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 :-)
@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.
Mainstream mathematics academics have never understood calculus. Only after I came along did they start writing: f(x+h)-f(x) = \int_x^{x+h} f'(x) dx It's not true at all that the second derivative represents an arithmetic mean ("average value" is meaningless nonsense). In the above equation (which is derived in one step from mean value theorem), ( \int_x^{x+h} f'(x) dx ) / h is the arithmetic mean. Similarly, f ' (x+h)-f ' (x) = \int_x^{x+h} f ' ' (x) dx implies that f ' ' (c) is the arithmetic mean of all the ordinates of the function f ' ' (x) in the interval (x, x+h). www.academia.edu/81300370/Mainstream_mathematics_academics_are_arrogant_and_incorrigible_ignoramuses_The_mean_value_theorem_IS_the_fundamental_theorem_of_calculus
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! 🙏
@@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)
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
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.
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! ❤
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.
Don’t stop making videos. You have a gift of making intuitive sense of equations. Don’t be stingy about showing each logical step in the derivation because that is where most explanations fail. It is usually the failure to show explicitly the trivial step that most new learners stumble
This is a very fundamental concept to understand laplace equation in potential flows also. I was just binge watching this video few days back. just two days before, my teacher introduced the laplace equation and asked the meaning of it. I was really able to explain reasonably well and he was happy. Thanks to you, very well put video. My best wishes for your future!
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.
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...
I wrote a comment explaining how I understand the derivatives. Well, I suggest you to read my comment but I will give you a hint: f(x)=x around x=0 is useful, one side is negative, the other positive, f(x)=x² around x=0 is useful because it shows only positive and f(x)=-x² only negative, but f(x)=x³ is flat around x=0, very little useful information and when going outside the surroundings of 0 it pretty much behaves as x being one side positive and the other one negative, item with the following functions powers of x.
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.
As a mech. engineering student I was busy enough learning just how to apply calculus rules to solve problems, but it's good to be able to see an explanation of math. fundamentals like this. There were some problems which could only be solved numerically l remember. If we had time we'd write a program for those in Pascal.
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.
Yo I just have to say as a physics major when you mentioned the wave function evolution over time and how as the particle moves outwards the curvature flattens as the second derivative was just an awesome light bulb switch in my brain for me so thank you sir!
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.
Funny enough for me it is the reverse. I like math a lot but I really hated physics because I couldn't grasp it. Physics felt more arbitrary and formulaic than math.
Niceee love the intuition behind the first derivative not contributing towards the curvature i took numerical approximations but this connected some dots
Very cool! I was thinking about how to think about the first derivative in this way and I'm thinking that it's like the average of the points on the positive side minus the average of the points in the negative side. I haven't done the analysis in the same way to verify that but I do really like this alternate way of thinking about derivatives.
14:50 The laplacian ∇²f is negative for positive crests of the wavefunction, so the wavefunction f is changed in the direction of a negative crest. According to the heat equation, this change scales with the wavenumber k = E / ħc. So higher energy -> higher wavenumber -> higher rate of oscillation.
