Recent content by Isaac0427

Thank you! I was not worried; I just curious as to how fundamental this empirical fact is to our theoretical understanding of particles.

Ok, this is making much more sense. I think the last follow-up question is this: I can see that if we somehow discovered a particle that was a mix of spin-0 and spin-1/2, this would force us to fundamentally reconsider our understanding of quantum mechanics as a whole. If we discovered a...

Right, I do know this. My question here was why that is the case, specifically if it was a theoretical necessity or an empirical observation. What I have gathered so far is that the superposition of integer-spin and half-integer spin is forbidden by theoretical necessity. The superposition of...

Right, I am talking about the term symbols -- i.e., for carbon ##^3D## is forbidden, since #L# is even and #S# is odd. I don't know how that works. Perhaps a mentor can move this post there, if it is appropriate.

Thank you for the answer. It leaves me with this question though: wouldn't the same logic prohibit, say, the state ##\frac{1}{\sqrt{2}}(|210\rangle + |220\rangle)## (expressing states as ##|nlm\rangle##? My understanding is that a state like that would be allowed. Additionally, I checked out...

Consider electrons in atom, and let's mostly ignore interactions between the electrons for now. What I mean by that is that the lowest energy level is the doubly degenerate 1s, then the doubly degenerate 2s, then the 6-fold degenerate 2p, etc. Textbooks like Griffiths use term symbols...

Hi all, it has been quite a while since I've been on here. I am curious, as the TLDR summary says, if it theoretically possible to have a particle in a superposition of states with different ##S^2## eigenvalues. For example, a particle being in the state...

Right, but I'm trying to figure out this with matrices/rotating coordinates to diagonalize ##A##. My understanding of what you wrote is that if you have ##\vec{y}=P\vec{x}##, this means (assuming ##P## is a constant nxn matrix) ##\partial_\vec{x}=P\partial_\vec{y}##. Is that correct? And if so...

I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I...

Force varies with distance from the center of Earth. Where does this come from? I may just be missing something obvious here.

Where does one find the solution? Wolfram Alpha didn't come up with anything. I could try to reverse engineer it, or in any case check a few things with the solution that I've wanted to look at.

That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations. I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...

And yes. I see though that it would all be meaningless if they were invertible matrices, as then the span of A would be all of ##\mathbb{R}^n##, and a vector in ##\mathbb{R}^n##'s projection onto ##\mathbb{R}^n## is just itself.

Ah, that's what I was missing. Ok, thank you.

Aren't they the same thing? If so, why would textbooks write the former? Ex: https://textbooks.math.gatech.edu/ila/projections.html or http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf or https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections Thank you!