As a beginner in quantum mechanics, I love how this ties so many things. Extracting coefficients by taking an inner product with an eigenfunction, the kronecker delta, fourier... Really makes the fourier transform look like something very natural ♡
This has been extraordinary for me!!!! I am a 65-year-old nerd who studied chemical engineering and is now trying to understand quantum physics. This is almost as good as gaming!!!!
@@PhysicswithElliot but you don't really have to multiiply by the complex conjugate to get the coefficient so why do that.?.and since I don't think anyone would think of thst anyway..it's not intuitive or obviously logical..so why do it thst way?
@@leif1075 Because sometimes you do have to multiply by the complex conjugate to get a real-valued coefficient, and in quantum mechanics, all such coefficients must be real-valued.
@@aloysiusdevadanderabercrombie8 what do you mean all the coefficients must be real valued..for the ones thst deal withvreal variables like momentum or velocity you mean..but what about those that Don't have to be real?
@@leif1075 Ok first let me ask you which coefficient you're referring to? I think I had assumed you were referring to a different coefficient. Perhaps you're actually referring to the psi_n coefficient? If so: The reason that we need to multiply by the complex conjugate in this case is so we can simplify the integral on the right side to 0 when n =/= m and 1 when n = m. This result is called the Kronecker delta, and it's important because we need the states with the same index to interact and states with different indices to cancel. It's only by multiplying by the complex conjugate that you can obtain this result. The exponential function that he's working with, e^(inx/R), is actually itself what is known as a set of eigenfunctions. Say you take some operator B̂, and you use it on e^(inx/R). If your result is then be^(inx/R), where b is some constant coefficient, then we say that e^(inx/R) is a set of eigenfunctions of the operator B̂. This means that when you measure the observable associated with the operator B̂, the result you get will always be associated with the state e^(inx/R) where n = 1, 2, 3,... However, the state of a quantum system (the wavefunction) can be _any_ well-behaved function, it doesn't necessarily have to be an eigenfunction of some operator. In fact, when we choose different operators, oftentimes we obtain different eigenfunctions, because quantum systems will have different possible measurements for different incompatible observables. Luckily, this isn't an issue. We know that the set of eigenfunctions e^(inx/R) form a complete set, meaning that you can represent any well-behaved function as a linear combination of every function in that set. This is why we have psi(x) = sum{psi_n e^(inx/R)}. This means the wavefunction can be represented as [psi_1 e^(ix/R)] + [psi_2 e^(2ix/R)] + ... where psi_n can be any complex number. Now that we know this, say we want to find psi_2. We might want to do this because you can use psi_2 to determine the probability that the result associated with the function e^(2ix/R) will be measured. The process of figuring this out is basically what he laid out in the video. We must have the Kronecker delta result for the integral on the right side in order to make sure we're only calculating the coefficient of e^(2ix/R) and not the coefficient of any of the other eigenfunctions in the set of eigenfunctions representing the wavefunction. This move where we multiply a function by a complex conjugate function and then integrate is a very common one in quantum mechanics, to the point where Dirac actually created an entire notation system to write it quicker. Look into bra-ket notation if you're curious. Hopefully this was somewhat clear? I'm not great at explaining things over a comments section.
Very well explained! I already knew Fourier Transform from my background, which partly entails signal-processing, which also pointed me into the direction of Wavelet Transformation. But I must confess that my intuitive grasp was not yet firm enough as to pretend that I really understood it all. This video is one of those rare gems that support many interested non-academics and engineers out there that are looking for good explanations. Thank you so much!
I have found the Fourier transform videos really interesting lately. I like that 3B1B, Veritasium, and you have all given a unique way to understand it.
This is exactly the kind of physics channel I been looking for all these years as someone who loved math but never went past multi variable calculus. Been trying to teach myself linear algebra to understand these videos better. Seems like that's when a lot of these things start to come together. But i feel I can still understand some basics just at my level.
In our theoretical physics 2 class we actually started out with the Fourier transform and are now working our way towards the momentum space wave functions
Indeed, you have the right idea professor. Most teachers nowadays thinks that understanding a concept requires knowing the tools first, which is totally wrong.. In reality you have to understand the PROBLEM FIRST, and then you start figuring out what each tool was made for.
