Recent content by lys04

In the last page of this image, the formula for the transmission coefficient, i'm not sure exactly what it means. The page says there is no reflection when the sine term is 0 cuz T=1), but for scattering states E>0 anyways? So won't it always pass through? Or is there a chance for a particle...

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...

Ok so you’re supposing s is the supremum of S, which when squared should be a number that is extremely close to 2, but not exactly since there is no sqrt 2 in Q. And since s is the supremum, there should be no number whose squared is bigger than it? But how is 2s/n+1/n^2 in S? Even though it’s...

is 2s/n+1/n^2 in S tho?

yes.

hmm I don't see where there might be a contradiction though..

Not really, is the goal here trying to find a contradiction to the assumption that s is a supremum?

I'm fine with the proof from the notes, its just that since you said it wasn't from first principles I was just wondering on how your proof, which comes from first principle works.

yes

Is this an assumption for contradiction?

properties of the rational numbers as in the distance d is irrational? (They used the fact that the difference between a rational and an irrational number is also irrational?) So does first principle refer to only using properties of the rational numbers? I had a read over your proof for this...

Sorry for the late reply. I was looking through my universitys notes and this is what they did, although it’s kind of a different example. Is this kind of similar to what I did?

Yes your right. So there is no q in Q that is equal to sqrt(2) since its irrational. For q>sqrt(2) I can do this: q=m/n >sqrt(2) for it to be an upper bound. Then using Archimedean principle, that for every epsilon>0 (Take this to be the distance between q and sqrt(2), this is an irrational...

q<sqrt(2) cannot be an upper bound based on the definition of the set though, because again q is rational here, and using Archimedean Principle the distance between q<sqrt(2) and sqrt(2) is irrational, so I can always find a bigger number that is in the set?

I'm not sure how I'm meant to find a contradiction for this. As in q=m/n is not an upper bound? Could I do this instead? Let q=m/n be an upper bound. Then q=m/n >=sqrt(2) But there is no q in Q that is equal to sqrt(2) since its irrational. so q=m/n >sqrt(2) for it to be an upper bound. Then...