When I ask my colleagues to name their favorite quantum mechanics textbook, they usually pick the one they used as an undergrad or in grad school. In the shifting fog of quantum theory, a physicist's first quantum textbook becomes an important lighthouse that guides future voyages. For me, my first quantum love will always be the English translation of the combined undergrad-graduate text by Cohen-Tannoudji, Diu, and Laloë, which is dry and neatly organized, like a delicious stack of graham crackers. But I know that only a fraction of my students go on to physics Ph.D. programs and much less pursue research in a quantum system subfield. For many students, the text for their intro course will be their last quantum exposure. So, what textbook is most likely to balance the needs of both populations?

Let's start with the obvious: David J. Griffiths is a great textbook writer. He has produced four widely adopted undergraduate textbooks known for combining a conversational style with sharp physical and mathematical insight. Paraphrasing singer-songwriter Steve Earle, who famously lauded underground favorite Townes Van Zandt above Nobel-laureate Bob Dylan: I'd stand on John David Jackson's coffee table in my cowboy boots and tell him that Griffiths' *Introduction to Electrodynamics* is the best E&M textbook in the whole world. Or on Landau and Lifshitz's ottoman, for that matter. Griffiths' E&M textbook communicates the beauty and power of a technically challenging subject to an undergraduate audience, and the current edition deserves its place of honor in the curriculum. The only drawback of that text (and any popular text) is the proliferation of solutions available on the internet for the “convenience” of time-constrained students.

If I don't stand on anyone's furniture to support the newly released third edition of Griffiths' quantum mechanics text, co-authored with Darrell Schroeter, it is not the fault of the book. Rather, I blame quantum mechanics. Unlike classical electrodynamics, essentially complete in its formulation since Maxwell's treatise in 1861, quantum mechanics eludes packaging into key formulas and the mathematical techniques to solve them. Although I balk at pronouncements like “nobody really understands quantum mechanics,” which mystify the subject for the public and mislead students about how much we do understand, I admit there are still some—let's call them difficulties—in formulating the foundations of quantum mechanics. Lacking consensus on the foundations of quantum mechanics, we also lack consensus on how to teach it.

Roughly speaking, there are two main approaches to teaching undergraduate quantum mechanics: waves-first or spins-first (other approaches include historical (an especially good fit for sophomore-level modern physics classes) and formalism-first (perhaps better for graduate quantum courses)). The Griffiths and Schroeter (G&S) text falls squarely in the waves-first camp. I would argue that if you want to teach a waves-first course, there is no better starting place than the first two chapters of the book, lightly revised and improved from the previous edition. I've taught the second edition several times and found that the impedance matching between the content and previous student knowledge allows clear signal transmission. The second chapter—especially when you include all the exercises at the end of the chapter—is an essential reference for one-dimensional quantum mechanics, a topic that's growing in relevance with increasing experimental capacity to engineer effectively one-dimensional systems such as solid state quantum wires or tight optical waveguides for ultracold atoms.

But I'm not sure whether waves-first is the right approach. The spins-first method also has merits. First, in many departments, physics majors have already seen Schrodinger's equation in the modern physics course in the fall of their sophomore year and then spend the spring semester studying waves and oscillations. If they take E&M before quantum mechanics, then they've had even more waves and partial differential equations. I don't want to leave them with the impression that doing theoretical physics is just learning more and more clever tricks to solve differential equations. Instead, I'd like to give them added practice with linear algebra that will pay dividends across a broader set of academic and career trajectories. G&S has linear algebra to be sure, and a nice summary appendix, but spins-first texts start with linear algebra and keep it in the center.

