Ch 4: What is an inner product? | Maths of Quantum Mechanics
ฝัง
- เผยแพร่เมื่อ 30 ก.ย. 2024
- 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:
quantumsensechannel@gmail.com
Thanks!
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For those who missed some fundamentals (like me): complex conjugate is flipping the sign of the imaginary part, so (3+1i)* = 3-1i.
Thank you so much for clarifying this!
-QuantumSense
It's worth noting that the more general idea of conjugation simply means that multiplying an element by its own conjugate reduces it to something in a simpler field - this is basically all Galois theory. Take, for example, the rationals extended by some square root, like sqrt(2). Nothing here is imaginary, but if I say that the conjugate of p + q*sqrt(2) is p - q*sqrt(2), then we can see that if we multiply these, we get p^2 - 2*q^2 (by difference of squares), and this is now simply rational, rather than being in this field extension.
Conjugation by changing i to -i is just a special case of this, where we are extending the field of real numbers by some square root of -1 (to get the complex numbers), and then conjugation by itself reduces the complex numbers back to the real numbers.
@Edward Huang if you want a simple summary of this (which is a much neater way to state it):
Conjugation in a field extension E/F is any element of Aut(E) that fixes F, and only fixes F (no more, no less).
You might see that half of this can be further simplified to "an element of the Galois group", so this is really a very fundamental thing.
In the case of complex numbers over real numbers, the Galois group is just C2; it has the identity function, and normal conjugation (i to -i). The identity fixes too much, so normal conjugation is the only choice.
@@quantumsensechannel Our Quantum Sensei :D
more people understand QM, more progress in basic science. You are pushing science forward.
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.
It was clarified at the beginning of the series that you are already expected to be moderately familiar with linear algebra. Thus, one should, in principle, already know what a complex vector space is.
got out-smarmed
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
3 brown 1 blue has a good linear algebra series. You should be able to find the full playlist on TH-cam
Sir please continue this series
Trying to think of a joke about series and continuousness
@@alejrandom6592something something epsilon-delta of time between episodes approaching 0 maybe
Kon clg e poris tui?
These videos just don't dissapoint, can't wait for the next one!
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.
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
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.
Hello! Thank you for watching!
In the next episode, this is exactly what we are going to clarify. In fact, we’re going to show how bra-ket notation is just a manifestation of the Riesz representation theorem, which provides the explicit connection between the inner product and the dual space, as you mentioned.
-QuantumSense
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.
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.
It quickly follows from applying the linearity of the inner product to each of the sides. Using the rules stated at 5:08 a proof might be something like this:
1. Apply rule 1 to each vector on the right. This extracts the sum over j.
2. Flip the product.
3. Apply rule 1 to the flipped product. This extracts the sum over i.
4. Flip the product.
It takes a bit of practice getting used to the sigma sum notation, but you can easily check this property for a sum of 2 or 3 vectors in each component of the product.
Okay I really like this series, but have you motivated why the complex numbers are necessary to use? I May have missed it 😅
Hello! Thanks for watching, this is a really good question.
I have not motivated the need for complex numbers in quantum mechanics yet, but I will in Chapter 13 when we derive the Schrodinger equation (sorry for the wait!). As a hint, we will find that the "i" in Schrodingers equation shows up because we need time evolution to conserve probability. So in other words, complex numbers are necessary in quantum mechanics because of the need to conserve probability. This is a fairly bold claim, but take my word for it for now!
-QuantumSense
It's not the deep reason, but you will soon see that due to quantum hijincks (Planck's results) you cannot turn the right hand side of a classical wave differential equation into a quantum one without plugging i. The derivatives will fail miserably otherwise.
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.
Hello, thanks for watching the video!
To be perfectly honest, there is no "need" for the length of a vector to be a positive real number. At the end of the day, math is invented by mathematicians, and we can define things however we want. But the point I wanted to make is that we *want* to be able to interpret the length of a vector in a way that makes sense to us. How would we interpret a length of 2 + 3i? We could take the complex magnitude, but now we're changing our formula for the length. So we want vectors to have a real length because it allows us to use all the understanding and intuition of lengths in a Euclidean vector space.
In short, we work with math in the way we define it, so might as well choose a definition that is most useful.
-QuantumSense
It is for the same reason that we consider the absolute value of a negative real number to still be a positive real number.