I took a class on Fourier Tranforms my junior year. Unfortunately our professor, while being a nice guy, was probably was the single worst teacher I've ever had. I left that class more confused and disinterested than anything else. I'm fortunate that videos like this exist. So many years later I've grown to comprehend and enjoy this part of math that initially confused and frustrated me. Thanks for the very informational upload!
Ok throughout my physics career in school, I was never truly able to grasp the reason and understanding of the uncertainty principle this really made since in terms of Fourier transforms. Thank you so much.
THANK YOU!!!! This has summed up about six consecutive months of trying to understand these concepts (and failing) and chipping away at it into something consice and beautiful and easy to understand. Thank you thank you thank you. 🙏🏽
For all of us teaching Fourier methods to engineers for circuits, signal processing and EM this is a fantastic way to bring in quantum mechanical examples as well.
I’m a second year chemistry graduate student and will be teaching quantum mechanics next semester. I will 100% be showing them this video alongside my own video “Demystifying Quantum Mechanics in 15 Minutes” Excellent stuff! I’m glad there are teachers like you in this world. Also, what program do you use to create to your animations. They’re’ beautiful. You’re both an artist and a scientist.
The other way to demystify QM is to eliminate the Copenhagen requirements for nondeterministic particle paths and the "measurement problem" (wavefunction collapse). Both are eliminated by the simple insights of David Bohm in his 1952 paper, only now beginning to be discovered by physicists.
@@david203, In 1952, Bohm recycled ideas discovered by Louis de Broglie in 1927. New Age folks, who like to use words like “quantum” and “energy” to mean “cool” and “spiritual” love Bohm. I don't think many serious theoretical physicists are embracing Bohm's ideas about pilot waves, non-locality, or implicate order. Bohm's ideas about determinism are attractive. It would be useful, if time allowed, to have give lessons on Bohm's and Bell's work, but I would not make either's work the central theme.
“Quantum mechanics demystified in fifteen minutes” seems just a tad presumptuous, if not arrogant. For a century the best minds have attempted to reconcile its implications with what the human mind perceives, and failed. What I mostly see is teachers, authors, and researchers parroting back what others have said, and demonstrating mathematical explanations, without altering the fundamental difficulties with it.
Top-tier physics internet content. I contribute to Elliot's Patreon and I encourage everyone who watches his videos and studies from his notes to do the same. Mainly because I want as much of this content as I can get. Elliot's videos have cleared up many residual confusions I've been living with for years.
As a telecommunication engineer... I was stunned when I 1st saw Schrodinger's wave equation. Stunned The frequency transform is just another form of a probability function. Actually we use the term probability density function & power density function interchangeably in Much of digital signal processing & communications theorems; that is all we care about to do filter designs🖐 your video highlights this and many other good ideas. BTW.Heisenberg's's uncertainty principle..arises well in wavelet theory too edited: by the way... you have explained how when sine waves are integer multiple... they make orthogonal functions believe it or not... this is the core idea behind OF..... 4G LTE technology ✌️ orthogonal is used to reduce noise generated by inter-symbol interference
Back in the 80's I did a lot of chemical spectroscopy. DSP was kind of new. But it was instructional to look at FFTs and time domain signals in almost real time. Writing code for FFT's or actual continuous integrals is also instructive.
@pbajnow that is another exciting field... Wavelet...which i explore extensively years ago... there is an Indian professor from IIT...has an excellent course online on youtube... it is being used in JPEG encoding. Others use it model brain waves and earthquakes.... Multi-rate processing is another rabbit hole...chears
The content is extremely good, but I have to say I'm very happy with all the equations being perfectly formatted. The fact you used \dif x (or \dx iirc from the physics package) instead of just "dx" makes me very happy.
I've been struggling to "remember" the QM that I never really got comfortable with as an undergrad 30+ years ago. RUclips is a wonderful thing, and this video is a great example of why. About half way through, I had a light bulbs moment. I've always been more comfortable understanding Fourier analysis in terms of acoustic or EM waves, but the probability waves of QM never felt natural. Your video builds to Fourier quite naturally but left me pondering, what is momentum in QM? I suddenly realized that it is directly analogous to frequency in sound or EM waves. This puts position in the role of (probability) amplitude. The ah ha was recognizing that momentum of an EM wave is proportional to frequency, and it all came together. I had great instructors and any shortcomings were my own. Still, I keep realizing that in a formal physics classroom, they drowned you in the math and glossed over the big ideas. Either that or they presented the big ideas as a result of 10 blackboards of math with a statement like "so clearly ..." even thought the chalk was still smoldering and I was heads down trying to copy the last three boards. It's leave thinking the proof had probably said something really important, but I didn't quite catch it. I like the math, but appreciate the context as well. Thanks for a great video.
