Lessons I wish I had learned before teaching differential equations [pdf] (1997)

Honestly your problem was that you didn't know any Classical Mechanics yet and you were assuming that the volume of recent developments made old books obsolete. Maybe in Biology, in Physics getting to recent developments would mean that you're familiar with Goldstein, Landau's Vol. I... Abraham-Marsden? Arnold? Those are old.

Often newer editions actually worsen textbooks and then only a few contemporary books become references in the long run. It's always been like this, there's tons of great books from the 70s that aren't used today and could definitely do. At least they're not ~1,000 pp. of waffle, which is what you usually get for your first textbook on anything nowadays.

The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as a way of rewriting the equation. What is a “differential”? Who knows. Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Who knows. But here are the steps to solve the equation; good luck on the test.

Another example is the quantum chemistry class I took (my first introduction to quantum mechanics). The professor mentioned something about how you can take a measurement and collapse the wave function of an electron; the probability of measuring its position at any location in space is non-zero. I followed up with some question about an experiment to confine the wave function collapse to certain regions of space which could be used to transfer information faster than the speed of light, which I thought wasn’t possible. My question was mostly met with a blank stare and something along the lines of “this course doesn’t cover that” before the professor moved on.

The most egregious example was a grad-level course on statistical mechanics. The professor mentioned something about the overall wave function of the system being a Slater determinant of individual wave functions. A student (not me) asked why we have wave functions for individual particles and a separate one for the overall system, and which was more “correct” fundamentally? The professor replied that the wave functions for the individual particles were more fundamental, and the overall wave function for the system was just an approximation. I replied, sure, the Slater determinant is a way of enforcing certain constraints that must be met for the whole system, but one of the defining features of quantum mechanics is that the function describing the state of the entire system is generally inseparable; without this we couldn’t have entanglement. The professor dismissed my response as incorrect and said students shouldn’t challenge professors on topics they don’t know anything about. This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).

Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.