The good news is that I have exposure to vector calculus, some linear algebra, and diff-eq (my personal favorite when it comes to math if only for it's applications in control functions). So then do the mathematics change from handling everything on an individual level (electrons in an orbital?) to treating it as a bulk? Dammit, stuff like this makes me appreciate physics so much and wish I had a grad level understanding of it.
Just saw your reply, sorry. I've had a realization of how I need to better manage my notifications, now I understand why I've lost some responses. Yep. A standard level graduate school education usually leaves the student with a firm understanding of only one electron in a Hydrogen atom, where there is similarly only one proton in the nucleus to pull on it (the Coulomb force dominates over gravity at this scale). The resulting solutions to the Schrödinger equation form a set of orthonormal/orthogonal spherical harmonics, describing the electron's spatial wave function (the probability distribution I mentioned earlier) that depend solely on the electron's energy level. As soon as you add more particles (protons or electrons) to the system, things get complicated fast. In my curriculum, I have also analyzed a system of two electrons, with no proton. That's also a rather unintuitive system due to a quantum mechanical parameter called "spin" that has no classical analog, i.e. there's no familiar way to think of it. You've just got to accept its existence and move on with your life (THIS IS DIFFICULT). But we came up with clever ways of treating systems of particles, and as you've guessed, the mathematics do indeed change. Handling systems of atoms and molecules is often best left to chemistry, which can be thought of as a result of quantum mechanics for large systems of particles. I brought up solid-state physics previously, which is also the application of quantum mechanical theory to a particularly interesting landscape; atoms connected in varying geometric lattice configurations via sharing of electrons. In undergrad, I took a solid-state physics course cross-listed as grad-level, but I'd have to take it about three more times to feel comfortable with all of the content. Do you have any links on control functions? :) I love me some physics, but maybe it's too much when you're spending 1/3 of your semester learning the culmination of several people's lifework. And that's why when I look at my homework from last month, I don't even remember doing it (or most importantly, how to do it).So then do the mathematics change from handling everything on an individual level (electrons in an orbital?) to treating it as a bulk?
Dammit, stuff like this makes me appreciate physics so much and wish I had a grad level understanding of it.