This is because particles are not really particles, but waves. Also there may be nonlinearity in qm due to wavefunction collapse. Composition of waves should break at some point.
Amazingly detailed, clear, and fascinating explanation of how Fourier analysis relates to the quantum wave equation. I never really understood it very well until now.
One thing I would like to add is that the complex polynomial function P(x) = x^n when restricted to the unit disc, is given as P(e^{it}) = e^{nit}. That is, approximating a function on the unit circle with complex polynomials _exactly_ corresponds to the discrete Fourier analysis of the function.
I got your this channel as a follow-up of a good teacher's video. You too done as a classic. The transformation from big Sigma to integral as you highlighted is some what partial because in very other situations one can derive similar relations with out following this formalism. I am eager to see those in your forth coming and other topics. Good things deserves good blessings.
Watching math like this all day, then coming back to the real world where you're reminded that many people still don't understand or fully comprehend mathematics. Immersing yourself in these types of videos makes you forget that many people still don't get these topics and how they apply to their everyday lives. Anyways, thanks for sharing this.
Welcome back and thank you! This was very interesting. When you explained the Fourier series around 5:20, you _made_ the function periodic by demanding it repeat itself outside of the bounds we consider. In the example at the end you took a wave function that is zero everywhere but inside the box. Are we still considering this to be periodic and repeating "behind infinity" or does the periodicity condition not apply when we take the infinite case? It is also interesting to not that this works the other way around. You can get Schrödingers Equation from Hamiltonian mechanics by Identifying the Poisson bracket with Fourier conjugates. Zap Physics did this in their last Let's Learn Physics stream on classical mechanics and it was really beautiful.
Thanks Narf! In the infinite space setup the wavefunction is not periodic, it's zero everywhere outside of that interval from -a to a. The area underneath |\psi(x)|^2 is fixed to one, remember.
@@PhysicswithElliot I'm not sure I understand. Perhaps I missed a step? The construction of the Fourier _sum_ relied on the function being periodic, with periodicity related to R. If we take R to infinity to get the integral that perodicity becomes infinite as well, no? Is the limit of periodicity going to infinity just the same as a non periodic function? Can we make that statement in general?
@@narfwhals7843 Yes the periodicity goes away in the limit. x ~ x + 2\pi R means that we identify each point x with every other point on the line that's shifted from it by 2\pi R. But when we send R to infinity, x isn't identified with any other points anymore. Then the discrete spectrum of Fourier waves becomes continuous, and all those waves interfere destructively outside of [-a, a] to make the wavefunction vanish instead of repeat.
Cannot wait to watch. Really interesting topic, that pops up quite often: In different lectures and an old submarine movie. Hopefully I get a step closer to understanding how to apply it.
I am studying quantum mechanics these days, and I really love your explanation about fourier transform by the system of particle on a ring. Thank you for making great video.
Always thought of the idea that all quantum mechanics courses should first teach the Fourier series and Fourier transform and all that stuff about signals and frequencies and distributions, and only after this they can hit the students with all that unintuitive facts about the double-slit experiment, the uncertainty relations, measurement processes, operators ...etc. And You assured the rightness of that idea.
Agreed. First time I learned about quantum mechanics was in pchem2 course and only because of my professor did I understand the basics. I don't recall the book (Atkins 6E) going into details of Fourier series and transforms. That professor covered some of it. It wasn't until I took a math elective on PDEs and variations calculus that I got into the nitty gritty as an undergrad. By the time I got into grad school, the math became so much easier lol
Thank you so much for explaining these wonderful subjects in such a simple and intuitive way! I would love it if you could try making a video on some of the less intuitive parts of thermodynamics/statistical mechanics :)
Thanks for making clear that momentum and position are two bases for the same vector space. Physicists often talk of “momentum space” and “position space” representations for a particle. But these terms should be replaced by the two above “bases” since they span the same Hilbert space.
