Ch 5: What are Dirac deltas and wavefunction inner products? | Maths of Quantum Mechanics
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- čas přidán 6. 01. 2023
- Hello!
This is the fifth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll dig into what the Dirac delta is and how we should understand it as physicists. We'll then use it to show where the inner product for wavefunctions comes from.
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
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By the way, if anyone would like to try and prove that sin(x/a)/(pi*x) is a Dirac delta in the limit, here’s some intuition into how we would prove that (and a quick mathstackexchange search reveals a multitude of rigorous proofs).
You might have seen that the Dirac delta is equal to the derivative of the Heaviside Step Function H(x), in the sense that it satisfies the integral property of picking out a certain function value. This is pretty easy to prove using integration by parts. So in other words, the *antiderivative* of a Dirac delta is a step function. Keep this in mind.
So here’s the idea: instead of proving that sin(x/a)/(pi*x) approaches a delta function in the limit, try proving that its antiderivative approaches a step function. Now you might notice that sin(x)/x has no elementary antiderivative, but that’s ok, just write the antiderivative as the integral from -inf to x of the limit (which you remember is the antiderivative via the fundamental theorem of calculus, right?). Now, show that taking the limit gives you a step function (this is the hardest part, but use the fact that the integral from -inf to +inf of sin(x)/x is equal to pi).
Great! Now to prove that sin(x/a)/(pi*x) is a Dirac delta: integrate a function against it, write the limit as the derivative of its anti derivative, you just showed that in the limit the antiderivative is a step function, so now you have the integral of the derivative of a step function times your function, which is easy to show gives you the Dirac property.
Now this isn’t the most rigorous proof, but it gives you an idea of what’s going on. In a way, what matters most about a Dirac delta is the fact that its integral equals the step function, even if the way we get to that step function is weird (like with sin(x/a)/(pi*x)). And a really cool way to “construct” Dirac deltas is to start with a function that approaches a step function in the limit, then take its derivative.
Anyway, this was a bit of a long comment, but I figured you might appreciate an outline of why this weird limit gives us Dirac behavior. Let me know if you have any questions!
See you all next chapter!
-QuantumSense
What do you think your thesis will be on?
@@FermionPhysics
Hello! Thank you for watching.
I work on research related to experimental dark matter detection. In particular, our group is further developing research into kinetic inductance detectors and their use in future dark matter detection experiments. I also spend quite a bit of time working on the upcoming SuperCDMS SNOLAB dark matter detection collaboration. So my thesis topic will likely be some combination of my work in those two arenas! I’m always happy to talk more about my research, so shoot me an email if you’d like to learn more!
-QuantumSense
Check the Dirichlet Kernel…. Another distribution that becomes a Dirac and closely and bit mysteriously related to the Sinc function … and the Dirac infinite gravy train…
Honestly, I am shocked by how useful this series is for both experienced and inexperienced physicists.
Though I know all these individual aspects of QM which you are presenting, I have never seen them sown together to form such a seemless and compact narrative. I
It is also visually stunning! Great job! If you keep this up, I can see you becoming the "3Blue1Brown of theoretical physics"!
P.s. I like that your backgrounds are dark, because I often watch videos late at night 😉
LET'S GO NEW VIDEO
CZcams has lots of math channels but very little good physics channels. You and 'Physics with Elliott' are top shelf. I know the videos take a long time to make with such richness and quality, but please keep them coming!
Eigenchris is also very good.
@@koenvandamme9409 never heard of him, I'll check it out. Thanks.
@@jamesbentonticer4706 you have to! Eigenchris is top tier imo, great way to self teach on stuff like Tensors, Diff Geo and General Relativity
Physics is relative. You probably can't appreciate some of the more complex physics videos simply because your knowledge base is still undergoing inflation.😉
Physics explained is also top tier
I’m so excited for this series to continue. Thanks for the rapid upload
0:00-Recap
0:38-Dirac Delta extraction
2:06-Nature of Dirac Delta
4:33-Problems with spike interpretation
7:34-Continuous orthonormal basis inner product
Glad to see that you are keeping this project alive!
He has already made all the videos and that's what makes me sure we are going to see this series to the end
Everything in the video (content, clarity, audio, visuals) is perfect. Honestly I have no suggestions to make.
Loving the quality, depth, and ofc upload interval. Very excited to see the rest of the series :)
Great video! I've not seen much that aptly explains this, been brushing up on quantum mechanics recently and this is a real gem! Wish I had this when I was learning it...
This is series is really nice, I think its all starting to come together
Your lessons are very encouraging to me! 😊
Awesome. I really look forward to the next video.
