Projectors,bars and kets - Lec 04 - Frederic Schuller

[@Patrick_Bruno]

All the fuss about criticising Dirac's notation sounds quite pedantic tbh. The is no need to distinguish multiplication of a vector by a scalar to be "from the left" of "from the right". Furthermore the notation |e>

[@HomoGeniusPDE]

I think the big criticism comes from the transition from finite to infinite dimensional basis. Ofcourse if you think of |e_i> as a column vector and

[@Newtonissac6]

I deeply love this lecture series. It's really amazing the way he provides the explanation for everything. I am in love with Dr. Schuller' s teaching. I would have loved to have any kind of supplementary materials for this lecture. Any particular textbook he if following or the problem sheets would be so helpful.

[@simonrea6655]

Hi Issac, if you are interested I am typing the lecture notes for this course mathswithphysics.blogspot.it/2016/07/frederic-schullers-lectures-on-quantum.html

[@carlesv1488]

Hi Issac. Take a look at Teschl's textbook "Mathematical Methods in QM". It's free in AMS website. Another textbook in the same style as Dr. Schuller's lectures is Valter Moretti's "Spectral Theory and QM".

[@millerfour2071]

51:02, 1:18:29 (kernel is hyperplane, orthogonal complement is 1d sub space), 1:24:18, 1:24:58, 1:36:00

[@lucasdarianschwendlervieir3714]

Excellent lecture. I'm guilty of using dirac notation ^^, but Schuller has convinced me otherwise. I also did not know there were only one separable hilbert space up to isomorphisms.

[@Newtonissac6]

Lucas Schwendler Vieira He proves it in the third lecture. Still pretty cool result regardless.

[@physicsdaemon]

Perhaps the main advantage is to be able to teach QM to undergrads, as most would have already known matrix algebra, and this translates directly to Dirac notation. Also for finite quantum systems this formalism would be perfectly fine and sufficient to follow lots of papers in quantum information.

[@danideboe]

Did prof. Schuller become Indian and called himself 'Aditya Bhandari'? (wtf)

[@chenardpierre8270]

Brilliant lecture, I hope more will follow.

[@littlekohelet940]

48:25 A map is continuous iff the preimages of closed sets are closed, so the proof that M^\perp is closed is immediate.

[@xrhsthsuserxrhsths]

Well, at 1:39:05 one could say that the map is between the elements of the hilbert space and the generalized elements of the hilbert space of shape \mathbb{C} (the complex numbers).

[@dingdinglhz001]

Awesome lectures! But I desperately want any supplementary materials, like problem sheets (and possibly solutions), lecture notes and/or textbooks!

[@seaset_]

have u found anything? I feel the same way

[@张子兼]

He has (perhaps in the first lecture) recommended the book "modern quantum mechanics" by J.J.Sakurai. What about trying that?

[@xLordOrix]

@@张子兼 unfortunately the book by sakurai is a little under the level of these lectures, its more of an undergrad textbook to be honest

[@张子兼]

@@xLordOrix Yes it is, so did my professor say and thus i guess it might be a good material for one who interests in the foundations.

[@sohanghodla7566]

@@xLordOrix Here are the notes: drive.google.com/file/d/1I7rIH7Rtm0cCKVuLNeWfFMdKurX123x5/view

[@kapoioBCS]

In 1:30:00 he says that H (x) H* is isomorphic to End(H) but in reality it is isomorphic to End(H*) except if dim H < oo. Which is not the case here.

[@abhishekkhetan]

H* and H are same as Hilbert spaces so End(H) and End(H*) are the same objects. I think what you meant was that the tensor product of H and H* is identified with the space of Hilbert-Schmidt operators from H* to H with the Hilbert-Schmidt norm.

[@tobiassugandi]

38:58 smooooth

[@brendawilliams8062]

It makes sense to help computer math with the expansion.

[@jimnewton4534]

At time 48m00s of Separable Hilbert spaces - L03 - Frederic Schuller (the previous video kzbin.info/www/bejne/a4CqdHytrZWjnqs) Schuller said he'd clarify the loophole in the definition of Schauder basis in the next class. He didn't seem to do this at the beginning of this video? Was that edited out?