Instant sub, this is the kind of crystal clear and intuitive but yet mathematically rigourous and detailed content that I'm looking for when it comes to explaining physics. You are right up there with Physics Explained as far as I'm concerned ! And of course 3B1B for the mathematical aspect. Bless the internet. Thank you very much for your work, I look very much forward to your next videos ! (before that, I have some catch up to do with your previous videos haha)
it is not mathematical rigorous. The eigenfunctions of the momentum operator are not square integrable. The operator is ill defined, thus the concluding statements are also not rigorous. The momentum operator like this has no eigenfunctions which are related to quantum physics. Operators in quantum mechanics are densely defined one the space of square integrale functions such that they have a complete set of generalized eigenfunctions.
@@youtubesucks1885 I should have said "more rigorous than the average". This video is supposed to popularize scientific ideas so it is a given it won't be 100% mathematically rigorous. I think the author is clearly aware of this but intentionally did not mention those deviations in ordre to not get too much in the way of the main ideas. I only had few classes regarding quantum physics so quantum operators are still untamed beasts for me :) Either I wasn't aware of this condition on the eigenfunctions or I forgot about it. Either way, thanks for the clarification. But I remember a lot of physics (not just quantum physics) is based on square integrable functions which is a rather nice condition in general compared to other purely mathematical problems where conditions can quickly and easily become more strict.
I’m sad because I never went beyond Algebra 2, and I’ve forgotten much of that, even. I love learning about the concepts of quantum physics, things like sterile neutrinos and penrose diagrams and Electroweak Symmetry Breaking, but this was just utterly beyond me to follow. Still, I didn’t get *nothing* out of it, and your style and delivery are delightful. Thank you for sharing.
OK I am a newbie in understanding ..but 2:14 seems like a derivative of where I turn out to be according to all the places I could have been? A bit like Humphry Bogart in Casablanca? but it has a dynamical component somewhere?
🎯 Key Takeaways for quick navigation: 00:00 🌌 Quantum Mechanics and Fourier Transform Introduction - Overview of the video's focus on the connection between the Fourier transform and quantum mechanics. 03:16 🔄 Particle on a Circle and Fourier Series - Describing a quantum mechanical particle living on a circle of radius r. - Introduction to periodicity, ensuring the wave function repeats on the circle. - Exploring Fourier series as a sum of Sines and Cosines for periodic functions. 06:40 🔍 Representation of Fourier Series in Exponentials - Introduction to expressing the Fourier series using exponentials. - Euler's identity as a key tool in transitioning between Sines/Cosines and exponentials. - Explanation of coefficients PSI sub n and the significance of wave number K. 09:43 🌊 Fourier Waves as Eigenfunctions of Momentum - Fourier waves (e to the ikx) as eigenfunctions of the momentum operator. - Significance of H-bar times the wave number (k) as the eigenvalue. - Explanation of quantized momentum values due to periodicity on the circle. 13:19 ➡️ Calculating Fourier Coefficients - Introducing the method to compute Fourier coefficients using integration. - Demonstrating the integral process to find the coefficients PSI sub n. - Linking the coefficients to the contribution of individual Fourier waves. 16:29 🔄 Transition to Infinite Space and Momentum Continuity - Discussing the transition from a particle on a circle to an infinite x-axis. - Elimination of momentum quantization as radius (R) goes to infinity. - Acknowledging the continuity in momentum values. 18:56 🔀 Fourier Series in the Infinite Radius Limit - Presenting the Fourier series formula in preparation for the infinite radius limit. - Introducing modifications in notation for the limit. - Simplifying the coefficients for the Fourier series in infinite space. 21:14 🌌 Fourier Transform Basics - The Fourier transform emerges from the Fourier series for a wave function on a circle in the limit where the circle becomes infinitely large. - Each Fourier wave represents a state of definite momentum, and the transform is a superposition of these basic waves. - The transition from discrete Fourier series to continuous integral is explained, highlighting the underlying concept's consistency. 23:06 🔄 Position and Momentum in Quantum Mechanics - The probability of finding the momentum in a window is related to the squared proportion of the Fourier wave, mirroring the position probability. - Position space wave function (ψ of x) and momentum space wave function (ψ-hat of K) are two representations of the same quantum state vector. - The relationship between position and momentum probabilities sets the stage for the uncertainty principle. 25:37 📊 Example: Fourier Transform of a Wave Function - Illustration of the Fourier transform for a particle localized in a position space interval. - Momentum space wave function (ψ-hat of K) graphically explained, showing how different Fourier waves contribute. - Introduction to the uncertainty principle by analyzing the width of position and momentum space wave functions in different scenarios. Made with HARPA AI
Love your video. It explains Fourier transform in such a simple way. Prof instead of using a square position function, is it possible to show us how to get the same result by a more general function.