Bro . Although this is a channel for quantum maths but pls do cover such micro but nuanced and important topics of math as well . Like topics of calculus - I think concept of limits and meaning of it's formulas is part of an abstract section of mathematics (at earlier levels of maths ofcourse. otherwise higher theoretical maths is nothing but abstract). Take up other such concepts from calculus , complex numbers like topics . Great video ofcourse. Subscribed ur channel . Cheers. 🎉🎉
In both mathematics and physics, the second derivative plays an important role in describing how things change and how fast those changes are occurring. Here’s a breakdown of what it does in both fields: In mathematics, the second derivative is simply the derivative of the derivative of a function. If you have a function f(x), its first derivative f'(x) tells you the rate of change of the function - that is, how quickly the function’s value is changing at any point. The second derivative, f''(x), tells you the rate at which this rate of change is changing. In other words, it measures how the slope of the function is changing as you move along the x-axis. Concavity: The second derivative helps determine the concavity of a function. If f''(x) > 0, the function is concave up (like a U-shape) at that point. If f''(x) < 0, the function is concave down (like an upside-down U) at that point. If f''(x) = 0, the function might have an inflection point, where the concavity changes. Acceleration: It gives insight into the acceleration of a function’s growth or decay. For example, for a position function s(t) of an object, the first derivative s'(t) gives velocity, and the second derivative s''(t) gives acceleration. So, it tells you how fast the velocity is changing. In physics, the second derivative often describes a physical quantity related to motion or force. Some examples include: Acceleration: As mentioned, the second derivative of position with respect to time gives acceleration. If an object’s position is described by s(t), then the first derivative, v(t) = s'(t), gives velocity, and the second derivative, a(t) = s''(t), gives acceleration. Acceleration tells you how quickly the velocity is changing. Forces: In classical mechanics, Newton's second law states that the force F acting on an object is proportional to the second derivative of its position with respect to time (i.e., acceleration): F = ma, where m is mass, and a = d²s/dt² is acceleration. Curvature of a Path: In a more abstract sense, the second derivative can describe how a curve or path bends in space. In the context of dynamics, it can also be used to describe the "sharpness" of an object's trajectory or how its path changes direction. Examples in physics: Projectile Motion: In the context of a projectile's motion, the second derivative of the height with respect to time would describe the constant acceleration due to gravity (which is a downward pull, represented as -9.8 m/s² near Earth's surface). Electromagnetic Waves: In wave theory, the second derivative can describe how an electromagnetic wave propagates in space and time, governing how fields change and interact. In summary, the second derivative gives us a deeper understanding of how things change over time or space, including acceleration, forces, and concavity, which are essential for understanding both the behavior of functions in mathematics and the motion and forces in physics.
Can you make a series of videos on various interpretations of QM? I have read the Helgoland and I love how Carlo has described the relational interpretation, would love understand the intuitions behind other interpretations!
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.
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!
Certainly gives an insight into what we thought was beyond us. This would help in lots of things_ perhaps even without being aware. The kind of logic is fuzzy but inherently available given a lead like this. You don’t need to fully grasp. In fact nobody fully grasp anything as all these are ultimately pointers. U r close to Eisenstein but not exactly there. Thank you genius
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 :)
Interesting! But why then would the gaussian spread out over time? Uncertainty principle means when the particle is localized in space it has access to higher momentum states. You're correct that the mean of the distribution is the classical momentum but you can't separate KE from momentum so I believe that yes indeed it's possible for the particle to have a large KE do to the uncertainty principle.
Where the first derivative is a tangent telling you the rate of change like the shift in change of state. The second is secant, a measure of curvature. In Hooke's law it focuses value from the field into the spring. If you are talking energy from the field subject to weak mixing, that angle applies to the secant to establish the focus of position=mass. Equilibrium for a set is defined by its curvature.
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.
O my Allah,,,what an explanation! Thank you brother for your hard work. the people who represents physics in a meaningful way, i respect. May Allah grant you.
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
My discord friend had to leave Manipur as well, I pray for you all. It’s stupid senseless violence, same story that has happened a thousand times before all over the world, lil details change but it’s the same group identity issue.
Thanks for the great explanation! You won't get any further in maths if you don't have an intuition for its laws and theoremes, which makes your video especially useful. Shame most manuals in maths don't have this policy being overly formulaic at the cost of intuition. P.S. I'm only slightly confused by you wishing us a quantum day, a superposition of which two states is it supposed to be? haha!
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?
Then what about the second time derivative in Newton's Second Law? Could this provide some insight into why we don't usually go beyond the second derivative of position in many physical laws?