I also got stuck here. Might be explained later on, but even creating the conjugate rule = doesn't seem to eliminate the possibility of complex lengths. I'm coming into this with near-zero knowledge of quantum physics, and don't yet understand why we care about complex vectors, but if we allow them, doesn't = i yield a complex length, assuming isn't complex itself? I'm sure this will be expanded on, but I was definitely confused there.
@@ABC_Guest The only thing we need is that is always a non-negative real number. We can be complex.
@@theemathas Ah okay, I didn't realize that was the intention. Thanks!
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.
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
and that is important because you no longer have to worry in which order you carry out the operations. First add and then take inner product of the sum or first take the inner products of the separate vector and then add. This independence of order (or its counterpart dependence of order) plays an important role in QM
Why the inner product commutativity has to be conjugated and not other, for example, anticommutatived?
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
Nice video, thanks :)
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.
Cool, didactic and engaging approach, man.
Thank you very much.
Thank you for making this series! The best explanations I've seen of this notation :)
. . . . . . . ** . . . . . . . . **
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?
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
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 ! )
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.
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.
Great video! Would you be willing to share your manim code for the animations?
Hello! Thank you for watching.
Each episodes is animated within a Jupyter notebook. I’m not sure where I would share those to be downloaded, but if you shoot me an email, I would have no issue sending over the jupyter notebook.
-QuantumSense
Just a small question: WTH?
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!
WTF , why your vedios are incredibly great and so exciting !!
Please continue sir
If the coefficient to a ket is indicative of the "probability" of an outcome, what does it mean for it to be a complex number in general?
These videos are just the best of the best, thank you !!
I CAN'T UNDERSTAND THIS VID BECAUSE IS VERY BAD EXPLAINED
These videos are just so well made and memorable, I love it
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.
Hello, thanks for watching.
In this episode, we are only considering the inner product of kets. In episode 6, we go into the bra, and we talk about the dual space, as you say. The inner product and the bra are independent objects, and you can have one without the other. The main connection between them is the Reisz representation theorem, which we talk about in episode 6 as well.
-QuantumSense
brilliant, that's the word to describe your videos
Eagerly waiting for the rest ...
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.
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.
*That notation isn't often used in most algebra courses.*
This is not true. Every linear algebra textbook that I have seen uses this notation somewhere.
i got confused a lot
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
He released a video before the series announcing it and I subbed. Assuming a lot of others did too
I did not understand the Right hand side linear property, can someone help me? I am trying to understand how this is valid expression
Its a definition not an equality.
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?
Because the two is fixed by the E2 on the left and thus the 2 in the delta on the right. Only matching indices will survive the sum, and only two matches the fixed delta index.
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?
A beautiful question. There is almost always a definition somewhere or some other foundational concept. In this case, somewhat amazingly, quantum physics was empirically discovered to operate with complex numbers. When you're first learning a subject, there's so many new concepts, it's easy to gloss over the implications of such a statement. It's huge and far-reaching. In this case, the math was created to work in this strange universe with strange behaviors. Dirac invented this shorthand notation so he could write a line of dense, precise mathematical language instead of 3 long-winded paragraphs to make a point. The answer to your question is: Describing the laws of nature is complex, and ultimately the notation is capturing it's subtleties so it's necessarily also complex, and thankfully (eventually), concise.
@@carywalker7662 Thank you for the reply, I guess this made more sense to me after I read Dirac's writings on the topic. It is interesting to think that complex numbers, something invented many years prior, can so beautifully be applied to understanding the workings of the universe.
Continue the series. Hearty congrats.
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?
Sorry. You lost me at 5:34. Poorly explained or possibly just went too fast.
I am working on it and making a little progress. I guess I need more experience with complex number manipulations.
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?
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
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.
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!
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
you are the best.....keep going .....
What a total legend you are
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.
Why all this was so convoluted before? Well, seems like waiting for you!
a left problem: What inner product does mean in quantum mechanics.
Keep up the good work
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.
So, just like how in relatively you use the metric to calculate the inner product of two vectors, I am 90% certain there is a similar object, namely the epsilon tensor, that you use to calculate the inner product of two state vectors or spinors. However I think this only applies in relativistic theories of QM not classical QM
The metric on Minkowski space is not an inner product. As you say, it can be zero or negative, which an inner product is simply not allowed to be. An inner product must be "positive definite". It is really not even a metric but a pseudo-metric. Which just means we allow it to be negative or zero. Also called "indefinite".
en.wikipedia.org/wiki/Minkowski_space#Pseudo-Euclidean_metrics
You can use similar calculations but the requirements of the result are part of the definition.
