What does the second derivative actually do in math and physics?
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- čas přidán 13. 04. 2024
- Happy Quantum Day! :) In this video we discover how we can understand the second derivative geometrically, and we derive a few physical relations using this intuition.
Link to the HQI Blog and their Quantum Shorts Contest: www.hqi-blog.com/contest
Derivation of Laplacian equal to average over sphere in 3D: isis2.cc.oberlin.edu/physics/...
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All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:
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Hi everyone! A quick note:
At 7:55 and onwards, there should be a vector sign over the input of the function: f(vector{x}), since now whenever we are talking about 3 dimensions, the input to the function is a coordinate in 3D space. Apologies for any mild confusion!
I remember I used to dislike when my professors would lazily forget to write vector symbols - but years later it seems I have become what I once despised, whoops.
Hope you all enjoyed the video!
-QuantumSense
It's a great video, but perhaps the visual and conceptual leap from 1D, a line plotted on a 2D graph, to a 3D scalar field was slightly glossed over? You covered it with the leap from charge density to scalar field potential but maybe just one more slide and line would have smoothed it over :-)
Very nice. I want to say that intuition is one facet one can apply to physics but very tough to apply to mathematics. But your explanation is fantastic where I also thought of many things concerning math intuitively. So, I want to say something about your clip with a point inside of the sphere where you call that point as something tangible. I can hold a sphere in my hands such as a basketball but I can never hold a point because a point has no dimension. So, when I see a point in your video I see a small sphere inside a big sphere which may be very misleading for viewers. A point having no dimension quantitatively is appropriately called x naught because is has no value. So, when you take a limit as dx goes to 0 and once the limit is reached we can only imagine that the limit has been exhausted at point zero qualitatively because at that point there is no dimension. And I have always thought that such points should have a separate notation for something imaginable and not real such as the wave function psi which is not a real wave. So, that's what my intuition tells me about points. Also, in your video you state at 8:58 minute that you showed the 3D case about the second derivative where the first case was in 1D. No, the first case is in 2D because you operate as x and f(x) which means you show a function in 2D displayed on x and y axes.
@@user-ky5dy5hl4dAgree with you. If I may suggest: Intuition is a guide to imagination of how the reality exists. Imagination is each person's view, and when we all concur using the precision of mathematics, then we are realigning our imagination to reality with precision. And when we accept internally this as TRUE, it becomes our intuitive perception, and an almost perfected view of reality. Then we take another step forward. It is why mathematics is precise, but Intuition is still learning based on existing knowledge.
@@raajnivas2550 Intuition + logic. Agree?
@@user-ky5dy5hl4d Intuition is what idiots use. Look up that word!
Return of the King
yes sirrrr W
The two towers >:)
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yep
I never understood why there was all this talk in my classes about the second derivative/laplacian being related to an average value, but no actual calculation/explanation was ever provided. Thank you so much for doing god’s work! 🙏
You did an entire physics degree without being shown? Not even in QM? Huh
@@jaw0449 I'm in the same boat actually
@@NormanWasHere452 you should go to your profs and and ask for derivations, then. That, or they’re expecting you to do the derivations on your own. No physics program should ever just give formulas (unless freshman courses)
LET'S GO DUDE. I got an 9/10 in quantum mechanics I thanks to you
how it's only been an hour since the vid's upload
@@aquaishcyanother videos
nice profile pic
@@squidwarg you too
Thank you! For this very clear and intuitive explanation.
This view really helps seeing the very deep philosophical connection to notions and axioms of locality in mathematical models. And it also makes the connections between wave equations and continuity equations very intuitive! ❤
The video content was quite insightful! Thanks for the upload. I hope you'll continue to do so in the future.
This is one of the finest educational videos I've ever come across! Please never stop making them!!
your narrative style is absolutely captivating!
really glad you returned , i was really fed by watching your videos on repeat , finally some new content
Welcome back bro
Thank you! What a great video! Multiple insights and new visualisations.
Great! The Schrödinger equation is postulated in many texts and one form to derivate it is using the path integral formalism, but you give a good argument about why it have the form that we know.
Hope there‘s a lot more to come from your channel! Love your work!
We need more channels like this! Subscribed
YOU’RE BACK!!! This is what we’ve all been waiting for, welcome back king 🙏🏻
You're back!
Edit: Changed the course of history from talking about his back, to the fact that he is back. You are welcome.
HE'S BACKKKK
This is such a great video, can't believe I've never looked at the second derivative like this. I'll definitely go and watch your series on quantum!
I've already read about how Laplacian can be interpreted as the difference between a point and the average of its vicinity, but your visuals nicely complement that picture. Nice work!
I think that's true if all second derivatives. After all, that's all a laplacian is. If I remember correctly, with scalars there is only one meaningful second derivative, but for vectors, 3 can be formed by permitting curl, div, and grad.
free education for a guy like me who can't pursue physics due to the conflict in Manipur and now here in hyderabad getting a free education for ba course hahaha
Adam seekers zindabad
Apostate prophet zindabad 🫡
@@sidheart7447Moron.
You know MIT and other universities offer all courses as open source/free online, right? You clearly have web access and desire to learn.
