XD sidney coleman just lighting up a cigarette at the beginning of lecture: youtube.com/watch?v=mnlHLd3UtYA

@alam Hi :-)

Hi:)

Can I move this to the problem solving room? We prefer the chat here to be about general principles rather than specific problems.

Ok

→ 5 messages moved to Problem Solving Strategies

is this an appropriate writing down of the Hamiltonian of an ideal scattering experiment?

i remark that I do not explicitly define what the Hamiltonian is between the asymptotic behavior and the bounds of the finite interaction interval $\mathcal{I}$

@alam hi

@VincentThacker so the map between objects is just $F(SO(2))=R^2$, right? I'm using $SO(2)$ to denote the one object of the group

i understand the map between morphisms.. the map between objects is weird becuz ive never seen a map of this form

i am also wondering if this is an appropriate interpretation of the $S$ matirx

in QFT we have these dirac deltas pop up that also encode conservation of (insert quantity) to my understanding

bleb well i think sakurai is saying that $1 - blah$ is the amplitude for no scattering and $blah$ is the amplitude for scattering

@SillyGoose yes

i don't understand how this quantity can be a distribution? at what point did we turn it into a distribution... it should just be an inner product i would have thought and so a complex number

but it is expected to be a distribution becuz it shud b delta in the absence of scattering

i should clarify that the incoming and outgoing state at this point is assumed to be a particle in a length $L$ box for now, but i guess i see your point once we take the $L \to \infty$ limit

okay i guess that makes sense. if the incoming and outgoing waves are plane waves then surely the $S$-matrix is not a discretely indexed matrix (in the momentum basis)

okay so it is really an artifact of the idealized set up

and then when we move into a real picture maybe we integrate twice: once over $k$ and once over $k'$ to get our packets

but the momentum space is discrete for qft in a box

i guess it depends on how u r setting things up

hm well i do have momentum space as being discrete at this point

so i have the kronecker delta indexed by $k$, and the dirac delta just appears from evaluating the integral in (72)

oh

but i think what you said makes sense bc ultimately i will take $L \to \infty$

that is how sakurai sets it up

so the $S$-matrix here is really like an $S$-matrix density or something

@RyderRude Yes

Is there any modern physics book which is not just plug the numbers and get the answer( I have books like Arthur beiser and paul tipler but they just throw formulas and in example number that have to be plugged and get the answer)

@SillyGoose the integral $\int dt'\mathrm{e}^{i\omega_{kk'}t'}$ is not convergent in the ordinary sense

It's the integral representation of the Dirac delta

More formally, you would understand this by learning about Fourier transforms of distributions

@SillyGoose I guess this png is related to your Phys.SE question. Out of curiosity, which ref. is the png from?