And vectors that have faster than light lengths are absolutely relevant objects. For example the "proper length" is the distance between two ends at an object at the same time(in its rest frame). This is not a distance that can be traversed slower than light. But it is a real, physical quantity.
@@narfwhals7843 ah, good clarification, thanks! I should learn to me be more careful with terminology
@@narfwhals7843 Thanks! I understand mathematicians like to have certain useful properties, but I'm not a huge fan of how they seem to often ignore the useful structures that just barely don't qualify. Is there a term for an inner-product like operation that uses a pseudometric rather than a positive definite metric? I think I've heard the terms "quadratic form" and "symmetric bilinear form," but those are both kind of a mouthful. One context I'm familiar with doesn't bother with the distinction and just calls it an "inner product" anyways.
@@angeldude101 It isn't ignored. It's just not an inner product. It is a bilinear form. Meaning it is linear in both arguments.
More specifically it is a "symmetric nondegenerate bilinear form".
Note that the inner product is not necessarily a bilinear form because it is _anti_ linear in the first argument.
But it _is_ a bilinear form on a real vector space because the conjugate doesn't change anything(as is shown at the end of this video). So the dot-product is a bilinear form.
The terms mean different, specific things. So it is very useful to know which is which.
Yes, they can be a mouthful, but it really is about specificity, not aesthetics of terms.
If you are told that the minkowski metric is an inner product and then learn what an inner product is you're going to be really confused by the negative values.
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 ...
I'm not sure I understand all your questions, but a complex number is a scalar. Complex numbers are sometimes represented on a 2d graph, but they're still scalars.
The inner product may be complex, but the square is real.
The kets are vectors of complex numbers. It seems matrices pop up in quantum field theory, but that's beyond my knowledge.
Learning a lot! Thanx! I'm starting to understand the Kroenecker delta! 😊
These videos are very well made. Any chance you could go into QAOA?
Fantastic videos, TH-cam 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 )
Hello! Thank you for watching.
And yes, there is a good reason, mainly being that I will cover bras in their own episode. But more importantly, I wanted to make it clear that the inner product has no connection to the bra, at least in isolation. In Chapter 6, we'll see that bras are their own independent object, and there is a very important theorem (the Riesz representation theorem) that connects bras to the inner product. So you are right, it is indeed a bra, but we will save that label for later.
-QuantumSense
4:03 could this be explained by | a +bi| = (a² + b²)^½ i.e. ((a+bi)*(a-bi))^½ ?
and here I am, binge watching linear algebra for quantum mechanics on a friday evening.
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.
Quantum mechanics, empirically, behaves in a manner that can only be explained with complex numbers.
orthogonal perpendicular {the same}
This is really good
The best, just the best
Why we apply some rule for linearity in order to getting a positive number?
Thank you so much for the videos , it is very clear and concise!!
first
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
The wiki article also mentions that in quantum mechanics it is conventional to use linearity in the second argument.
Really good explanation for a topic I would have considered too dull for a youtube video... great job!
Super
great episode! Really ties a lot of important concepts together!
Excellent explanation for switching a Ket
uhhh what?
This is a masterpiece, thank you very match.
wow 😮
Good work, please continue
this is the greatest video in all of history
Thank you for these awesome videos.
The relationship between bras and kets is technically a little more involved than just complex conjugation.
Hello, thank you for watching!
And yes, absolutely! I dive into this later in the series, where I’ve dedicated a full video into what bras are.
-QuantumSense
@@quantumsensechannel It’s a good explainer series but probably does need a disclaimer that you’re simplifying a lot for didactic purposes :)
can we write =a where v and u are our vector . if no than why ?
I believe it's a*
ty
Sir how i(-i)=1
i^2 = -1 as i = √-1 so -i^2 is simply -(-1) which is equal to 1 that is how it is 😅
At 7:14 knowledge hit
3rd
awesome!
At 3:23, is the issue not that the ket is imaginary?
Quantum physics, by experiment, works in complex numbers. Our notations must therefore support them.
amazing!
Wow！！！！！！！！
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.
this was the hardest part to understand for me, I didn't understand but i will continue to watch for the rest