This is very well made. I have been waiting a long time for a good video explaining this at sufficient depth. Appreciate your hard work. If I can make one request: while describing equations, can you use what the variable represents instead of the letter? For example instead of saying "p = h bar over i" say "momentum equals Planck's constant divided by root of -1 which we denote as i" or something like that. I usually pause the video and think this through myself but occasionally I am not sure what a variable represents and am stuck. The repetition of what variables represent will promote conceptual understanding. This is not a criticism, just a humble request. Your videos are great and I make time in my schedule to ensure I can watch them.
Cool! There are more things that should be said about the Fourier transform and the theory of representations in QM, but it will be more fun with the Dirac's Bra - Ket notation.
The Schrödinger equation cannot be modified to work on curved spacetime. Instead, you need to look into Lagrangian densities that account for curvature in spacetime, and derive the quantum field theories from there.
@@angelmendez-rivera351 im pretty sure you can, like for example instead of using the normal laplacian you use the laplace-beltrami operator (which is basically the laplacian but generalized to (pseudo) riemannian manifolds) but im not sure of the details
As I saw the time running out on the video I honestly got worried that we weren't gonna get to Heisenberg in time 😅 but we got a nice taste of the principle in this video
just now, after ten years of studying quantum mechanics, I can say that I understand what the wave equation means. Please any one tell Fayman about it. , and if anyone is related to Roger Penrose, send him this video, he was experiencing the same problem . Thank you so much.
I like the old background better (the white one with a box that has a grid, and has equations outside of it). I like the new fonts that you are using, though.
This channel is phenomenal, absolutely top-class explanations of such deep concepts. It’s a real treat to see such an elegant presentation of mathematics and physics where the derivations and results follow so seamlessly. I hope y’all continue this channel for years to come, these videos are a gift to the world. This nuclear engineer sincerely thanks y’all for the content, keep it up!👍
I feel like this video was sent to me from the gods. The proportion of people who know a lot about rigorous QM and don't know a lot about Fourier transforms has got to minuscule. This has been an absolute treat
Can Somebody please explain, why we replaced psi_k with 1/√2πR • psî(k). 19:01 Thinking of it, psi_k is a constant whereas psî(k) is a function. How we can exchange them? What am I missing here? Also, 1/R is a sensitive variable here. (Going to be ‰ in further steps) So why (is it allowed to do so) include it into the replacement term??
Hi! You've made an outstanding video which not only explains the Fourier series and integrals in an excellent matter, but also shows the maths behind it. However, I do have a question. At 16:30, how are we able have k under the summation sign and not n=-inf to inf like we had earlier? Thanks!
21:21 isn't this a bit hand wavy? If k depends on R, and R is in the denominator of the exp argument, shouldn't the exponential go to 1? (exp(1/inf)) why does the k just stays there?
I think something that's worth clarifying here is that the form of the momentum operator can be inferred by matching the generator of translations to the de Broglie Hypothesis. You *need* the de Broglie Hypothesis to build quantum mechanics "from the ground up" or otherwise assume uncertainty or the momentum operator axiomatically. I think this is important to note because the de Broglie Hypothesis is essentially an intuitive guess that ended up being instrumental for the success of quantum mechanics.
De Broglie's theory tried to justify a local view of QM, to answer some of Einstein's feelings about QM. Bell's theorem shows that any local view of QM must be wrong on basic principles. Entanglement also shows that a local view must be wrong. If you want to find a more sensible interpretation of QM, look into David Bohm's paper of 1952, which describes QM as deterministic and nonlocal.
@@david203 That's not the point I'm making here. I'm saying that the form of the momentum operator is either taken itself to be axiomatic or derived from the De Broglie Hypothesis along with considering the translation operator's generator. This is not a statement about interpretations of quantum mechanics.