U don't have to do anything with education, u all have to do is a propaganda.Those who are funding u, will leave u useless after sometime.
mate youve killed this video! Such a complex idea explained so concisely
Excelent video, it really gave me a new perspective on the second derivative. I wonder why the third, and other higher order derivatives are so rare in physics compared to the first and second...
Hi, I found your channel just yesterday. I did check out all your videos. I don't know how to express my love and respect towards you. I'm an undergrad student from Bangladesh. I am really interested in quantum computing. I want to learn more. And your channel seems to be a great resource for people like me. Keep up good work.
I feel so proud of being able to follow your lecture!
Never heard this way of thinking about the 2nd derivative, provides great insigt, thank you.
Nice Video as always!
wow i just found gold(en content) in this channel! thank you so much keep making more this is amazing
Bro what have you made! Beautiful!
Excellent video. Thank you!
Great work man :) don't stop to make videos its really helpful !!
Glad you're back.
Nice coverage of topic. Thanks. Subscribed. Cheers
THE KING HIMSELF RETURNED! (thx for good video btw)
Fantastic upload, maybe a series on second quantization in the future like your first one on QM?
As a physics major, you are carrying my ass through QM and modern physics.
Cheers! You’re amazing!!
Excellent! Thank you.
Thanks for the simplified version of seeing QP
This is an great video. I have a BSc in Mathematics, and I never knew about this
Love to see you using manim
We missed u bro !! Welcome back
That's what I least expected. Thank you.
No way. I actually understood everything. Thank you man
Sensacional!! Fascinante!!! Congratulations from Brazil
wow, what, an upload? big fan
Please make more and more videos on Physics and Math.❤️
Nice bro , that was actually great (also inspired me to create a video on some qm topic )
Thanks bro
Keep making these type of videos
Honestly I hate math, mostly because I was forced to cram formulas to pass exams. But this video opened my eyes to the practicality of it, now I love math a little bit more. So thank you, currently binge watching your playlist on Math for QT.
You were forced?
The heat equation is twice differentiated in space and once differentiated in time because it accurately captures the dynamics of averaging over spacetime.
Twice differentiating in space can be intuitively explained by Feynman's ball average approach. The rate of change towards the average is represented by the Laplacian.
I believe that the single differentiation in time is due to the fact that heat changes are only affected by the past. Since the present is not affected by the future, only the rate of change in one direction is considered in time, resulting in a single differentiation.
Look at "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... not Feynman's approach. It was well known before Feynman.
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1 minute in and I’ve already liked and subbed!
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A beautiful lesson indeed.
no way ! was waiting for it
10 mins ago? welcome back!
Finally you came back :)
Wow. I'm not so good with math yet this is insightful. Kudos
Yessss!!!!🤩🤩🤩🤩 These are the BEST videos ever!!!🤩🤩🤩🤩🤩🤩
Loved the video! You are really an amazing presenter.
One thing that I *will* bite the bullet for is calling Laplacian *the real* second derivative in 3 dimensions. The full second derivative is really a bilinear form, also represented as the 3x3 matrix (hessian) of all possible second order partial derivatives, which the laplacian is just the trace of. There are other second order differential operators that you could get from it.
Excellent video
Wowow so much calculus lore!!!😳😳😳 Great video ❤️❤️
He is back!!!!!! 🔥
Welcome back!
Super information thanks sir
Great Video! Appreciate the effort you take in explaining all these things to enthusiasts! Must have been a lot of effort in the editing as well, Could you please tell me which tool/platform do you use to edit videos like these with equations and numbers flowing around the screen? I would love to create something similar very soon!
As a 15 year old.All of this looks so cool!
Our Quantum Sensei is here!!!
Wow, a new video after 9 months. I miss you Bro..
4:35 I always thought that arround = indout. Perhaps they are equal
I can't grasp the physics part coz lack of relating knowledge, but the second derivative part really amazed me, didn't think about how it related with average.
very cool. Thanx !
I now know what happens when I 《f》around and find out. Thank you!
great video !
I like it so much and it's very good.
We missed you!
nice video: I think the big question for a folowing video is this one: How this "averaging" intuition of the 2nd derivative is related to the "aceleration" intuition of the 2nd derivative when time is the studied variable?
Excellent
I think that Feynmann was talking about the Cauchy integral theorem. He stated he didn't need to know the center value just the value on the exterior ball.. that is exactly the Cauchy integral theorem -- you average the surface of the ball and you have the center value
FINALLY HES BACK
This reminds me if my Numerical Analysis class in undergrad...good times!
Interesting take 🙂
The video from Feynman, which one is it? Or what was his lecture about?
super interesting video
You are changing the world ♾️
Ah, another Feynman enthusiast, I see! Really, he was just an incredible person, every person who ever had the chance to be taught by him was blessed.
And, of course, great video, and very much needed for a lot of people who passionately care about these things.
Most physicists admire Feynman second to only Newton himself. He represents the joyful genius and the spirit of scientific curiosity
As I said in another comment, I saw the same concept in "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... so, I don't think was a feynman's idea.
