Guys, say we have the Schrödinger equation, $$ i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}+V\Psi. $$ Now my book says that if $V$ is a function only of $x$, we can solve the Schrödinger equation by the method of separation of variables. I don't see how this is the case though. Why wouldn't it be possible that $V(x,t)=\alpha(x)\beta(t)$, for instance? Or should I just take my book for their word?
Bishop-Gromov says that if $\text{Ric} \geq K(n - 1)$ for a constant $K$ then you have $\text{vol}(B_{r}(p))/\text{vol}(B_{r}^{k}(p_{K}))$ is nonincreasing or something @Mike
(It's getting late) I think I have some rough idea on how cantor function can jump despite it is horizontal lines almost everywhere, cause I am seeing infinite convergent sequences popping up again when playing with $N_x > 1$. Zee said the jumps are lesbegue measure zero so I suspect it might be once again had something to do with the weirdness of rationals. But first for tomorrow, I need to re-convince myself again by proving that the cantor function is continous. Hopefully by doing so I can see how infinite number of steps can make this naively look discontinous staircase continous by so…
@Ted but then that would mean that if $V(x,t)$ is not a function of $x$ only, we can't solve it by separating variables. And that was exactly my question; how come?
First of all, @Sha, your logic is faulty. It's an implication, not an if-and-only-if. But I'm guessing separation of variables will get all tangled up in that case. Try it.
@Ted no I meant $(A\implies B)\iff (\neg B\implies\neg A)$. I don't really know how to try it for now, but maybe later, when I'm a bit more accustomed to it
@Mike btw Neves started saying something about gauge theory that went super far over my head, what's a good place to learn the very basics of that stuff, might have time to look a bit when classes end
I can give you access to my reference collection which includes this stuff but I still need to upload the gauge theory stuff. Salamon's notes are online for free though.
@DanielFischer So it seems. Well, I'm not going to desist saying things (since I do not think the community is seriously offended by any of it) out of being threatened by flags (obviously not made in good faith), I guess.
I think I'll cap it at reviewing some PDE stuff, familiarizing myself a bit more with the Cartan game for that bryant paper, and looking at the gauge stuff
The most important difference is the bubbling phenomenon Uhlenbeck discovered first with harmonic maps, whereby you need to compactify the relevant moduli spaces due to certain conformal reparameterization properties of the objects of study. This compactification is important in so many kinds of geometry (that rest on Donaldson-ey or Floer-ey ideas) that it's really beautiful to see it in the easiest contexts.
I do not know it personally, @EricSilva. Jeanne Clelland also just wrote a new book the AMS published. I reviewed it in great detail pre-publication. They're all Robert Bryant students.
But that's exactly why I think you should start with Seiberg-Witten - you can avoid these hard technical points and see how to extract invariants from well-behaved differential equations.
@Mike I'm probably gonna review the stuff in Taylor/Evans, I've got a few weeks before I have to attend an analysis summer school on PDE so i figure i should properly review fundamentals
I would long ago have been dead were it not for some talented doctors and surgeons. First heart issues, then cancer. So I very much appreciate good doctors.
Guys, not sure if this is the right place to ask (but I'll try anyway); say we have the following ODE: $$ i\hbar\frac{1}{\phi}\frac{d\phi}{dt}=E. $$ The general solution is $C\exp(-iEt/\hbar)$. Now my book says that "we might as well absorb the constant $C$ into $\psi$ (since the quantity of interest is the product $\psi\phi$)." Therefore $$ \phi(t)=e^{-iEt/\hbar}. $$ I don't really understand why we can "absorb" $C$ into $\phi$. Do mean that when we write $\phi$, we actually mean $1/C\cdot\phi$?
So, what, the method is to pick a nice choice of local frame in a good choice of normal coordinate system, and then realize that everything tensorial is determined by this local frame and its derivatives?
Most likely, the point is that you have 2 differential equations $\phi$ and $\psi$ which both have solutions $A \dots$ and $B \dots$, but since you care about the product $\phi \psi$ , there's no point in having it equal $AB ....$. A single constant suffices, and you choose to put it with either $\phi$ or $\psi$.