That's a good question. It probably has more to do with the fact that at an introductory level, there is no need to go beyond the 2nd derivative, because there are worlds of applications of concepts that do. However, it's not true entirely that we are limited to 2nd derivatives in real applications. There are reasons you want to consider the 3rd derivative, since it governs your reaction to changes in the forces that act on you. Abrupt changes in the 2nd derivative are very uncomfortable to experience, which we call jerk. We also call the 3rd derivative of position, jerk, to account for it. Roads and roller coaster tracks are designed with the 3rd derivative in mind, so that they don't abruptly change curvature, but rather ease into the curve. The fundamental laws we are accounting for in this application, only really go up to the 2nd derivative. But consequences of these laws, can arise from the 3rd derivative. Another application of higher order calculus, is Euler beam theory. The derivative chain proceeds as follows: (0th derivative) elastic curve y(x), (1st) slope theta(x), (2nd) curvature kappa(x) and bending moment M(x), (3rd) shear force V(x), (4th) distributed loading w(x). This again, is really just a compounding of multiple physical laws that only depend on up to second derivatives, so it may not be what you have in mind. There is a physical law that does depend on the 3rd derivative of position, called the Abraham-Lorentz force. This is a body's reaction to emitting electromagnetic waves, as a result from interacting with its own motion.
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.
Man, I was afraid that you were gonna forget about the heat equation. Using this reasoning it just means "the temperature at a point wants to approximate that of the surrounding points", as in a cold point surrounded by hotter points will get hotter. I think it is the absolute best example of this, because once you explain it like that it becomes trivial.
What programs do you use to create this video? And also what mic do you use ? Such a clear sound. I'm looking to make a science course, but I'm a total still in the discovering phase of how everything works.
About the second exercise (I could be royally wrong): in QM, particles with higher energies (hence, frequencies and smaller oscillation time) have lower wavelengths. That means, due to smaller oscillation times the uncertainty in time is low and that in Energy is high. That entails that if we apply the energy operator again, we will get an overall negative value... I was thinking of connecting both the Heisenberg Uncertainty relations together but something just doesn't commute. Would love to know the explanation...
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 :-)
@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.
@user-ky5dy5hl4d Intuition is what idiots use. Look up that word!
Mainstream mathematics academics have never understood calculus.
Only after I came along did they start writing: f(x+h)-f(x) = \int_x^{x+h} f'(x) dx
It's not true at all that the second derivative represents an arithmetic mean ("average value" is meaningless nonsense). In the above equation (which is derived in one step from mean value theorem), ( \int_x^{x+h} f'(x) dx ) / h is the arithmetic mean.
Similarly,
f ' (x+h)-f ' (x) = \int_x^{x+h} f ' ' (x) dx implies that f ' ' (c) is the arithmetic mean of all the ordinates of the function f ' ' (x) in the interval (x, x+h).
www.academia.edu/81300370/Mainstream_mathematics_academics_are_arrogant_and_incorrigible_ignoramuses_The_mean_value_theorem_IS_the_fundamental_theorem_of_calculus
That's okay. I always go around in a state of mild confusion.
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)
Return of the King
yes sirrrr W
The two towers >:)
😂😂
Yes🎉🎉🎉
yep
You're back!
Edit: Changed the course of history from talking about his back, to the fact that he is back. You are welcome.
In college/university, professors/TAs regurgitate a lot without much explanation! Thank God there is TH-cam and channels like this!
I couldn't agree more
If you read all the book and do more problems outside what is assigned, this kind of intuition will come. Your professors can't force you to think
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
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.
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! ❤
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.
You're content has really motivated into learning more about quantum mechanics and physics. Keep doing what you're doing!
This is one of the finest educational videos I've ever come across! Please never stop making them!!
The video content was quite insightful! Thanks for the upload. I hope you'll continue to do so in the future.
Welcome back bro
HE'S BACKKKK
Don’t stop making videos. You have a gift of making intuitive sense of equations. Don’t be stingy about showing each logical step in the derivation because that is where most explanations fail. It is usually the failure to show explicitly the trivial step that most new learners stumble
YOU’RE BACK!!! This is what we’ve all been waiting for, welcome back king 🙏🏻
As a physics major, you are carrying my ass through QM and modern physics.