I found this series yesterday, 4 AM, 3 F local temp in NY… Really usefully… Full of insights that may surprise many people studying this for years, including me.
Excellent series. Please keep them coming in.
Nice videos so far, please keep going -- definitely good and helpful content!
Ingenious presentation !
beautifully done
Really Helpful ! Great series...
Looking forward to your next video.
Thank you so much for these videos !
Oh my God
After so many years it finally clicked in my head. (2:42)
Thank you for the awesome explanation and visuals
Really high quality explanation, thank you
I recommended this series to my professor :P
Very good work!
There are channels on education with high quality content but no consistency and low quality with high consistency but both only I saw only with this channel
Phenomenal explanation keep it up sir love you from India🇮🇳
great explanation
Yep, definitely gonna subscribe to that channel.
Amazing
great content
Very good video, thanks :)
Im curious to know, how you decided to do this series?
I mean, what map or curriculum did you use. Its so gooood!!!
Weldon ....keep going...
Love it
The δ function is a functional which maps each function f(x) onto the number f(0).
The δ function is not an element of a Hilber Space but it may act on Hilbert Space vectors if the Hilbert Space is defined as a set of square integrable functions.
Yes, that is pretty much what it all comes down to.
Thank you. Physicists really complicate their lives. If only they took the time to understand Lp spaces and a bit of functional theory...
@@nestorv7627 Quantum physicists do in fact understand Lp spaces. There is just no point in actually treating the Dirac delta as a functional, in practice.
@@angelmendez-rivera351 nope.
A standard inh. diff. equ. in math. phys. may be:
L(f/dx)y(x) = f(x),
f.i.:
L: some "Hamiltonian",
y: state function
f: some coupling
1) solve: LG(x) = δ(x)
G: Greenfunction
2) Perform Fourier trans. of above
L(k)G(k) = exp(ikx)
L(k): polynomial in k
3) G(k) = exp(ikx)/L(k)
4) find G(x) by inv.Fourier transf.
5) now, for the solution we have:
y(x) = G(x)*δ(x)., convolurion of G with delta.
#####
Lp and their dual spaces is NOT enough.
Above:
L(d/dx)
God I would have LOVED to see these videos back when i took QM1
I feel like the “big spike” interpretation doesn’t fail with sin(x/a)/(pi*x) either, because the image of points around the origin tends to a null measure set and so are “negligible” i.e. if you integrate on a finite interval that doesn’t contain the origin, the limit tends to zero!
Correct me if I said something off
clear and concise video, one of the best ever seen on the web.
One question: why delta(2.71 -x) but not delta(x - 2.71) ? I suppose delta(2.71 -x) and delta(x - 2.71) have different meaning.
Just in case you still don't know the answer; He explains this in the video. Basically, it is irrelevant whether you use x-a or a-x; the result should still be the same.
Brilliant. Can you also talk a little bit about rigged Hilbert spaces ?
A good angle to see QM.
I think to be consistent with the notation you have been using you should have either put the psi(x) inside the bra, or, having it outside have made it a conjugate. The way you presented it, the conjugate appeared out of nowhere.
Hello, thank you for watching.
Yes, I think I agree completely. In truth, I debated my notation in that section quite a bit, only because I had not yet discussed the bra (which I go into next episode), so I didn’t want to present it as a linear combination of bras. So I figured I would try and represent the left side of that inner product as a linear combination with normal coefficients, introducing the complex conjugate through anti linearity of the inner product. I agree that it may be somewhat weird, but I wanted to avoid presenting something that we hadn’t introduced yet.
-QuantumSense
@@quantumsensechannel But you have introduced
Hello,
Ah yes, I see what you mean. I think I had decided that it was too confusing to have something like “dx |psi(x) x>”, since the x inside psi and the x in the ket are labeling different ideas. I 100% agree it is an imperfect notation on my part, but at the time I felt it was a good compromise (although in retrospect, there’s probably a way I could have worked around it).
-QuantumSense
@@quantumsensechannel It is just a minor quibble, but when trying to explain things to people the first time, it can cause them to have trouble with understanding what is going on. You might want to add a comment to clarify what you meant to avoid confusion.
I know you said you are going to talk more about operators in later chapters, but will you talk about the Poincaré algebra?
Would you be interested in doing a series on QFT?
try watching quantum physics professor Jean Bricmont on Bell's Inequality. He debunks QFT. thanks
How could one practice these concepts? Could you give "homework assignments" or just some physics-maths tasks to help solidify our understanding? Thank you!
