Arm-Waving Quantum Mechanics

Nobody really understands quantum mechanics. As an instructor who views this subject through a chemistry lens, I purposefully arm-wave my way through some of the mathematics (where I deem it not crucial for the students). I occasionally arm-wave my way through a physical analogy of what’s going on – in my experience students find this helpful despite the limitations of any (and every) analogy. But I’ve never asked students to arm-wave their way through learning quantum mechanics. It sounds like asking the students to recite rubbish.

 

Turns out asking students to arm-wave can be helpful. I stand corrected by a delightful article titled “Waving arms around to teach quantum mechanics” by Kelby Hahn and Elizabeth Gire, published in the American Journal of Physics (Oct 22 issue, DOI:10.1119/5.0073946). Here’s the abstract.

 

The authors discuss using a “spins-first” approach to teach quantum mechanics, something I could see a physicist doing but not a chemist. We have different goals in teaching quantum mechanics to students, which is why these classes are offered separately for physics and chemistry majors. I have had the experience of explaining why to an administrator who once asked me why they can’t be combined into a single class (that would have higher enrollment), but I won’t bore you with those details. Let’s talk about arm-waving instead!

 

The article outlines five activities they use which are hierarchically scaffolded, each new one building on the earlier ones. The authors also clearly discuss what the students should know ahead of each activity, what to emphasize, where students get confused, and they share “noteworthy anecdotes” on each activity. The orientation step gets each student to stretch out their left arm to represent an Argand diagram so they can bodily (or kinesthetically) learn three ways to represent complex numbers. I do wave my arm when first introducing polar coordinates, but I should get students to stand up and do it too! Here’s the first picture.

 

In subsequent exercises, students pair up to get a sense of how spin operators in Cartesian space can be represented as linear combinations. Then they learn about relative phase. At some point, they get to representing time evolution of quantum states, which now requires rotating their arms. A culminating activity is to transition from spins to a wavefunction. This last activity is particularly clever because they have to form larger groups to handle higher spin states and eventually when they start thinking about “position” of a quantum “particle”, they start to visualize what it might mean to be a “continuous” wavefunction.

 

I’m unlikely to implement this approach in my quantum chemistry course – because my goals are quite different. Even though I can read bra-ket notation, I don’t introduce it to my chemistry students. And I don’t spend too much time on spin states in my course; we use it for spectroscopic notation, for selection rules, and when discussing Hund’s rule; it also gets embedded in our mathematical representation of the Pauli Exclusion Principle via the Slater determinant. That being said, I really liked the approach used by the authors, and it motivated me to rethink how I introduce spin in my course. I’m reminded that students do get confused by the spin operators, and it takes them some problem-set practice to figure out how they work and why they are useful. There’s likely some gist that I can borrow from the waving arms approach to help my students, and I will ponder this a bit more for when I teach quantum again next fall. I’ve also been thinking on spending an entire class period on the O₂ molecule – its triplet ground state and its surprising kinetic stability. Arm-waving may not be good after all.