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

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

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

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

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.

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

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.

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.

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

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!

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

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.

Sir please continue this series

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

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

What a total legend you are

Continue the series. Hearty congrats.

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

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

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 ! )

great episode! Really ties a lot of important concepts together!

brilliant, that's the word to describe your videos

Learning a lot! Thanx! I'm starting to understand the Kroenecker delta! 😊

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.

This is a masterpiece, thank you very match.

Thank you for these awesome videos.

this is the greatest video in all of history

Thank you so much for the videos , it is very clear and concise!!

Good work, please continue

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!

Keep up the good work

Really good explanation for a topic I would have considered too dull for a youtube video... great job!

WTF , why your vedios are incredibly great and so exciting !! Please continue sir

Eagerly waiting for the rest ...

you are the best.....keep going .....

Nice video, thanks :)

This is really good

awesome!

Hi, I am reviewing your quantum series to help prepare myself for teaching some PChem lecture material. Your video production makes the slides nice and fluid, and learning from them is great. The e^x graph simulation with each successive term changing the graph is pretty cool. I am curious how you accomplish morphing your slides/sentences into the next.

Excellent explanation for switching a Ket

im in cyber security and i want to understand quantum better. these are the best videos I have seen on quantum math so far. I think it would help if you made a video explaining quantum symbols and the meanings behind each one too just a suggestion.

amazing!

I’m really enjoying this series and can’t wait to get through the rest Also why do you have more subscribers than total views lol

wow 😮

At 7:14 knowledge hit

this was the hardest part to understand for me, I didn't understand but i will continue to watch for the rest

Super

I hope you will be able to answer my questions thank you so much for your channel again. I definitely got lost in the weeds there for a few minutes. God bless! Should you always wish to define the inner product with itself by length of a vector as the square root of the inner product itself. Does the complex conjugate still apply if its multiplication or division by flipping/reversing it or is that only for addition & subtraction? Why use a complex conjugate on the imaginary numbers in step 4? What did you flip in step 5? Is the complex conjugate like the ultimate reversal uno card? It seems like the complex Conjugate turns everything to the opposite as negative to positive & vice versa, is that correct? Do you put the complex conjugate where you always move something? Is the left slot inside the kets to the left of the addition sign or outside the sign, if so with the middle bar between the kets would the antilinear always be to the left of it or is it possible that it could be on the right despite being on the left side of the addition sign? Also are your examples just for this situation or would this apply everytime magnitude is referenced? What are you when you mean orthonormal/orthogonal i get it means perpendicular but are you referencing a graph or a location of something? I stopped asking questions because im just in a mind fog after 6:15 I will more questions after I understand the 1st 5 minutes.

Hey, great video! I have a question, in order to generalize the dot product, why do we add the extra 'complication' of complex numbers? This makes the left and the right sides behave differently. I understand that this is a consequence of defining our vector space over the field C, but any idea why we need to do this?

Why the inner product commutativity has to be conjugated and not other, for example, anticommutatived?

These videos are very well made. Any chance you could go into QAOA?

ty

at @8:04 why do you assume that both phi ans psi can be expanded by the same E? is it because they are for the same system? or is it true in general

Third term to fourth term. Has he justified previously why the conjugate of the inner product of phi and i*phi is equal to the conjugate of i times the conjugate of the inner product of phi and phi or am I just thick and missed something obvious? Presumably same applies if we replace i in that statement with any complex number? Great course so far btw

Oh my god, the Kroeneker delta is just a discretised Dirac delta! Never made that connection until now. So the Dirac delta's translations also form an orthonormal basis for all functions on R^n? Is that possible? Wouldn't the inner product break since the stipulation is that it's "length" must be 1? EDIT: nvm I thought about it a bit more; you always have to add infintensimally scaled deltas anyways, and it's possible for convergence because of the triangle inequality - I guess that's what the Cauchy-completeness is for. Oh, it's in ch. 5 anyway lmao

and here I am, binge watching linear algebra for quantum mechanics on a friday evening.

The dot product is a measure of "parallel-arity,” not perpendicularity. The more parallel two vectors happen to be, the more NON-ZERO their mutual dot product will be.

Bra-ket notation is not inner product: inner product takes 2 vectors to a scalar, while bra-ket takes a dual vector (bra) and a vector (ket) to a scalar.

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

Quick question: in 8:11, "the inner product of a sum of vectors is the sum of the individual inner products", which rule is that? Does it come from linear algebra / the definition of an inner product / property of sums / the orthogonal-ity of the basis? If it's explained in an earlier part of the video, could you give the time step? Thank you.

7:13 I was very confused why linearity means we can move the inner product into the sum until I realized linear means it has the distributive property

Again, excellent but a side issue: I think it would be useful to point out why one uses dummy indexes for the sums (i.e. why a i and j are used for the two different bases.) That would confuse some who are not familiar with that notation. That notation isn't often used in most linear algebra course.

