Ch 4: What is an inner product? | Maths of Quantum Mechanics

Hello!

This is the fourth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll derive some intuition for the inner product, and understand why it is a useful tool in quantum mechanics.

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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♪ xor by Patricia Taxxon
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[kisaragi-hiu]

For those who missed some fundamentals (like me): complex conjugate is flipping the sign of the imaginary part, so (3+1i)* = 3-1i.

[nocomment]

I can't believe it's taken me this long in life to see the connection between the Kronecker delta and orthonormal basis.

[narfwhals]

I was going to comment that the dot product _is_ an inner product for real vector spaces. But using it as a pedagogical tool to show the power of the Kronecker Delta was a really nice touch!
I am very impressed by these videos so far.
Though you have left it a bit up in the air what it means to have a complex vector space.

[tobyayres]

i have tried learning linear algebra and quantum mechanics in the past, however i have had much struggle and haven't ever got very far. i just got recommended this video and lots of the stuff i learned makes sense now. this approach to teaching is truly amazing the way i was present allowed me to understand it so quickly this single 10 minute video has taught me more that i would have learned in a day. thank you

[christian]

These videos just don't dissapoint, can't wait for the next one!

[pureatheistic]

Love the series. I absolutely agree too many of the best media on the quantum like to avoid the math for fear of losing people. You showed you can throw the math at the wall and have it stick intuitively. Loved it.

[stevenschilizzi]

This is an amazing series. There’s not a word too many nor a word too few. I’ll be watching several times over until it has all sunk in well.

[Farligefinn]

These videos gives are giving *great* intuition and builds up the needed tools without too little or too much formalism. I'm currently watching this as a companion piece to Quantum Mechanics: The Theoretical Minimum. I can highly recommend it.

[HighWycombe]

I was happily following this series, thinking how easy it all was and not paying full attention, then half way through this one I got a wake up call !! I had to go back and follow it more carefully. This episode seems to be key. It tells us how the INNER PRODUCT is different to the dot product. We get the introduction of multiplying vectors with complex conjugates, and the ket notation. All the stuff I need that's been stopping me from properly understanding the mathematics of Quantum Mechanics. This episode is KEY.
(Now I understand the inner product, I'm starting to wonder what the outer product might be ! )

[it]

0:00-Recap
0:38-Usefulness of dot product
1:54-Inner product definition
5:18-Antilinearity
6:13-Kronecker delta, orthonormal condition and coefficient extraction
7:56-Inner product of 2 vectors

[zdbpthf]

HI! I've only just started watching your videos but I want to compliment you on how clearly you are presenting the topics! I'm a non physicist with no background in quantum mechanics and have watched numerous other videos on the subject and get lost within minutes! Thanks so much for making this intimidating topic seem so much clearer! Looking forward to watching all your videos!

[peki_ooooooo]

more people understand QM, more progress in basic science. You are pushing science forward.

[kichelmoon]

These videos are just so well made and memorable, I love it

[davidhuo]

Thank you very much, most people care how to use the math to solve problems more than what exactly it is and how to deduce the formula.

[e.s.r]

Thank you for making this series! The best explanations I've seen of this notation :)

[pacificll]

These videos are just the best of the best, thank you !!

[jullyanolino]

Cool, didactic and engaging approach, man.
Thank you very much.

[MaxxTosh]

Okay I really like this series, but have you motivated why the complex numbers are necessary to use? I May have missed it 😅

[themercantilistannihilator]

this is the greatest video in all of history

[domingo]

Ok dude my eyes feel like crazy googly eyes as you go all over the screen with new things. I am loving the detail you put in here but you might have to go back & add more information if you can. Thanks again for such an in-depth youtube channel we all really needed it! God bless!