Blog

As I go into my third round of teaching the second semester of advanced quantum mechanics I’ve been giving some thought into what to do with the discussion sections. These are standard part of many upper level courses at University of Massachusetts Amherst, but there seems to be no consistent way of using them. The most recent paper that I read on student challenges in learning degenerate perturbation theory, really helped me see the parallels between teaching Quantum Mechanics for second and third year majors compared to teaching of introductory physics to students in their first year. As mentioned in my last post, in both of these cases students lack what we would call quote correct unquote? Intuition about the subject. In the case of introductory physics the alternative conceptions of motion and forces are well documented. In the case of quantum mechanics, on the other hand, students is lack of Prior intuition is completely understandable given the removal of the subject from their everyday experience and the famously non-intuitive nature of quantum mechanics as a subject.

This parallel between introductory physics and advanced quantum mechanics presented in the paper suggests a way in which to use these discussion sections: use them for students to complete tutorials. As detailed in the paper described in my last post, there are now several different tutorials for quantum mechanics such as the quilts. Thus, the discussion could be a good time for students to work through some of these tutorials in the company of both myself and a graduate ta. This would serve an additional function in helping graduate TAs learn how to apply active learning pedagogies to more advanced courses.

I just finished reading Challenges in sensemaking and reasoning in the context of degenerate perturbation theory in quantum mechanics¹ (could not be a paper cast as there is far too much mathematics for that format!). Not only did the paper give some good insights into the active teaching of time independent degenerate perturbation theory, but I also gained an improved appreciation of the parallels between teaching quantum mechanics at the 3rd/4th-year level and introductory physics at the first year level.

Active Learning of Perturbation Theory

As has been discussed elsewhere, I have been teaching a QM II class (Physics 564 at UMass Amherst) for the past two springs, and will be teaching it again this coming Spring 2025 semester. This course is based on chapters 5-7 in Griffiths and Schroeter’s excellent textbook² with a much more lengthy discussion of symmetries through the lens of an introduction to group theory than is present in that text. Much of this group theory material is from Matthews and Walker’s Mathematical Methods of Physics³ as well as my own notes⁴.

As I have been iterating the course, I have been adding more and more active learning activities: an approach which I think works well in a more advanced class such as this. In the first pass, there were occasional activities as I refreshed myself on the material. Starting with the second pass, I began to incorporate some activities of my own devising starting with the opening sections of the course. However, I was unable to add them to all the topics in a single semester. In particular, the discussion of perturbation theory was still predominately based on a traditional style lecture.

This step-wise approach, however, led to an interesting, if expected-in-retrospect, result which reemphasized my commitment to active learning; students’ exam scores were noticeably lower on those topics which did not have as many active learning exercises. Thus, for the coming semester, I have committed to incorporate even more active-learning exercises in my discussion of perturbation theory. Moreover, the materials of Christof Keebaugh, Emily Marshman, and particularly Chandralekha Singh (the authors of the paper) seem to be a good starting point.

The parallels of teaching quantum mechanics compared to introductory physics

The main thrust of the paper, from my perspective was the parallels between the teaching of quantum mechanics and the teaching of introductory physics. In both cases, the students are novices to the subject. This novice identity applies to both the physics content, and the mathematical language which is used to describe the physics.

In introductory physics, many of the students are seeing the subject for the first time and the challenges students face in developing a Newtonian perspective are well documented in the literature. In addition, many of the students are simultaneously new to the mathematical practice of calculus: having just completed it or being co-enrolled at the same time as introductory physics.

Similar conditions apply to students first “real” exposure to quantum mechanics in their 3rd/4th year. Here I refer to courses based in linear algebra as opposed to the introduction that many students first get in a 2nd-year modern-physics class. Just as with introductory physics, many students do not enter with any sophisticated conceptual picture (quantum mechanics is famously non-intuitive after all!). Moreover, quantum mechanics, being based in linear algebra is mathematically fundamentally different than the basis in calculus and differential equations which characterizes prior courses such as classical mechanics and electricity+magnetism.

As a consequence of these parallels, students in introductory physics and quantum mechanics show several similar behaviors:

1. Their conceptual schema are only locally, as opposed to globally, coherent. As a consequence, their answers to deeply related questions may not be internally consistent and may even be mutually contradictory.
2. Students in both courses do not always check their results for reasonableness.
3. In both courses, students can get stuck in “math mode” or “physics mode” when solving a particular problem, but struggle to integrate the two perspectives.
4. Students in quantum mechanics exhibit some of the same novice problem solving strategies which they have “grown out of” in the context of classical/calculus-based physics including engaging in with Tuminaro and Redish call the “recursive plug-and-chug” epistemic game,⁵ as well as memorization and recourse to authority.

These are all facts to keep in mind as I move forward to preparing my course for spring 2025.

1. Keebaugh, Christof, Emily Marshman, and Chandralekha Singh. “Challenges in Sensemaking and Reasoning in the Context of Degenerate Perturbation Theory in Quantum Mechanics.” Physical Review Physics Education Research 20, no. 2 (November 5, 2024): 020139. https://doi.org/10.1103/PhysRevPhysEducRes.20.020139. ↩︎
2. Griffiths, David J., and Darrell F. Schroeter. Introduction to Quantum Mechanics. 3rd edition. Cambridge: Cambridge University Press, 2018. ↩︎
3. Mathews, Jon, and R. L. Walker. Mathematical Methods of Physics. 2d ed. New York: W. A. Benjamin, 1970. ↩︎
4. These are from courses I took with Prof. Jonathon Feng at University of California – Irvine as a Ph.D. student. ↩︎
5. Tuminaro, Jonathan, and Edward F. Redish. “Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games.” Physical Review Special Topics – Physics Education Research 3, no. 2 (July 6, 2007): 020101. https://doi.org/10.1103/PhysRevSTPER.3.020101.
   ↩︎