What's the best way to introduce physics undergrad students to quantum mechanics for the first time?— Sean Carroll (@seanmcarroll) April 10, 2017

As someone who’s been wrestling with this question for 30 years, I perked up at this tweet, and not only voted but even tweeted a couple of responses. It’s a fascinating question!

The second answer is the traditional one, and there are many good arguments for it: a solid experimental basis in phenomena that are easy to demonstrate; vivid images of wavefunctions for building intuition from classical waves; and a huge array of practical applications to atomic physics, chemistry, and materials science. The down-side is that the mathematics of partial differential equations and infinite-dimensional function spaces is pretty formidable. Mastering all this math takes up a lot of time and tends to obscure the logical structure of the subject. Especially if your main interest is in the new field of quantum information science, this is a long and indirect road to take.

Hence the alternative of starting with two-state systems, which are mathematically simpler, logically clearer, and directly applicable to quantum information science. The difficulty here is the high level of abstraction, with an almost complete lack of familiar-looking pictures and, inevitably, no direct connection to most of the traditional quantum phenomena or applications.

A fundamental challenge with teaching quantum mechanics is that it’s like the proverbial Elephant of Indostan, with many dissimilar parts whose connections are difficult for novices to discern. From various angles, quantum mechanics can appear to be about Geiger counters and interference patterns, or differential equations and their boundary conditions, or matrices and their eigenvalues, or abstract symbol-pushing with kets and commutators, or summing over all possible histories, or unitary transformations on entangled qubits. Stepping back to get a view of the whole beast is challenging even for experts, and bewildering for “blind” beginners.

I think most physicists would agree that an undergraduate degree in physics should include some experience with

*both*wave mechanics and two-state systems. Carroll’s Twitter poll, though, asks not what a degree program should*include*, but how we should*introduce*physics students to quantum mechanics. That’s a hard question, and one’s answer could easily depend on any number of further assumptions:- Who exactly are these “physics students”? Students taking an introductory course, which may be their last course in physics? Typical undergraduate physics majors? Undergraduate physics majors at Caltech? What’s their math background?
- How long an introduction are we talking about here? A single lecture, or a few weeks, or an entire course?
- Will this introduction be followed by further study of quantum mechanics? In other words, is the question
*merely*about the order in which we cover topics, or is it also about the totality of what we should teach, and what we can justifiably omit, when we design a course or a curriculum? - Are we constrained to use existing resources, including textbooks, instructor expertise, and locally available lab equipment? Or are we dreaming about an ideal world in which any resources we might want are magically provided?

Due to all these ambiguities, we should interpret the poll results with caution. Carroll’s interpretation was that the winning second option “probably benefits from familiarity bias. I’ll call it a tie”—so I infer that his own preference is to start with two-state systems. I agree that some respondents were probably biased in favor of what’s familiar, but I also suspect that Carroll’s Twitter followers have more interest in fundamental theory, and less interest in atoms and molecules, than would a random sampling of physicists. I also wonder if some respondents weren’t biased in favor of what’s

*un*familiar: it’s easy to suggest a radical curricular change if you’ve never actually tried it out and had to live with the unintended consequences. Carroll himself is currently teaching an advanced quantum course that emphasizes two-state systems, but as far as I can tell he has never taught a first course in quantum mechanics for undergraduates.
No professional quantum mechanics teacher should be completely unfamiliar with the two-state-systems-first approach, because it’s used, more or less, in Volume III of the

*Feynman Lectures on Physics*, published in 1965 (thirty years before Schumacher and Wootters coined the term qubit!). I say “more or less” because Feynman actually starts with two-slit interference and other wave phenomena, and then he introduces a*three*-state system (spin 1) before settling into a lengthy treatment of spin 1/2 and other two-state systems.
There are also some well-known graduate-level texts that begin with two-state systems: Baym’s

*Lectures on Quantum Mechanics*(1969) and Sakurai’s*Modern Quantum Mechanics*(1985).
At the upper-division undergraduate level, the earliest text I know of that takes the two-state-systems-first approach is Townsend, which first appeared in 1992. Several others have appeared more recently: Le Bellac (2006), Schumacher and Westmoreland (2010), Beck (2012), and McIntyre (2012). Instructors who want to take this approach in such a course can no longer complain about the lack of suitable textbooks.