“If you can explain what you have learned …you are SPECIAL !! “That’s what you are brother
Can you add topics like Lie Group, algebra, perturbation, commutator / operator, Baker-Campbell-Hausdorff formula, etc ?
Im in the last semester of my bachelors in physics currently. Your video series gives many questions I had that I found some loose answers to a more well-formulated answer, and I truly appreciate it!
I have some more questions I asked myself, that would maybe be worth being turned into videos:
Why are commutators important?
Why can two commuting operators be diagonalized simultaneously?
Why should the angular momentum operator be defined the way it is?
Hello! Thank you for watching.
The first two questions of yours already have a dedicated chapter on my channel. As for the last question, we will answer this after we derive the schrodinger equation. Glad to hear you’re enjoying the series!
-QuantumSense
@@quantumsensechannel Oh, i didnt check the playlist too well, my bad.
I have a further question, not directly related to the video series but more towards the choice of applied vs theoretical physics. Which path did you take? And why?
Hello,
I am currently getting my PhD in experimental physics. Although I love math and the theory behind how we describe our universe, I prefer the variety of work I get to do in experimental physics. Depending on the day, I wear the hat of a mechanical engineer, electrical engineer, nanofab engineer, computer scientist, machinist, data analyst, statistician, etc - It feels like every day of my PhD I am doing and learning something new, and that is so rad, and makes me excited to go into lab every morning.
And in the evenings, I get to indulge in the theoretical math by working on videos like these. Theoretical physics research is much more rigorous and thorough, and I think I prefer it more as an evening hobby.
Of course, theorists would argue the opposite! And I know many who would hate the thought of having to solder a circuit or design CAD models for hours on end. So to each their own; part of the joy in physics is finding out which niche allows you to express your passion most brilliantly. But as long as you have a passion for physics, I guarantee there is a niche here that will support it.
If you have any other questions about my career in physics or my PhD, let me know!
-QuantumSense
@@quantumsensechannel Are you in the semiconductor industry by any chance? Im currently working on my bachelors thesis, trying to improve medical MRI, and most of the things im doing for that is computational physics (EM simulations, Bloch equations) - i found myself liking that kind of work a lot, since it combines math, physics and IT which I all like a lot. I dont mind soldering, in fact I like electronics a lot - im mostly interested in semiconductors for that part.
I currently have to decide between doing a theoretical or an applied master, and I'm really unsure which path I should choose. All of the things I mentioned so far are applied physics topics, but I know some people that are doing the theoretical master at my university and QFT and general relativity do sound interesting as well...
Do you have experience with those topics as well? And how relevant are they in applied physics? Would it be possible to do computational work on quantum mechanical topics, like perturbation theory, or is that field studied enough already?
Im really worried about making the wrong choice, because I dont want to end up in a job I dont like, and my biggest wish is to have an impact on the scientific community, making some breakthrough in a field that I think is interesting
The thing you brought up about doing things for a hobby really inspired me though, I've always had projects that I follow for myself, not linked to university, and you telling me about it made me realized that I will always be able to keep having my own projects, so I could study general relativity and so on in my free time, even if its not my main dedication - I should have the mathematical background to understand it.
Oh, and another question that popped into my mind - Do you have a discord server for this youtube channel?
"Physicists make loose justifications all the time!!" - Quantum Sense
Woah Bro 🤣👌
Is there a good book with exercises, you recommend?
Seems that the delta function should equal 1/dx when c-x=0. This links to the dx in the integral allowing us to cancel it out regardless of the size of dx. It also allows us to do math since infinity ♾️ is not an object one should treat like a number.
i always imagined the dirac delta to be 1 at one point is there something wrong with this? defining it as infinity at one point is looking for trouble imo.
on second thought, maybe in my definition it's not the derivative of the heaviside function, idk
@8:48 Does wave function of 'x' need to have conjugate? I can see that it is not inside of the ket. So why does it need to have conjugate?
I know you touched on it in an earlier video but I still don’t quite understand. Can you elaborate on why “dx” comes before the integrand rather than after?
Hello, thanks for watching! I’m pasting my response to a similar question in another video:
In truth, there is no real reason to put the dx in front or in the back, or even in between - at the end of the day, it’s notation.
That being said, theoretical physics books tend to follow this notation of putting the differential in front. One reason is that it immediately makes it clear what you are integrating over. In physics, sometimes we integrate over a lot of variables (sometimes infinitely many, as we’ll see when deriving the quantum path integral!), so putting the differential in front helps you keep track of what you’re integrating against.