Second, and more importantly, starting with finite-dimensional systems like spins gets to the true heart of quantum mechanics faster. Beginning with waves in a quantum class reveals important physics, but classical waves also have modes that interfere. The key difference between quantum mechanics and classical mechanics is the structure of probability amplitudes and measurements. While the first chapter of G&S is a great introduction to continuous probability distributions, starting spins-first gets you to the heart of the alternative logic of quantum measurements right away. David H. McIntyre's 2012 text *Quantum Mechanics* uses this organization, starting with Stern-Gerlach experiments (and gedanken-experiments of successive Stern-Gerlach devices). This approach was pioneered by Feynman in Volume 3 of the famous series and is also effective in J. S. Townsend's 1992 text *A Modern Approach to Quantum Mechanics*.

Another advantage of the spins-first approach is that it provides quicker access to contemporary physics research. Recent research articles on quantum systems show that many use techniques not much more sophisticated than Pauli matrices and two- and three-level systems. With proper guidance, undergraduate students interested in quantum information theory, atomic, molecular, and optical physics, and other quantum subfields will be able to apply their finite-dimensional quantum mechanics skills to current problems. Somewhat speculatively, because spin systems are the smallest (qu)bits of the new quantum technology, starting with them also prepares students for the possible future era of quantum engineering.

Waves-first or spins-first, a challenge for any quantum text is how to introduce formalisms like bra-ket notation, self-adjoint operators, and all that. The spins-first approach has an easier job because the states and operators for finite-dimensional systems can be represented as matrices and the instructor can (carefully and ideally) appeal to students' geometrical intuitions. The waves-first approach of G&S has a harder path for students to follow. For example, the distinction between an operator and its representation on a particular basis is a lot easier in finite-dimensional systems; McIntyre uses a specific notation $\u2250$ to highlight the distinction. The handling of formalism and representation was a weaker feature of the previous edition of the G&S, and this edition contains improvements. For example, compare section 3.6 in the two editions, which has been renamed from “Dirac Notation” to “Vectors and Operators” and contains some welcome expansions and clarifications.

However, even with the improvements, the treatment of finite dimensional systems still feels marginal compared to the main narrative. Spin systems are not discussed until the fourth section of Chapter 4 “Quantum Mechanics in Three Dimensions,” after a clear presentation of what should be much more technically challenging material: spherical coordinates, the hydrogen atom, and orbital angular momentum. Spin systems are finally introduced by analogy with orbital angular momentum, but multiple notations and representations for spin-1/2 systems are introduced that confuse kets, spinors, and their representations. The presentation doesn't reveal the simplicity and power of the subject and doesn't demonstrate why the language of spin-1/2 systems is so useful for all two-level systems.

The biggest improvement to the third edition is the addition of a new Chapter 6 “Symmetries and Conservation Laws.” It adds a deeper discussion of unitary operators which realize that symmetry transformations show the connection between symmetry operators and degeneracy, making the textbook a more complete reference. A similar successful revision in the second half of the book is Chapter 11, “Quantum Dynamics,” which not only retains the best parts of two previous chapters but also adds material on Fermi's Golden Rule that rounds out the subject more fully.

Given that the choice of a student's first quantum textbook may have lifelong ramifications, it's a heavy responsibility to set a student's quantum path. I assuage my guilt by requiring my students to pick a second textbook to provide them a counter-narrative to the course's main text. All semester I ask them to journal or blog their way through it, comparing the ideas and presentation. To students who complain that I am doubling their reading load, I say that as they advance in any field and in life, they will have to explore the same subject from multiple, complementary perspectives. I tell them that it is a good practice and a metaphor for complementarity in quantum mechanics itself. At the end of the semester, they write an AJP-style book review comparing their alternative textbook to the main text of the course.

Through this assignment, I have evaluated over 20 quantum textbooks, and here is my final recommendation: If you go waves-first, you should choose this new and improved edition by Griffiths and Schroeter. But I encourage you to consider going spins-first for your primary text and using this excellent book (or dear old Cohen-Tannoudji, Diu, and Laloë) as your waves-first secondary text.

*Nathan L. Harshman is an Associate Professor of Physics at American University. He publicly admits that he is a mathematical physicist. He specializes in symmetry in quantum mechanics with applications to particle physics, quantum information theory, and trapped ultracold atoms*.