Ch 3: Why do we need a Hilbert Space? | Maths of Quantum Mechanics

Hello!

This is the third chapter in my series "Maths of Quantum Mechanics." In this episode, we'll find that infinity brings up a few issues within our quantum framework, and we'll see how a Hilbert space fixes them.

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:

Music:
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♪ Sine by Patricia Taxxon
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[tomtomtomtom]

I can’t believe that having known the Taylor series of e^x for so long I’ve never realized that it’s an example of an infinite series of vectors that does not converge to a vector. The metric space of polynomials is not complete but a Hilbert space is!

[meofamily]

I am delighted to encounter a mathematically literate explanation of modern physical theory that is both elegant and clearly explained.
You are a true find, sir.

[Kevinfreddo]

I love the illustration with the infinite series of polynomials converging to a nonpolynomial which is out of the vector space. I wish I had that type of illustration when I learned about Hilbert space

[alessandroc.]

The polinomial example was enlightening for understanding the completeness. This series is great!

[patrickwang]

Thank you so much. We were tought the definition of a Hilbert space but were never explained as to why we'd need it. I guess it's inherent within the definition but when you get bombarded with 1000 definitions at once stuff like this completely flies under the radar.

[it]

0:00-Recap
0:57-Dimension of vector space
2:03-Problems with infinite dimensions
5:39-Hilbert spaces and Cauchy completeness

[martlock]

So… we resolved the issue of apple pie by just deciding to expand the definition of rocks to include apple pie? 🤔

[roccokainka]

Beautifully describes QM in just the right amount of detail. Keep it going👌, very good work

[cosmicconquerer]

Just discovered your channel. You are an amazing teacher, no exaggeration!

[megiddo]

AMAZING !Thank you so much for the effort of making such content. I'm currently laerning QM all over again and even when I thought I mastered it, I realize that I didn't have these intuitions and I was kind of just brute forcing the math ^^.
THANK YOU !

[jayadevjoshi]

I went to 10 blog posts, and 10 or more videos to understand the Hilbert Space but non of them made sens to me.. and Finally God rewarded me with your Video in suggestion. Now I know what the Hilbert space is and why its essential when we convert classical ML's feature space to QML feature space :D. Thanks for your awesome videos :D

[dennisbrown]

Good definition on the Hilbert space - something often just accepted in a Quantum course.

[flmbray]

Wow... what an easy way to understand Hilbert space! Great job!

[kenjih]

This video is awesome! I've always wanted to understand what the Hilbert Space is and finally I can understand the idea. Thank you so much for creating this series of videos!

[anju]

Really, it gives a depth of understanding regarding Hilbert space

[physicsbutawesome]

Really good visualisation of Hilbert Space, and why it is needed.

[RurczakKurczak]

Questions:
- If an infinite polynomial can converge to an exponent, why can't we just say that the exponent is in fact an infinite polynomial? And the same about any other function, that we can take a Tailor series of.
- Our rule we added is exactly the Caushy completeness condition. But the Hilbert space has also the inner product condition, which we seemingly don't need for now. Why do we add it then? Why can't we assume that our space is just Caushy complete, without an extra inner product?

[christian]

Cauchy completeness is something I never understood fully.

[surenazolfaghari]

Finally, I understood with a good example, why we need Hilbert space. Thank you.❤

[curtiswfranks]

This is the best explanation of this concept which I have ever had the pleasure of learning from. Wow.