Cheers! You’re amazing!!
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
got 9.1/10 so was up?
This is a very fundamental concept to understand laplace equation in potential flows also. I was just binge watching this video few days back. just two days before, my teacher introduced the laplace equation and asked the meaning of it. I was really able to explain reasonably well and he was happy. Thanks to you, very well put video. My best wishes for your future!
really glad you returned , i was really fed by watching your videos on repeat , finally some new content
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.
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...
I wrote a comment explaining how I understand the derivatives. Well, I suggest you to read my comment but I will give you a hint: f(x)=x around x=0 is useful, one side is negative, the other positive, f(x)=x² around x=0 is useful because it shows only positive and f(x)=-x² only negative, but f(x)=x³ is flat around x=0, very little useful information and when going outside the surroundings of 0 it pretty much behaves as x being one side positive and the other one negative, item with the following functions powers of x.
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!
Intro with a Home song? You already got me sold!
Phantastic video! Never saw such a clean and straight-forward explanation on the 2nd derivative!
Thank you! What a great video! Multiple insights and new visualisations.
your narrative style is absolutely captivating!
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.
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 mech. engineering student I was busy enough learning just how to apply calculus rules to solve problems, but it's good to be able to see an explanation of math. fundamentals like this. There were some problems which could only be solved numerically l remember. If we had time we'd write a program for those in Pascal.
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.
We need more channels like this! Subscribed
What an elegant and simple explanation of the second derivative. Subscribed.
Yo I just have to say as a physics major when you mentioned the wave function evolution over time and how as the particle moves outwards the curvature flattens as the second derivative was just an awesome light bulb switch in my brain for me so thank you sir!
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?
Funny enough for me it is the reverse. I like math a lot but I really hated physics because I couldn't grasp it. Physics felt more arbitrary and formulaic than math.
Hope there‘s a lot more to come from your channel! Love your work!
mate youve killed this video! Such a complex idea explained so concisely
Niceee love the intuition behind the first derivative not contributing towards the curvature i took numerical approximations but this connected some dots
wow i just found gold(en content) in this channel! thank you so much keep making more this is amazing
Very cool! I was thinking about how to think about the first derivative in this way and I'm thinking that it's like the average of the points on the positive side minus the average of the points in the negative side. I haven't done the analysis in the same way to verify that but I do really like this alternate way of thinking about derivatives.
14:50 The laplacian ∇²f is negative for positive crests of the wavefunction, so the wavefunction f is changed in the direction of a negative crest. According to the heat equation, this change scales with the wavenumber k = E / ħc. So higher energy -> higher wavenumber -> higher rate of oscillation.
Bro . Although this is a channel for quantum maths but pls do cover such micro but nuanced and important topics of math as well .
Like topics of calculus - I think concept of limits and meaning of it's formulas is part of an abstract section of mathematics (at earlier levels of maths ofcourse. otherwise higher theoretical maths is nothing but abstract). Take up other such concepts from calculus , complex numbers like topics .
Great video ofcourse. Subscribed ur channel . Cheers. 🎉🎉
Liked and subscribed for glorious intro alone.
Never heard this way of thinking about the 2nd derivative, provides great insigt, thank you.
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
In both mathematics and physics, the second derivative plays an important role in describing how things change and how fast those changes are occurring. Here’s a breakdown of what it does in both fields:
In mathematics, the second derivative is simply the derivative of the derivative of a function. If you have a function f(x), its first derivative f'(x) tells you the rate of change of the function - that is, how quickly the function’s value is changing at any point. The second derivative, f''(x), tells you the rate at which this rate of change is changing. In other words, it measures how the slope of the function is changing as you move along the x-axis.