Great explanation as always ! I just had a small question. Is the continous model using the fourrier transform consistent with the observtions of quantized momentum ? How can k be continous if momentum is quantized (discrete) ?
The strong force is said to act over a short range. Does this imply that quarks have a finite range of probability where they could be found and thus potentially have discrete momentum states rather than continuous momentum states?
The probability density curve is the modulus squared, ie absolute value squared, partly because we hope to measure actual quantities, the wave function is complex
This is almost incomprehensibly beautiful on a number of levels.
As a beginner in quantum mechanics, I love how this ties so many things. Extracting coefficients by taking an inner product with an eigenfunction, the kronecker delta, fourier... Really makes the fourier transform look like something very natural ♡
This has been extraordinary for me!!!! I am a 65-year-old nerd who studied chemical engineering and is now trying to understand quantum physics. This is almost as good as gaming!!!!
let me break it down for you.
They just applied statistical mechanics to an atom.
You seriously give 3B1B a run for his money, this is fantastic. I can’t wait to see more!
Thanks Maxx!
@@PhysicswithElliot but you don't really have to multiiply by the complex conjugate to get the coefficient so why do that.?.and since I don't think anyone would think of thst anyway..it's not intuitive or obviously logical..so why do it thst way?
@@leif1075 Because sometimes you do have to multiply by the complex conjugate to get a real-valued coefficient, and in quantum mechanics, all such coefficients must be real-valued.
@@aloysiusdevadanderabercrombie8 what do you mean all the coefficients must be real valued..for the ones thst deal withvreal variables like momentum or velocity you mean..but what about those that Don't have to be real?
@@leif1075 Ok first let me ask you which coefficient you're referring to? I think I had assumed you were referring to a different coefficient. Perhaps you're actually referring to the psi_n coefficient? If so:
The reason that we need to multiply by the complex conjugate in this case is so we can simplify the integral on the right side to 0 when n =/= m and 1 when n = m. This result is called the Kronecker delta, and it's important because we need the states with the same index to interact and states with different indices to cancel. It's only by multiplying by the complex conjugate that you can obtain this result.
The exponential function that he's working with, e^(inx/R), is actually itself what is known as a set of eigenfunctions. Say you take some operator B̂, and you use it on e^(inx/R). If your result is then be^(inx/R), where b is some constant coefficient, then we say that e^(inx/R) is a set of eigenfunctions of the operator B̂. This means that when you measure the observable associated with the operator B̂, the result you get will always be associated with the state e^(inx/R) where n = 1, 2, 3,... However, the state of a quantum system (the wavefunction) can be _any_ well-behaved function, it doesn't necessarily have to be an eigenfunction of some operator. In fact, when we choose different operators, oftentimes we obtain different eigenfunctions, because quantum systems will have different possible measurements for different incompatible observables. Luckily, this isn't an issue. We know that the set of eigenfunctions e^(inx/R) form a complete set, meaning that you can represent any well-behaved function as a linear combination of every function in that set. This is why we have psi(x) = sum{psi_n e^(inx/R)}. This means the wavefunction can be represented as [psi_1 e^(ix/R)] + [psi_2 e^(2ix/R)] + ... where psi_n can be any complex number.
Now that we know this, say we want to find psi_2. We might want to do this because you can use psi_2 to determine the probability that the result associated with the function e^(2ix/R) will be measured. The process of figuring this out is basically what he laid out in the video. We must have the Kronecker delta result for the integral on the right side in order to make sure we're only calculating the coefficient of e^(2ix/R) and not the coefficient of any of the other eigenfunctions in the set of eigenfunctions representing the wavefunction.
This move where we multiply a function by a complex conjugate function and then integrate is a very common one in quantum mechanics, to the point where Dirac actually created an entire notation system to write it quicker. Look into bra-ket notation if you're curious.
Hopefully this was somewhat clear? I'm not great at explaining things over a comments section.
Very well explained! I already knew Fourier Transform from my background, which partly entails signal-processing, which also pointed me into the direction of Wavelet Transformation. But I must confess that my intuitive grasp was not yet firm enough as to pretend that I really understood it all. This video is one of those rare gems that support many interested non-academics and engineers out there that are looking for good explanations. Thank you so much!
