Spinors for Beginners 19: Tensor Product Representations of su(2) [Clebsch-Gordan coefficients]

Videos on Clebsch-Gordan Coefficients:

0:00 - Introduction
2:45 - Direct Sum vs Tensor Product
7:19 - Multi-particle systems
8:27 - Tensor Product of Lie Algebras
12:45 - Tensor product of su(2) reps
15:21 - Eigenvalue Operator
17:39 - Ladder Operators
20:50 - 2x2 = 3+1
24:09 - Casimir Operator
26:24 - Clebsch-Gordan Coefficients
29:28 - 3 and 4 spinor products
32:40 - Weight Diagrams
35:12 - Building tensors using spinors
36:59 - Larger Tensor Product Reps.

[giovanileone]

Literally the best math for physics content on YouTube!

[derickd]

This is a fantastic video! I wish this had been my introduction to this topic. Even just one passive viewing, , without commiting all details to memory, nonetheless removes all the mystery and so, upon rereading, one would no longer see this all as black magic but rather be confident in one's ability to understand it!

[Impatient_Ape]

Really good! Bravo again! This series should be required viewing for all graduate students taking the QM course sequence. Things may have changed in the years since I was a grad student, but for me, there wasn't any stress on learning tensor products or Lie algebra representations. But now, with quantum computing being drenched in tensor products, it's irresponsible to fail to teach this to students. (For those who don't know, QM courses tend to be taught by senior professors who use antiquated lecture notes and older textbooks.)

[junghoonkwon]

Always thanks for your works. In my view, these viedo serise are best way to learning lie algebra representaion theory 👏👏👏

[benwhite]

if only i saw this before my Quantum final, this video is a great explanation and builds more around the CG coefficients

[stleisink]

Great video, I used to be only able to mindlessly read the coefficients, now I understand them!

[wafikiri_]

Surprisingly, watching this video has enforced and enriched (by analogy) the theory of cognition I'm building (which I'm beginning to call something like multidimensional cognitive algebra) even though I know Lie groups and Lie algebras only work in continuous spaces and the cognitive ones aren't. But the tensor sum and the tensor product act similarly. Thanks a lot!

[JohnSmall]

All these videos are excellent. Just what I need to revise QM and QFT

[UPu-tqzm]

Incredible work, keep it up. Whenever you have a video covering material i am currently learning in uni i am very happy

[RicardoZamora-osdu]

HI, thankssss you have helped me so much. I have a question, how do you know that the form of the singlet is that? in 21:10 . You say that you miss it, but, there is a form of find it? or make the computation to arrive to him. Thank youu

[franks.]

32:44 How is Mark Thomson's graph algorithm easier to understand than standard physics of spinning tops aligning or anti-aligning?

If I have two 1/2h spinners I can
A. line them both up in the same direction to get a 1h angular momentum spinner with 3 orientations (m)
B. flip one of them into the opposite direction to get effectively 0 angular momentum with 1 "orientation" (only m=0).
= 3+1 = 4 states

Then if you get a third 1/2 spinner, you can use it to
A. lengthen (strengthen) the 1h combination from above to 3/2h (4 orientations)
B. shorten (antagonise) the 1h to 1/2h by spinning in the opposite direction (2 orientations of the result)
C. lengthen the singlet state above to 1/2h again (also 2 orientations)
= 4+2+2 =8

Then if you get a fourth 1/2 spinner, you can use it to
A. lengthen the 3/2h to 2h (5 orientations)
B. shorten the 3/2h to 1h (3 orientations)

C. lengthen each of the 1/2h to 1h (two times 3 orientations)
D. shorten each of the 1/2h to 0 (two times 1 orientation)
= 5+3 +3+3 +1+1 =16

[karkunow]

Great one! Took some insights with me today:
- About the right way to Lie-algebrize the Casimir operator
- Visual weight diagram multiplication
- Clebsch-Gordon tables as change of basis.
Thanks!

[kennethferrari]

Wish this video existed when I was in undergrad… I missed the lecture and felt like I missed a semester

[alexarnold]

I don't know if you'll get to it, but at 12:10 this feels very similar to finding the whole hamiltonian in a multi-particle Hilbert space. Any relationship? Maybe since hamiltonians act as the generators of the time evolution operator?

[Alain_Co]

Thanks, It's so interesting, I start to feel something about what is the spin, but I've reached my limits (the ladder operator, need to be studied again). Sure I'll need to go back to College...
Maybe for my retirement.

[BakedAlaska]

Especially appreciate the spin 3/2x1/2 and 1x1 examples at 37.29. Opportunities for self-affirmation, sort of.

[GeoffryGifari]

One last question:
If we start by tensor-producting two algebras g1⊗g2 and exponentiating it to get members of the Lie group, is it related to the Lie group members of the individual algebras?

Can we write exp(g1⊗g2) = exp(g1) ⊗ exp(g2) for example?

[mockingbird_proxi]

Damn I got flash backs back to the time where I was way to lazy to evaluate the Klebsch-Gordan coeffs on my own <3

[CCequalPi]

Genuine question because Ive been watching your videos for years and I've never been able to figure this out. Is this your real voice or a generator??

[theunknown]

The captions in this is less... deranged than the last video. 8/10 stars