Ancient greeks have used intuition and reasons
If you know space ecuation, the first der is speed and second is acceleration.
So second der is the dynamic of the function.
Your demionstration is excelent model.
Cong.
Nice idea about the average on the ball!
But must correct the misleading idea in the QM part - localized particles in position is equivalent to large uncertainty in conjugate (momentum) space, like you said. But this does not translate to necessarily large kinetic energy. The equivalence principle is for the mean of the distribution, and this would be the "classical" kinetic energy of the particle, which does not change due to variance. This explanation was a stretch, but you could explain this exactly with the diffusion equation, which the Schrodinger equation is just a specific case of :)
This is an awesome video! Thanks for making these!
However, there is one fundamental part that I don't seem to understand, and I was hoping somebody could help me: That is the relation to the ball for 2D: We look at the 2nd derivative at x0, with f_around calculated based on x0 minus dx and x0 plus dx. That all makes sense to me, but that is not necessarily a circle centered on x0 (as pr the illustration at time 4:35). So how does the circle become a part of this?
Grateful for any help.
Got a 2/10 on my second QM problem set. Ended with a 100% on the final and just pulled a 100 on a QM2 midterm! Would love more advanced quantum, but you gave me such a good basis :D
The next step is second quantization - redefining the non-relativistic fixed particle mode to a framework capable of analyzing relativistic many body systems in which the number of particles in a system are no longer fixed. There are quite a few approaches to this, the most common and most utilized framework being quantum field theories appropriate for the different types of fundamental interactions and particle properties.
Extending to the Fock space - the Hilbert space completion of the symmetric and antisymmetric tensors in the tensor powers of a single particle Hilbert space is standard to incorporate creation and annihilation operators of quantum states that change the eigenvalues of the number operator by one, analogous to the quantum harmonic oscillator. Something that becomes more important in QFTs.
You may have already been introduced to some of the fundamental aspects of this approach, as the natural extension beyond a Junior/Senior undergraduate QM course is the introduction of different QFTs, with particular emphasis on QED.
hey, could you tell me what app you use to make these great videos? thx.
Long awaited
Sure, you can connect connect kinetic energy to the position-momentum conjugate duality, but a better way in my opinion is through the derivatives of the action. Because it's well known that the derivatives of the action wrt a variable are conjugate to that variable. but if you really want to intuit the uncertainty principle, you can just look at the Einstein and deBroglie relations: E=h f and p=h lambda, then you realize the duality is baked into these variables maintaining the constancy of the action, h.
Welcome back, on world quantum day!
Just a quick observation:
States in QM are expressed in terms of a complex vector space. Complex numbers permit expression as 2x2 matricies over a Real number Field.
Your first derivative intuition is really just a scaling factor ... the Determinant of a 2x2 matrix gives this scaling factor.
The second derivative intuition is like a divergence ... the Trace of a 2x2 matrix is this (for example SL2(R) Lie algebra is 2x2 matricies with zero trace)
The Schrodinger equation is fine for doing chemistry. However, I wonder if there is utility in building an intuition of the Dirac equation using your intuition approach and the matrix algebra. I wonder if there is a geometric intuition on the Clifford algebra commutator [A,B] and notions of adjoint and self-adjoint. For example [x,p]=ih/2*pi implies a deBroglie wave equation where lamba=h/p ... I wonder if your intuitive approach could give a deeper understanding of the Heisenberg uncertainty principle?
Wicked insight! From feynman!
Even for heat equation, this is the most intuitive tool I've ever used to understand the temperature distribution. What a great explanation. I was wondering how you could understand the Newton's second law using this though.
if the distance an object has travelled in the past dt is less than the distance it will travel in the next dt, it means the object is acccelerating
U got a sub with this one...
I truly wish I knew what he was talking about. We only got up to IROC in high school, so he’s describing a topic that i haven’t even been introduced to.
Thank you for the beautiful video. Just a question: increasing the uncertainty in the particle's momentum simply means that the particle samples a larger subset of its momentum space, and if the Gaussian distribution becomes flatter and flatter, the particle should explore high and low momentum values in the same way and kinetic energy should follow the same trend. So I don't see the link you make between the uncertainty principle and the momentum gain! Did I misunderstand something?Thank you.
The Heisenberg Uncertainty Principle states that there is a fundamental limit to how precisely we can know both the position and momentum of a particle simultaneously. As the uncertainty in momentum increases, the particle's kinetic energy can vary more widely, reflecting the broader range of momentum states the particle can occupy. As the Gaussian distribution flattens, the particle is more likely to explore both high and low momentum values in a similar manner. This exploration of a larger subset of momentum space is reflected in the kinetic energy of the particle.
@@IamPoob Thank you for the answer. Suppose this argument is true. If I consider the test case of an exciton spatially and tightly confined in a quantum dot, the consequence of the argument will be that the kinetic energy of the exciton will increase according to your interpretation of the uncertainty principle, this increase will ultimately lead to a certain renormalization of the energy and the eigenstate of the exciton will change... But this conclusion is false since the exciton remains in its ground state in the spatial confinement of the quantum dot.
6:00 Iff you can take the derivative from both sides, that isn't always possible but often holds in physics.