Don't blindly apply, UNDERSTAND Bra Ket Notation with this! | Quantum Theory
HTML-код
- Опубликовано: 20 июл 2024
- 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
Related Videos:
The Stern-Gerlach Experiment: • Know This, If You Want...
Why John Bell Deserved a Nobel Prize: • Bell's Theorem Clearly...
Schrödinger's Uncertainty Principle & its Relation to Heisenberg's: • How Schrödinger Impro...
Maccone-Pati Uncertainty Relations:
• The Quantum Triviality...
Quantum Theory Playlist: • Quantum Theory
If you have any questions or suggestions for future videos, please leave a comment here or shoot me an email at abidebyreason@gmail.com.
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!
you are welcome. Glad it cleared things up for you!
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.
thank you for the kind note and recommendation!
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: yields a matrix, |a>|b> (or
yes, that's one way to look at it that can be useful in certain circumstances. I think you also meant the other way around, yields a number and |a>
That's actually not exactly true: kets and bras are elements of an abstract Hilbert space and its dual space, respectively. Row and column vectors are merely their representations in the isomorphic space C^n
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!
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!!
Glad you were able to get something out of this video!
In my experience, Hilbert Spaces were only briefly mentioned in one of my undergrad math classes (Real Analysis) but came up a bunch in QM. It seems most common to not really get into Hilbert Spaces (as a math student) until taking graduate level Functional Analysis. Dual Spaces are much more approachable independently though. Any good upper division Linear Algebra textbook should address them.
Thank you so much for the kind note!
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.*
Thanks for the comment!
This is actually a really good example of how attempting to make things rigorous in a way that would satisfy mathematicians actually DOES become cumbersome. Just 2 brief questions to address would be:
1) Is the sum infinite or finite and does what I did in my "proof" still work if the sum is infinite?
2) Also, how does one interpret what's in the sum as a tensor product? Since a tensor product like f tensor g would need to have 2 elements as an input like (a, b) where f tensor g (a, b) = f(a) g(b). But here there appears to be only 1 input element, |psi>.
There are more things that need to be addressed. All of this works because of the rigorous math behind it, but I do think in general mathematicians would prefer to just stick with a more conventional notation. That of course doesn't mean that there aren't exceptions or that this notation isn't incredibly useful!
Such a clear explanation! Excellent work. 🎉
Thank you!
Very clearly put ! Going through the math itself, it's hard to get an appreciation of its significance.
Let’s go 💯💯 great vid
Such a great video! I love it because I have a background in mathematics and am now learning about Quantum Computing
Awesome! Glad you liked it.
Excellent presentation, thank you.
you're welcome!
Great video!
Thank you!
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 works, because =
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
Or should I write ?
If I think of
thanks for the suggestion. I plan to make a video on operators in the near future, so stay tuned!
Think my braincell has just blown a fuse. I will have to come back to this after a good night's sleep.
Thanks❤
you're welcome!
Cant we only consider continuous linear functionals for the Riesz Rep theorem?
yes you are correct, the dual space of bras will only be the linear functionals that are continuous.
bro what is all these spaces I only took linear algebra
This is applied linear algebra :)
The name makes me highly uncomfortable.
i love brasseire-ketamine notation :)
@@frank_calvert ok that's better