Why we apply some rule for linearity in order to getting a positive number?

Great video! Would you be willing to share your manim code for the animations?

At 8:02, how can you expand both the bra and the ket in the same orthonormal basis? As far as I understand, they're not in the same vector space, since the bra is part of the dual space.

orthogonal perpendicular {the same}

So far you were discussion vector spaces as an abstract concept, but suddenly when definition inner product, you assumed that the field is the set of complex numbers. Is there a general definition of conjugate applicable to any field F?

can we write =a where v and u are our vector . if no than why ?

Fantastic videos, RU-vid wasn't around when I was learning QM. Any reason not to refer to the the left row vector adjoint to the ket vector as bra? (other than keeping it simple to keep us all [ me included ] on board )

Why all this was so convoluted before? Well, seems like waiting for you!

4:03 could this be explained by | a +bi| = (a² + b²)^½ i.e. ((a+bi)*(a-bi))^½ ?

Wow！！！！！！！！

Why does the length of a vector have to be a positive real number? I find it weird that complex vectors can’t have complex lengths.

One simple question at 7:40 why exactly do we know that we have to put 2 where i is? Why not 1 or something else?

at 2.04 you say "the inner product is a map that takes in 2 vectors and spits out a number which might be complex" this confuses me. because in my mind a complex number is a vector... or at least is not a scalar. later on, you add a rule to inner product to ensure the output is a scalar : A.B == B.A*. where A* is the complex conjugate of A if you include the conjugate rule, does this mean that the final definition of inner product is a map that takes in 2 vectors and spits out a real number. also are the vectors the MEASURED values from experiments, where the value is quantized. if so, are the KETS matrices of Real numbers I LOVE YOUR WORK. you are a superposition of 3b 1b and paul dirac ...

What about Minkowski space? The Minkowski metric has non-zero vectors with 0 magnitude, namely light-like vectors. Meanwhile, vectors with _negative_ squared magnitude are possible when given faster-than-light quantities. Naturally, nothing can reach FTL as far as we're aware, so those aren't really a problem as long as you stick to valid Lorentz transformations.

In the first three videos everything was so well motivated, but here I got pretty lost. Why do we allow complex scalars? Wouldn't it be much easier to just stipulate scalars are real? The basic kets (possible states) themselves are real valued, no? So since the scalars play the role of probability, quantum systems somehow require complex probabilities? That doesn't make a lot of sense to me but maybe that will somehow work out with the wave function? Or maybe I'm just being dense and missing something super obvious? Anyway, I've been trying to think this through for like 20 minutes and I guess I'll just move on to the rest of the series. But it would have been nice if you had said a word or two about why we have to deal with complex scalars in the first place.

1 thing I don't really understand about math, is how some of these conclusions are reached. For example, at 4:00, you say the issue is solved via adding the condition that when you flip the inner product, you add a conjugate to it. Is this just problem solving or is there a reason you do this specifically?

At 3:23, is the issue not that the ket is imaginary?

I didnt got on 8:20 sec " rememebr that the right slot is linear so we pull out the b coefficient; on other the other hand left slot is antilinear ....... -fresher undergrad

I did not understand the Right hand side linear property, can someone help me? I am trying to understand how this is valid expression

The relationship between bras and kets is technically a little more involved than just complex conjugation.

3rd

Sir how i(-i)=1

first

For each Vector Space V there exists a dual Vector Space V*, the space of linear maps of all elements - vectors - of V into the real - or complex - numbers. If V = V* the Vector space ist selfdual. The - unique - Hilbert Space is selfdual So for each pair of elements f,g ε H there exists a - unique - map (f,g) into a number. In Quantum Mechanics |(f,g)| is the transition probability of a state g into a state f.

The mathematician went to his therapist to try to get in touch with his inner product. Mrs. Hilbert wanted a divorce because she needed a space of her own. Euler had a bad identity crisis. Then there was the manic-depressive physicist who was having a very bad dipole moment. You can calculate the sum of the squares all of the time, all of the squares some of the time, but you cannot calculate all of the squares all of the time. There was the sad story of the 78 year old mathematician who had not been in his prime for 5 years. When the great mathematician Euler died, he was given a very fine oilergy.

a left problem: What inner product does mean in quantum mechanics.

. . . . . . . ** . . . . . . . . ** What is☝☝☝THIS or ☝☝☝ THIS?? I often see this notation in mathematical writings. To me, they both look like inner products, but with THREE inputs. How do you go about evaluating these? What is the proper interpretation of this notation?

Just a small question: WTH?

uhhh what?

Sorry. You lost me at 5:34. Poorly explained or possibly just went too fast.

I CAN'T UNDERSTAND THIS VID BECAUSE IS VERY BAD EXPLAINED

Have you mixed up the left and right sides? On wikipedia it is defined the other way round: = a* + b* en.wikipedia.org/wiki/Inner_product_space#Basic_properties