[@jihongzhi]

His claim at 39:50 that "If H is separable, then M is separable," which he claims is obvious, is not so obvious to me. He mentions a "sub basis" as justification, but even though H must have an orthonormal (Schauder) basis, it's possible that none of these basis vectors are in M. We could start projecting them into M, but then we will eventually find that some are redundant and have to be eliminated. So we start eliminating a possibly infinite number of vectors from an infinite sequence, and need to make sure we are left with a still-orthonormal Schauder basis, and nothing seems too obvious at this point. Anyone have a good justification for this claim?

[@hengruizhu1086]

you can do it as long as it is countable

[@UnforsakenXII]

I need to see how to write the resolution of identity with bra-ket notation then! Edit: Probably spectral theorem, huh.

[@moebutamoebuta5723]

At 40:56, he said “only closed sub space is a Hilbert space” is not true. On the other side, a Hilbert space contains all its limits point hence must be closed. I’m confused. Can anyone clarify it to me?

[@BoudabraMaher]

At 1:27, shouldn't be ?

[@ziadfakhoury4194]

Isnt what he presents as Riesz's lemma the riesz representation thoery instead

[@jupironnie1]

Other than gaining knowledge, i have yet to see how we improve our understanding of QM i.e. physics. After all Dirac is showed that both Heisenberg and Schrodinger views could be shown as part of Linear Algebra. I take Algebra to be an axiomatic basis knowledge base.

[@teretx566]

Maybe I am missing something here but his "proof" at 27:28 looks like utter nonsense. He talks about an increasing sequence bounded from above but he actually shows a finite sum from m to n. What is the point? This sum exists, of course, but he needs to show that it becomes smaller as m and n become bigger. That is what Cauchy requires. His argument is totally circular. Where am I wrong?

[@RolReiner]

I noticed this too. The proof makes sense if you take that sum from 0 to n instead of from m to n. He might have confused between the general term of the series of the partial sums (which is the sum from 0 to n) and the cauchy statement (which needs || Sn - Sm || < epsilon, where Sn and Sm are partial sums from 0 to n and m respectively)

[@teretx566]

@@RolReiner I think you are right, after all the proof exists, of course. Let's give the prof a break ;-)

[@omegapirat8623]

I think the reason why many physicists are using dirac notation is because they rather want to perform calculations instead of mathematical proofs and you are faster if you are handling the formalism more intuitively instead of mathematical rigor.

[@aeroscience9834]

There's a typo in the title. You wrote bars not bras

[@ActionPhysics]

any element of Hilbert space has finite norm , it follows from the completeness property of norm ,,, right or not ?

[@jackozeehakkjuz]

It follows from the definition of norm.

[@mouatadidlhousain3207]

Great lecture. think you.

[@henrywang6931]

My objection to Dr. Schuller's criticism to Dirac notation is that unless your field of research is on the mathematical foundation of quantum theory, it is unnecessarily cumbersome to maintain mathematical rigor. A working physicist can produce good physics by just using Dirac notation, even a theoretical physicist rarely needs the mathematical rigor as presented in these lectures (of course, unless you are a badass like Dr. Schuller who's researching quantum gravity!).

[@chasebender7473]

how can you say this when thinking like this is part of why physics has stagnated in a certain sense? Clearly better mathematical literacy can only improve research, although it is time consuming to learn

[@antoniolewis1016]

I was pretty skeptical about his criticism of Dirac notation, and when I heard his explanation, it felt very pedantic (as he said so himself "I'm being mean" at 1:33:18). The biggest problem with this is the following: Physicists don't have to communicate only with mathematicians! They also have to deal with engineers and chemists! And those people barely (if at all) understand linear algebra. That's why physicists sacrifice the beauty mathematicians desire without delving into dual spaces vs. inner products too much. Further, most physicists approach infinity as the limit of something finite, so proving things directly on infinity isn't appealing.

[@SkyFoxTale]

Antonio Lewis I thought the most substantive reason he gave was that it seems intuitive but could easily lead you to false conclusions if you follow such intuition too far.

[@jonathandreckz]

I disagree with you. By just introducing the Dirac notation then you have to deal with more complicated structure at the end. And I feel sometimes like that, the is no need to add more structure to something unless this structure provides a better understanding on the theory. I don’t see that is the case for the structure you have to add in order to property work with the Dirac notation.

[@Mezmorizorz]

The argument is wrong because of what he says near the end. You are almost always working with an object that makes all the "problems" not problems. The only person who isn't is going to be a mathematical physicist, and at that point, sure, go ahead and not use it.

[@sayantansaha9047]

german elitism..... it never dies.... :P