As I am going through teaching P132 – Introductory Physics II: What is an electron? What is light? I have noticed a good instructional goal that I did not consider back at the beginning of the semester when I was first planning out the course: developing an appreciation that nonsensical mathematical results can still possess physical meaning. In P132, there are several topics where the formulae can give nonsense answers. Two more straight-forward examples include Snell’s Law n1 sin θ1 = n2 sin θ2 and quantization conditions requiring integers.
For Snell’s Law, if n2 < n1 then the possibility arises of sin θ2 > 1 which, of course, implies that total internal reflection occurs. Similarly, when considering transitions of quantum energy levels for particles in 1-D boxes, the initial and final energy levels must be characterized by an integer n. Transitions with non-integer values are impossible. In both of these cases, however, students struggle with interpreting these seemingly nonsensical results.
Learning the basic tenet that a “broken” equation giving “nonsense” still is trying to tell you about the world is an important idea. Using the mathematical limitations of modern models is a critical aspect of determining modern research directions. Using an example from my own background in particle physics, the fact that the WW cross section exceeds 1 at the TeV energy scale was part of the key motivators given for LHC construction: we knew that something new had to happen at this energy. Turns out, the new thing was the Higgs boson.
Given this realization, I think that including this goal explicitly and developing activities to further this goal would be a good addition to future versions of this class.