Concavity: The second derivative helps determine the concavity of a function. If f''(x) > 0, the function is concave up (like a U-shape) at that point. If f''(x) < 0, the function is concave down (like an upside-down U) at that point. If f''(x) = 0, the function might have an inflection point, where the concavity changes.
Acceleration: It gives insight into the acceleration of a function’s growth or decay. For example, for a position function s(t) of an object, the first derivative s'(t) gives velocity, and the second derivative s''(t) gives acceleration. So, it tells you how fast the velocity is changing.
In physics, the second derivative often describes a physical quantity related to motion or force. Some examples include:
Acceleration: As mentioned, the second derivative of position with respect to time gives acceleration. If an object’s position is described by s(t), then the first derivative, v(t) = s'(t), gives velocity, and the second derivative, a(t) = s''(t), gives acceleration. Acceleration tells you how quickly the velocity is changing.
Forces: In classical mechanics, Newton's second law states that the force F acting on an object is proportional to the second derivative of its position with respect to time (i.e., acceleration): F = ma, where m is mass, and a = d²s/dt² is acceleration.
Curvature of a Path: In a more abstract sense, the second derivative can describe how a curve or path bends in space. In the context of dynamics, it can also be used to describe the "sharpness" of an object's trajectory or how its path changes direction.
Examples in physics:
Projectile Motion: In the context of a projectile's motion, the second derivative of the height with respect to time would describe the constant acceleration due to gravity (which is a downward pull, represented as -9.8 m/s² near Earth's surface).
Electromagnetic Waves: In wave theory, the second derivative can describe how an electromagnetic wave propagates in space and time, governing how fields change and interact.
In summary, the second derivative gives us a deeper understanding of how things change over time or space, including acceleration, forces, and concavity, which are essential for understanding both the behavior of functions in mathematics and the motion and forces in physics.
I never knew that second derivative was so helpful in 3d quantum mechanics kudos to you ❤❤
Can you make a series of videos on various interpretations of QM?
I have read the Helgoland and I love how Carlo has described the relational interpretation, would love understand the intuitions behind other interpretations!
This is an great video. I have a BSc in Mathematics, and I never knew about this
This entire video is absolute gold. Thanks bro.
Nice Video as always!
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.
Great work man :) don't stop to make videos its really helpful !!
Sounds like 3Blue 1Brown- no one else could offer such a intriguing explanation Thank you
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!
I feel so proud of being able to follow your lecture!
… thinking about the Riemann tensor and curvature … enjoyed your entire series … fun actually
No way. I actually understood everything. Thank you man
Superb crystal clear concept on second derivatives and its evolution with examples from physics 🙏❤ congratulations sir🙏🌹
Bro what have you made! Beautiful!
Just what I need when it’s 4am
Certainly gives an insight into what we thought was beyond us. This would help in lots of things_ perhaps even without being aware. The kind of logic is fuzzy but inherently available given a lead like this. You don’t need to fully grasp. In fact nobody fully grasp anything as all these are ultimately pointers. U r close to Eisenstein but not exactly there. Thank you genius
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 :)
Interesting! But why then would the gaussian spread out over time? Uncertainty principle means when the particle is localized in space it has access to higher momentum states. You're correct that the mean of the distribution is the classical momentum but you can't separate KE from momentum so I believe that yes indeed it's possible for the particle to have a large KE do to the uncertainty principle.
6:36 this is what we were taught to find the maximum or minimum value of a function
Where the first derivative is a tangent telling you the rate of change like the shift in change of state. The second is secant, a measure of curvature. In Hooke's law it focuses value from the field into the spring. If you are talking energy from the field subject to weak mixing, that angle applies to the secant to establish the focus of position=mass. Equilibrium for a set is defined by its curvature.
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
Glad you're back.
O my Allah,,,what an explanation! Thank you brother for your hard work. the people who represents physics in a meaningful way, i respect. May Allah grant you.
1 minute in and I’ve already liked and subbed!
thank you so much for your science, hard work and generosity
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
@@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.
