Lectures on Quantum Mechanics, Steven Weinberg, Cambridge U. Press, New York, 2013. $75.00 (358 pp.). ISBN: 978-1-107-02872-2

Steven Weinberg, a Nobel laureate for his contributions to the standard model of elementary particles, has a well-deserved reputation as a writer who draws on great depths of physical insight to produce exceptionally clear prose. Until now, his books have been intended either for a general or advanced audience. For general readers, his books include *The First Three Minutes: A Modern View of the Origin of the Universe* (Basic Books, 1977) and *Lake Views: This World and the Universe* (Harvard University Press, 2010). For advanced readers, he has written *Gravitation and Cosmology* (Wiley, 1972), the three-volume *Quantum Theory of Fields* (Cambridge University Press, 2000), and *Cosmology* (Oxford University Press, 2008).

Weinberg now turns his attention to a core subject in physics with *Lectures on Quantum Mechanics*, a text based on a year-long course he has taught to first-year graduate students. The book begins with a 27-page “Historical Introduction” that concisely and elegantly summarizes the development of quantum physics, including an explication of Werner Heisenberg’s matrix mechanics and its equivalence to Erwin Schrödinger’s wave mechanics. We also find some little-known historical tidbits, such as who coined the word “photon.”

The detailed discussion of quantum mechanics then begins with the Schrödinger equation for a particle in a central potential. Even in such well-worn territory, Weinberg finds interesting twists. For example, he neatly shows how the separability of the wavefunction follows directly from rotational symmetry. He also comments on how the energy levels of different atoms influence the cooling rates of astrophysical gases. This pattern of detailed mathematical exposition enlivened by examples—sometimes surprising ones—of physical phenomena continues throughout the book and is one of its chief merits.

Next, Weinberg introduces the mathematical machinery of state vectors, Hilbert space, observables, and transition amplitudes, with an emphasis on the role of symmetries. Strikingly, though, Paul Dirac’s bra–ket notation is eschewed almost entirely, because, as Weinberg explains in the preface, “for some purposes it is awkward.” That may be so, but given its ubiquity in the physics literature, it is disappointing that students will get so little exposure to it in this book.

Both students and experts will be particularly interested in the section “Interpretations of Quantum Mechanics,” which discusses the Copenhagen, many-worlds, and decoherent-histories interpretations in some detail. Weinberg’s striking conclusion, which he admits is “not universally shared,” is that “today there is no interpretation of quantum mechanics that does not have serious flaws.” That will no doubt provoke further debate, and the section is a good primer for those who would like to follow future developments.

From here, the book moves on to material that is mostly standard for a graduate-level course. But Weinberg presents it with a high level of rigor and clarity and with numerous discussions of related physics not always found in other textbooks: for example, magic numbers in nuclei, how the parity of the pion was determined, and the existence of both left- and right-handed sugars as an example of symmetry breaking. He also gives a complete treatment of the quantization of the electromagnetic field using Dirac’s formalism for constrained systems. Calculations of radiative transition rates in atoms then take us back to the phenomena that originally prompted the development of quantum mechanics. The book concludes with a chapter on “Entanglement” that contains derivations of several forms of the Bell inequalities and an all-too-brief discussion of quantum computing.

In the 24 October 2002 *New York Review of Books*, Weinberg wrote of the tension between “cultures of the image and cultures of the word.” He declared, “I am an unreconstructed believer in the importance of the word, or its mathematical analogue, the equation.” This book clearly reflects that belief, as it contains not a single figure: There are no pictures, diagrams, plots, or graphs of any kind. Furthermore, Weinberg is completely comfortable with dense notation and expects that his readers will be as well: In the chapter “General Scattering Theory,” we encounter an *S*-matrix element labeled with 14 subscripts. These aspects of the book, along with a relatively modest set of end-of-chapter problems, may temper its appeal as a primary textbook, especially for students with limited preparation.

Overall, *Lectures on Quantum Mechanics* must be considered among the very best books on the subject for those who have had a good undergraduate introduction. The integration of clearly explained formalism with cogent physical examples is masterful, and the depth of knowledge and insight that Weinberg shares with readers is compelling.