Is Higher Mathematics Crucial in Understanding Quantum Mechanics?

[Lavabug]

I'm an undergrad currently taking my first course in QM. We've just about finished our chapters on the mathematical formalism and it has been making me pose a lot of questions. We use a lot of vocabulary/definitions that I suspect come from higher algebra, topology, & functional analysis all which are alien to me at this stage in my curriculum. Terms like compactness, completeness, orthogonal functions to name a few, I feel like am expected to take them on faith without really knowing what they mean/why they're used in QM (I can see the parallelism between the cartesian unit vectors i,j,k and cos/sin functions in the interval [0, 2pi], but that's about it). Our suggested textbook (Cohen's) doesn't aid me much other than giving me more precise definitions of what is talked about in class.

I pretty much felt the same way in analytical mechanics when we covered Hamilton-Jacobi theory and my last MM course on integral transforms. Integral transforms was taught in a plug n chug manner and I didn't quite know what significance swapping a problem into a "reciprocal space" had, other than to make some PDE/ODE problem simpler...

Am I getting ahead of myself? Is there anything I can do that can remedy my situation? I get the impression that this gets even worse in more advanced subjects like QFT, which I would really like to take at the grad level. Should I just worry about completing the course in QM before pondering all of this? Feel free to shake the curiosity out of me if you feel I risk failing my course from spending time on these questions haha.

 

[johng23]

There's only so much mathematical background a class can include while still covering enough physics. Definitely keep trying to read up on the background mathematics when you can, but make sure you can solve the physics problems first and foremost, even when you have to take some mathematical statements as "recipes". If you are learning this physics for the first time, you are going to need to revisit it many times in the future anyway, so each time you can gain a deeper understanding of the math. You continue learning about a subject long after the class ends, so you don't want to be TOO idealistic; when you graduate you want to have the strongest understanding possible with good grades, otherwise you may not have a chance to apply the understanding to anything.

I am in grad school and I have a similar problem. I am often more tempted to read about the fundamentals than to try to make research progress with my current level of knowledge. Trying to take my own advice.