Another more satisfying reason is that it helps enforce the idea that integration is almost like an operator. The integral sign and the dx together form one object, and you “apply” it to a function to get a number , “apply” to a function to get a Fourier transform, etc.
Again, it’s just notation so don’t get caught up. But it’s used a lot on upper level physics, so it’s worth getting used to.
Thanks again, hopefully this sort of answered your question!
-QuantumSense
One question regarding notation: Since inner products are defined for vectors, what is the meaning of the inner product between two scalars x_i , x_j shown above < x_i | x_j > = dirac_delta(x_i - x_j) at video time instant 8:01 ? Further to this question, at video time instant 1:24 above, the right part of the equation shows an inner product between scalars: . Again, is not inner product defined for vectors? For scalars, would not the result then equal 2.71*x to be consistent with defintion of products?
Could the big spike interpretation be valid if we consider that over a certain region when x ≠ y , we choose such a region such that the integral of this region is zero. And that these regions cover the domain of the delta function except when x=y ? That way could we interpret any dirac delta function to change into a big spike function? And justify this by stating that as we are dealing with orthonormal basis, it wouldn't matter what we do with the dirac function over any other region than x=y.
So for the dirac delta for the fourier, could we interpret the other sin like waves outside of x=0 to cancel each other and get a big spike function ultimately.
So we could have our cake and eat it too. I hope my abstract ideas are clear... I don't know how to state it rigorously
This has been an absolutely brilliant series so far, but I'm afraid this episode completely lost me. I had never heard of the Dirac Delta before landing here, and unfortunately by the end of the episode I wasn't much enlightened. It seems I'm missing some pre-requisite background. There are better introductions to the Dirac Delta elsewhere. (Such as Parth G.) I'll be watching those and coming back. I'm still committed to this series as my way into the mathematics of QM.
I don’t understand why it was necessary to jump through so many hoops. I tend to think of vectors as lists of information. If the items in this list are continuous rather than quantized, shouldn’t we just think of this entire video as getting the 2.71st component of a vector? At that point it becomes obvious that the inner product will involve multiplying functions, one of which is a complex conjugate, and (not summing, but) integrating over the vector space, which I assume has to be the same vector space for both functions. Otherwise, the inner product would mean nothing but nonsense.
Since we are in an orthonormal basis, I think it makes sense to assume normalization is baked in, but if it weren’t, my sense is that we would have to divide the function by the square root of the integral of its square.
Are you making these at the pace that they are coming out or were they saved up for continuous release?
Hello!
These episodes have all been completed in advance, and I am just releasing them at a regular schedule. I’ve been working on this series for over a year and a half in my free time, since the animations and script writing do take a bit of time.
Thanks for the support in watching the video!
-QuantumSense
@@quantumsensechannel Yah, that makes sense, these are really high quality and just in time for me as I start quantum next semester.
I had a quick question if you have the time:
When we take = 𝚿(x,t) ,where S(t) is the wavefunction free of basis, what does "x" mean in this context, I understand what the inner product does generally and I understand that we are taking S(t) into the basis of position eigenfunctions, but I don't understand what taking the "x component" in that basis consists of intuitively.
@@gollygaming139
Hello, thank you for watching.
If you haven’t watched episode 2, I recommend you do. In that episode we see that the wavefunction is the the coefficient function in front of each position ket. So psi(x) means that psi returns the coefficient corresponding to the position ket x, when we expand our quantum state in the position basis. So you sometimes see things like psi(x) = . This just says the same thing: the wavefunction is what you get when you project into the position basis and get the list of coefficients.
Now when we look at quantum dynamics, these coefficients change in time, hence we get psi(x,t).
Let me know if this didn’t clear it up!
-QuantumSense
@@quantumsensechannel So I guess to clarify my question. I don't really understand what |x> means in the vector space that the wavefunction lives in. Is it a collection of basis vectors, a single vector, etc. I'm trying to visualize the transformation that is occurring when we take the coefficients of the wavefunction in the position basis.
Thank you so much for the help, I watched you second episode and while it was really helpful, it didn't quite clarify this point for me.
@@gollygaming139 hello!
Ah, I see. When we write |x>, we mean to write the single ket (vector) that corresponds to the quantum state of being in position x (whatever x may be). In chapter 7, we’ll show that these position kets actually form the eigenvectors of the position operator. So |x> would be the single eigenvector of the position operator with eigenvalue x. I agree the labeling is weird, but that’s the convention. Let me know if there’s still any questions!