But at the lower-division level, where most students

*first*encounter quantum mechanics, the pickings are still slim. Nobody actually teaches out the*Feynman Lectures*. You could try to use a few chapters out of one of the more advanced books (McIntyre would probably work best), or you could use Styer’s slim text*The Strange World of Quantum Mechanics*(2000, written for a course for non-science majors), or you could use the new (2017) edition of Moore’s introductory*Six Ideas*textbook (which inserts three short chapters on spin and “quantum weirdness” in between electron interference and wavefunctions), or you could try Susskind and Friedman’s*Theoretical Minimum*paperback (2014, an insightful tour of the formalism with little mention of applications—see Styer’s review here).
I suspect that the time is ripe for someone to write an otherwise-conventional sophomore-level “modern physics” textbook that introduces quantum mechanics via two-state systems and qubits before moving on to wave mechanics. I really wish Moore would expand his Units R and Q into a more complete “modern physics” text!

Personally, I’ve had a soft spot for spin ever since I took a quantum class from Tom Moore in 1982, at the end of my sophomore year (after a conventional “modern physics” class) at Carleton College. This half-term class was mostly based on Gillespie’s marvelous little book, which lays out the logic of quantum mechanics for a single spinless particle in one dimension. But Moore departed from the book to introduce us to two-state and three-state spin systems as well, even writing a simple computer simulation of successive spin measurements for us to use in a homework exercise. The following year I saw more spin-1/2 quantum mechanics in the philosophy of science course that I took from David Sipfle, using notes prepared by Mike Casper, probably inspired by the

*Feynman Lectures*. So when I took Casper’s senior-level quantum course after another year, I was well prepared.
A few years later, while procrastinating on my thesis work during graduate school, I converted and expanded Moore’s computer simulation into a graphics-based Macintosh program. Moore and I published a paper about this program, and how to use it at various levels, in 1993. From there the concept made its way into Moore’s Six Ideas course, and also into the Oregon State Paradigms curriculum and McIntyre’s book. Last year I ported the program to a modern web app.

I recount this history mainly to establish my credentials as an experienced advocate for, and contributor to, the teaching of quantum mechanics via two-state (and three-state) spin systems. So you may be surprised to know that on Carroll’s quiz I actually voted

*against*this approach and in favor of starting with the traditional wave mechanics. And in my own teaching I’ve actually never started with spin systems: I’ve always started with one-dimensional wave mechanics in both upper-division quantum mechanics and sophomore-level modern physics. In calculus-based introductory physics I teach a little about wave mechanics and don’t really cover two-state systems at all. My reasoning is simply that for these students, in these courses, the balance of the pros and cons listed above seems to weigh in favor of starting with wave mechanics.
Meanwhile, I think there are opportunities to improve on the way we teach wave mechanics. One serious drawback with most wave mechanics text materials is their relative neglect of systems of more than one particle. As a result, students tend to develop some misconceptions about multiparticle systems, and don’t hear about entangled states—an important and trendy topic—as early as they could. I’ve recently written a paper on how to address this deficiency, with some accompanying software to help students visualize entangled wavefunctions.

My bottom-line opinion, though, is that the best answer to Carroll’s question depends on both the students’ needs and the instructor’s inclinations. Back in 1989, Bob Romer published an editorial in the

*American Journal of Physics*titled “Spin-1/2 quantum mechanics?—Not in*my*introductory course!” But he hastened to clarify: “not in*my*course, thank you, but maybe in yours”—enthusiastically encouraging instructors to innovate and to follow whatever teaching plan they believe in. I wholeheartedly agree.
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