sections. With reasonable elisions a one-year course would need to proceed at the brisk but attainable pace of two sections per week. The table of contents enumerates the topics covered in each section. As in Weinberg's other books square boxes are used to separate successive topics; the reader who is able to check all these boxes will be well educated indeed. In many respects Weinberg goes deeper and farther into the subject than other comparable textbooks. His book will appeal to precocious and ambitious students. But other students may find the standard texts more accessible. There are some nice problems that accompany each chapter but instructors will almost certainly have to augment that pool with additional problems. There are few subjects in physics where figures are more dispensable than graduate quantum mechanics but inexplicably some reviews have still complained about the absence of figures in this book.

Regardless of whether it is widely adopted as a graduate text, Weinberg's book is a magnificent resource. It will educate and inspire generations of physicists. It sets forth what Steven Weinberg believes students should know about quantum mechanics. Anyone who is serious about the subject should pay heed.

*Harsh Mathur is a theoretical physicist at Case Western Reserve University. His research interests include condensed matter physics, cosmology and fundamental physics*.