My discord friend had to leave Manipur as well, I pray for you all. It’s stupid senseless violence, same story that has happened a thousand times before all over the world, lil details change but it’s the same group identity issue.
Hope you’re well now
wow, what, an upload? big fan
That's fire man! Great lesson
I am 56 and that was a trip to memory lane... Good video
4:35 I always thought that arround = indout. Perhaps they are equal
Thanks 😮. Facinating
8:17 i think, that 6 is just 3! Over there. For 2nd dimen. It will be 2!=2.
U r not wrong
Thanks for the great explanation! You won't get any further in maths if you don't have an intuition for its laws and theoremes, which makes your video especially useful. Shame most manuals in maths don't have this policy being overly formulaic at the cost of intuition.
P.S. I'm only slightly confused by you wishing us a quantum day, a superposition of which two states is it supposed to be? haha!
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?
Awesome video dude. Please what software did you use to make the animations?
Really amazing video
That's what I least expected. Thank you.
Omg the legend is back😭👏
Nice coverage of topic. Thanks. Subscribed. Cheers
10 mins ago? welcome back!
Wowow so much calculus lore!!!😳😳😳 Great video ❤️❤️
3:57 wait how is this a ball in one dimension? What do the axes stand for?
A "ball" in one dimension is just an interval.
Then what about the second time derivative in Newton's Second Law? Could this provide some insight into why we don't usually go beyond the second derivative of position in many physical laws?
That's a good question. It probably has more to do with the fact that at an introductory level, there is no need to go beyond the 2nd derivative, because there are worlds of applications of concepts that do. However, it's not true entirely that we are limited to 2nd derivatives in real applications.
There are reasons you want to consider the 3rd derivative, since it governs your reaction to changes in the forces that act on you. Abrupt changes in the 2nd derivative are very uncomfortable to experience, which we call jerk. We also call the 3rd derivative of position, jerk, to account for it. Roads and roller coaster tracks are designed with the 3rd derivative in mind, so that they don't abruptly change curvature, but rather ease into the curve. The fundamental laws we are accounting for in this application, only really go up to the 2nd derivative. But consequences of these laws, can arise from the 3rd derivative.
Another application of higher order calculus, is Euler beam theory. The derivative chain proceeds as follows: (0th derivative) elastic curve y(x), (1st) slope theta(x), (2nd) curvature kappa(x) and bending moment M(x), (3rd) shear force V(x), (4th) distributed loading w(x). This again, is really just a compounding of multiple physical laws that only depend on up to second derivatives, so it may not be what you have in mind.
There is a physical law that does depend on the 3rd derivative of position, called the Abraham-Lorentz force. This is a body's reaction to emitting electromagnetic waves, as a result from interacting with its own motion.
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.
Our Quantum Sensei is here!!!
Wow, a new video after 9 months. I miss you Bro..
Love to see you using manim
Man, I was afraid that you were gonna forget about the heat equation. Using this reasoning it just means "the temperature at a point wants to approximate that of the surrounding points", as in a cold point surrounded by hotter points will get hotter. I think it is the absolute best example of this, because once you explain it like that it becomes trivial.
Excellent video. Thank you!
Yessss!!!!🤩🤩🤩🤩 These are the BEST videos ever!!!🤩🤩🤩🤩🤩🤩
Great video. Heisenberg's Uncertainty Principle explained with KE. Thanks
YOU ARE BACK!
What programs do you use to create this video? And also what mic do you use ? Such a clear sound. I'm looking to make a science course, but I'm a total still in the discovering phase of how everything works.
About the second exercise (I could be royally wrong): in QM, particles with higher energies (hence, frequencies and smaller oscillation time) have lower wavelengths. That means, due to smaller oscillation times the uncertainty in time is low and that in Energy is high. That entails that if we apply the energy operator again, we will get an overall negative value... I was thinking of connecting both the Heisenberg Uncertainty relations together but something just doesn't commute. Would love to know the explanation...