This question was posted on mathoverflow (here) without too much success.
I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" and the related "The Yang-Mills equations over Riemann surfaces".
The following statement serves to explain...
@Slereah yeah, the "packets" terminology and popular depictions of photon energy kinda make it seem like even a free photon must have energy a multiple of some "lowest-energy packet"
$\hat{\mathbf{H}}(t) = \mathbf{U}^\dagger\hat{\mathbf{H}}_0 \mathbf{U}$ where $H$ is the Hamiltonian and $U$ is the time evolution operator. Since the two operators commute, it can be shown that $\hat{\mathbf{H}}(t) = \hat{\mathbf{H}}_0$. But we do have time dependent Hamiltonians, right?
@Yashas not really - how would "it evolves unitarily" tell you that the system obeys the Schrödinger equation and not some other equation whose solution is a unitary evolution?
@ACuriousMind I can derive SE with this assumption: $\mathbf{U} = \mathbf{I} - \frac{i}{\hbar}\hat{\mathbf{H}}$. Except for the constants, the assumption can be arrived at with some reasoning like probability must be conserved and time evolution should allow composition.
@Yashas The equation is wrong (that is the infinitesimal approximation to time evolution, the actual $U$ is not equal to the r.h.s., the r.h.s. isn't even unitary) and I don't see what it has to do with my point anyway.
Yes but can't I can take the limiting case with a summation of small $dt$ which would give $\frac{dU}{dt}$? I am not sure if I have done some mathematical nonsense but I somehow get SE from the limit.
"Time evolution is unitary" reasonably can lead us to believing that there is an infinitesimal generator of time evolution (via Stone's theorem)
it alone cannot tell us that that generator should be the Hamiltonian
i.e. $\partial_t \psi = A \psi$ for some operator $A$ is what you get from "time evolution is unitary". That $A$ is the Hamiltonian (times some constants) is additional information
alas, if you're not thinking about quantization then there's no content in saying "the Hamiltonian", so then there's really no difference between "the Schrödinger equation" and the generic notion that there is an infinitesimal general of time evolution
@JohnRennie This one is just a repost of their previous closed question. I wasn't sure whether to flag it as a duplicate since it says to do so if "This question has been asked before and already has an answer" (which the duplicate question doesn't)
@NiharKarve yes, add the tag. Hopefully someone else will be along to approve the edit. If I addthe tag the system won't then let me close the question.
Would it be possible to analyze exponential graphs using calculus, different series(such as Laplace transform etc) and in case you wanted to know, I am doing newton’s law of cooling.
Yes but the idea is that I deriving the equation of newton’s law of cooling but my high school teacher said that it will be a bit easy to derive an equation that could easily be found online so therefore I am thinking of how to make my report a bit more difficult by adding more analysis. Would it be possible? (Any help from you would be very useful.
Well, I'm not sure how much extra "analysis" you could do on the law of cooling itself, it's not a particularly interesting equation, in that a graph of "rate of cooling" against any of the input variables would just be a straight line
You could look at the Biot number, that's written just under it on Wikipedia, might give you something more to talk about
In this question, after getting a few answers and some discussion (which honestly didn't make what I needed), I myself came to know the answer from a book.
What should I do? Delete it, or answer it myself?
@ZeroTheHero I will ask the user to be more specific. Is there anything specifically that needs to be explained in more detail? I'm trying to help clean up the unanswered queue. This is the question:
I am trying to understand space groups in crystallography. In International tables for crystallography, for a nonsymmorphic space group, they list some symmetry operations. 8 of them are listed under the (0,0,0)+ set and 8 in the (1/2, 1/2, 1/2)+ set. What does this mean? Are there 16 operations ...
I'm finding scarce information online talking about this, is it abuse of notation to raise and lower the indices on the Lorentz transformation matrices? I.e. $${\Lambda^\alpha}_\beta g_{\alpha\gamma}\overset{?}{=}{\Lambda}_{\alpha\gamma}$$
I feel like they aren't tensors and so the answer is yes, but I have seen this done a few times in (what I would consider) fairly reputable sources
In this question, after getting a few answers and some discussion (which honestly didn't make what I needed), I myself came to know the answer from a book.
What should I do? Delete it, or answer it myself?
