Thoughts on the mathematics behind quantum mechanics
Thoughts on the mathematics behind quantum mechanics
Let me preface this by saying I am not at all well versed in QM, but I've just recently finished a course that took us through most of Griffiths and I have some thoughts.
The thing that blows me away the most about QM isn't necessarily all the standard "weirdness", like tunneling and superposition and whatever, but rather that all of this weirdness can be accurately described by (mostly) pre-existing mathematics. Linear algebra was a growing, but already established field, and the concept of abstract linear vector spaces, and later Hilbert space, came just before and evolved alongside QM.
The fact that we discovered a mathematical language before finding out what that language could describe just blows my mind. I understand this isn't the first time the maths came before the physics, but considering how groundbreaking and unintuitive the theory is I think that fact is quite exceptional. Even more so is that we can use this mathematical formalism to derive physically observable phenomena. I don't know if you get what I mean, but wow. I think that's nuts haha
Well research is always done off existing foundational theory. But one thing I know about modern physics is people are bad at explaining their work. I remember working through some of a book on statistical mechanics and there was two equation lines directly adjacent to one-another and it took me two and a half pages of work to show how to get there. I will hand it to them, though, it was terse.
Yeah I've been through a couple of textbooks that do the same. I think Griffiths QM can be pretty egregious in that regard, actually.
Are you high? I kid, I kid.
I know what you mean. Physics without calculus is kind of useless. Physics with calculus is elegant and powerful. Although you are talking about QM and Hilbert spaces, the same thing goes for Newtonian mechanics and Newton/Leibniz, as well as E&M.
Actually I was just talking yesterday with my husband about relativity vs string theory, and the same thing: one produces (edit: spelling) falsifiable predictions, the other doesn't. And Einstein was a real ballsy dude predicting gravitational lensing, which is so completely counter-intuitive. It speaks to his incredibly deep grasp of what his theories actually are (I know that seems like "duh" but clearly he wasn't just doing math).
Yeah I agree, it's pretty incredible. I am no mathematician, in fact I'm pretty far from one, so it often blows my mind that there are/were people out there than can read through the numbers and make deep connections between seemingly abstract mathematics and physical reality. I've found that I generally approach physics intuitively, meaning I try to intuitively understand a system before deciding on what relevant maths or concepts I'll need to solve/quantify the system. I think that stands in contrast to those who start with the mathematics, and maybe tease out an intuitive understanding ex post facto.
Also, nice username!
I am right there with you, I use math but I develop an intuitive understanding of the system generally. And thanks re: username! I work in nuclear materials.