-QuantumSense
I had the fanciful idea (probably wrong for all sorts of reasons) that you could define the Dirac delta as 0^x so long as A^0=1 for _all_ A (so, 0^0=1). I do realize many consider 0^0 to be undefined like x/0. (But my Windows calculator tells me that 0^0=1. 😃) And, obviously, 0^(x-a) to set the unitary value to whatever a you want.
I'm not super experienced with quantum mechanics, but I think this definition wouldn't work. The dirac delta shouldn't be equal to 1 at 0, it should be equal to infinity as, normally, the contribution that a single point of a finite valued function makes to the integral is pretty much just zero. Being equal to one would mean that its contribution would be negligible.
@@MarceloKatayama That's a good point, but I have seen it defined as vanishing everywhere but zero (or some _x-a_ where _a_ is a constant) where it has the value one. Often, it's not integrated but used as a multiplier to allow only certain values through an equation.
But to your point, it's sometimes defined as an "infinitely" sharp gaussian so it can be differentiated or integrated.
How would you define a conjugate function psi(x)*?
Hello, thank you for watching.
psi(x) is defined as a function from the real numbers R to the complex numbers C. Since the output is a complex number a+bi, we can take the complex conjugate which is defined by a-bi. So we take whatever the output of psi is, and make the imaginary part negative.
-QuantumSense
At 8:00, don't you need to add to the right side of the equation an integral sign and a dx? Otherwise, when i = j, the equation gives us 1 = infinity. Adding the integral sign and dx would fix that, so that the equation gives us 1 = 1 (when i = j). Perhaps I either misunderstand or nitpick.
Hello, thank you for watching.
And yes and no. Note that for the continuous orthonormal basis, we no longer have = 1; we instead have = infinity. This is known as Dirac orthonormalization. So in that regard, we wouldn’t have 1=infinity, and everything is consistent.
That being said, what are we to make of = infinity? Rigorously and formally, it means that these continuous states aren’t *actually* in the Hilbert space. Physicists sweep this under the rug using all this Dirac toolwork, but the formal math still tells us that these states spell trouble. There are ways around this (eg expanding our space to instead be a rigged Hilbert space), so it isn’t the end of the world.
Let me know if this didn’t address your question!
-QuantumSense
@@quantumsensechannel Ah, I see. Dirac orthonomalization. Makes sense. Thank you!
LEGGOGOGOG
At 7:56, that is not right. Dirac delta is a distribution function. It could only work if you would integrate the right part from -infty to infty. Also, you are putting constant and a function to equal
At 5:24 I think you meant to write lim(x->0) instead of lim(a->0). The latter would keep the same amplitude at every point
Hello! Thanks for watching.
And not quite, we want a function of x out of this, and lim(x->0) would just give us a point as a function of a. One way to think of it is that as a->0, 1\a approaches infinity. So we are multiplying the argument of sine by a bigger and bigger number, which squeezes the graph as shown in the animation. And also note that the amplitude is modulated by 1/x. Let me know if this doesn’t clear it up!
-QuantumSense
@@quantumsensechannel It doesn't really clears up no. Even if the argument goes to infinity, the sine function is still restricted to [-1,1] so I don't understand why the amplitude is getting bigger than (1/xpi) in the middle. Except if you did some graphic squeeshing and it's not actually getting bigger.
Well now that I think about it, maybe the middle corresponds to x=0 and that's why it gets bigger but there is no x-axis on your graphic so I can't say
Hello,
Yes, the middle of the graph indeed corresponds to x=0. So although sin(x/a) approaches zero at x=0, so does the x in the denominator. So to figure out the value in the middle, you need to take a formal limit as x-> 0. This is pretty easy to do, and you’ll find that the limit equals 1/(a*pi). So as a->0, the point in the middle approaches infinity.
-QuantumSense
@@quantumsensechannel I actually know this limit (I'm graduated in theoretical physics) I was confused due to the absence of x-axis. Everything's fine then !
Yeah i have to agree not that i found it too confusing i think it is best to have some sort of axis at all times😊
More correctly, delta(x) is a distribution instead of a function.
The definition is still lacking. You should've just said that the delta function**al** maps f(x) to f(0)
goofy ahh continuous linear functional on a frechet space
so you are a talking atom?
aren't we all really talking atoms?
@@evilotis01 f*ck
hi, fellow Carbon
Most people?
Walkie-talkie messed up wavefunctions!
🤫😉🙃🥴
this guy just seems to be regurgitating stuff he was told at a high priced university. Seems to know nothing about what these things really mean.
lol why so mad bro
maybe this episode but not the others.
Could you please explain why these physical objects can be described using complex numbers? was it just brute force, or is there some physical background?