Don't blindly apply, UNDERSTAND Bra Ket Notation with this! | Quantum Theory

This is the fourth video in my Quantum Theory playlist. I give a detailed explanation of Bra Ket Notation (aka Dirac Notation) and highlight why it works due to the Riesz Representation Theorem.

0:00 Introduction
0:36 Inner Products vs Linear Functionals
1:57 Dual Space vs Hilbert Space
2:26 Riesz Representation Theorem explained
3:36 Bra Ket Notation explained
6:23 Example of the usefulness of Bra Ket Notation
7:55 Conclusion

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[BrianDLee]

This is the clearest explanation anyone has ever given about anything. Made one of the most confusing things to me clear as water. Thank you sir!

[celestial]

This is phenomenal! I just finished a class in quantum information theory this past semester. The use and manipulation of bra-ket notation quickly became a necessary skill for the course. This video is definitely going to be my go-to recommendation for anyone I meet who is interested in getting into the subject.

[renatovenegas]

Suddenly everything makes sense, great video pal

[davidrtx]

I'm a math major in undergrad right now, and this video is so good, you have such great and clear presentation skills! I still haven't gotten to a formal class that's taught Hilbert and dual spaces, but I've seen the little hints towards it in my textbooks and in my own explorations. I've tried looking into it, but it hasn't felt as independently approachable compared to other subjects for me. I never had a deep understanding of the 'why' in the abstract algebra classes I've taken (probably bc I haven't taken a QM class it seems), but this video made something click in my head. Thank you for sharing!!

[rishabhsolanki]

Such a clear explanation! Excellent work. 🎉

[pranayagrawal]

I don't know why but this makes much more sense after having learnt tensors . I think the Riesz Representation theorem is similar to the correspondence between the vector space V and the dual space V* (covector space). Interestingly, again the corresponding covector to any vector u in the vector space V is the u._ operator. Lovely explanation by the way!

[lolmanthecat]

6:00 "if you are trying to do rigorous math proofs this notation will probably only be cumbersome and sometimes even cause confusion" *proceeds to make a rigorous proof of the fact that the sum of tensor products of the element of a basis with their respective dual basis elements is the identity.*

[kgblankinship]

Very clearly put ! Going through the math itself, it's hard to get an appreciation of its significance.

[ribamarsantarosa]

Me, as an engineer, just see bra as row vectors and and kets as column vectors, and the (double) bar as a sort of dot product: <a|b> yields a matrix, |a><b| yields a number and (7:59) |a>|b> (or <a|<b|) aren't commutative LoL

[tobiassteindl]

Such a great video! I love it because I have a background in mathematics and am now learning about Quantum Computing

[garyknight]

Think my braincell has just blown a fuse. I will have to come back to this after a good night's sleep.

[drdca]

One thing which I think might be nice to elaborate on, is, how to write how linear operators act on these.
Like, if one has a Hermitian operator H, then <psi | H | phi> works, because <H psi, phi> = <psi, H phi>
But what if I have some operators that aren’t self-adjoint, and I want to apply them to the two vectors?
Do I write <A psi | B phi>
Or should I write <psi A| B phi> ?

If I think of <psi| as an element of the dual space, then I suppose the composition of that with a linear operator A would be <psi| A
and...
Hm, I suppose this would be, <A^* psi| ? Is that how we would say it?

This is something I should definitely already know...

[krumkutsarov]

Doesn't the Riesz representation only work for BOUNDED linear functionals? If not then it shouldnt be the "dual space" but the "algebraic dual space"

[ColbyFernandez]

Cant we only consider continuous linear functionals for the Riesz Rep theorem?

[Student-jsqy]

bro what is all these spaces I only took linear algebra

[CosmicHase]

The name makes me highly uncomfortable.