I have found the Fourier transform videos really interesting lately. I like that 3B1B, Veritasium, and you have all given a unique way to understand it.
The amount of intuition this video provided was remarkable. Thank you.
This is exactly the kind of physics channel I been looking for all these years as someone who loved math but never went past multi variable calculus. Been trying to teach myself linear algebra to understand these videos better. Seems like that's when a lot of these things start to come together. But i feel I can still understand some basics just at my level.
In our theoretical physics 2 class we actually started out with the Fourier transform and are now working our way towards the momentum space wave functions
Incredible description, incredible graphics,incredible explanation. Simply beautiful. So much work must have gone into this labour of love. Thank you.
Indeed, you have the right idea professor.
Most teachers nowadays thinks that understanding a concept requires knowing the tools first, which is totally wrong..
In reality you have to understand the PROBLEM FIRST, and then you start figuring out what each tool was made for.
I took a class on Fourier Tranforms my junior year. Unfortunately our professor, while being a nice guy, was probably was the single worst teacher I've ever had. I left that class more confused and disinterested than anything else. I'm fortunate that videos like this exist. So many years later I've grown to comprehend and enjoy this part of math that initially confused and frustrated me. Thanks for the very informational upload!
Did we have the same teacher? Very similar experience here ahahah
Amazing to get a deeper understanding of the Fourier transform, very thorough as always. THX!
Glad you liked it!
Ok throughout my physics career in school, I was never truly able to grasp the reason and understanding of the uncertainty principle this really made since in terms of Fourier transforms. Thank you so much.
Outstanding use of visuals, along with how the transitions happen, when the quantities are varied.
THANK YOU!!!! This has summed up about six consecutive months of trying to understand these concepts (and failing) and chipping away at it into something consice and beautiful and easy to understand. Thank you thank you thank you. 🙏🏽
For all of us teaching Fourier methods to engineers for circuits, signal processing and EM this is a fantastic way to bring in quantum mechanical examples as well.
Simplicity from complexity? I would never have believed it.
Very well done !
I’m a second year chemistry graduate student and will be teaching quantum mechanics next semester. I will 100% be showing them this video alongside my own video “Demystifying Quantum Mechanics in 15 Minutes” Excellent stuff! I’m glad there are teachers like you in this world. Also, what program do you use to create to your animations. They’re’ beautiful. You’re both an artist and a scientist.
Thanks Bobby! I did the bulk of the animation in Keynote
The other way to demystify QM is to eliminate the Copenhagen requirements for nondeterministic particle paths and the "measurement problem" (wavefunction collapse). Both are eliminated by the simple insights of David Bohm in his 1952 paper, only now beginning to be discovered by physicists.
@@david203, In 1952, Bohm recycled ideas discovered by Louis de Broglie in 1927.
New Age folks, who like to use words like “quantum” and “energy” to mean “cool” and “spiritual” love Bohm. I don't think many serious theoretical physicists are embracing Bohm's ideas about pilot waves, non-locality, or implicate order.
Bohm's ideas about determinism are attractive. It would be useful, if time allowed, to have give lessons on Bohm's and Bell's work, but I would not make either's work the central theme.
“Quantum mechanics demystified in fifteen minutes” seems just a tad presumptuous, if not arrogant. For a century the best minds have attempted to reconcile its implications with what the human mind perceives, and failed.
What I mostly see is teachers, authors, and researchers parroting back what others have said, and demonstrating mathematical explanations, without altering the fundamental difficulties with it.
Teach them all academics and NOT ideologies, Bobby...🇺🇸 😎👍☕
Top-tier physics internet content. I contribute to Elliot's Patreon and I encourage everyone who watches his videos and studies from his notes to do the same. Mainly because I want as much of this content as I can get. Elliot's videos have cleared up many residual confusions I've been living with for years.
Thanks Joel!!
As a telecommunication engineer... I was stunned when I 1st saw Schrodinger's wave equation. Stunned
The frequency transform is just another form of a probability function. Actually we use the term probability density function & power density function interchangeably
in Much of digital signal processing & communications theorems; that is all we care about to do filter designs🖐
your video highlights this and many other good ideas.
BTW.Heisenberg's's uncertainty principle..arises well in wavelet theory too
edited:
by the way... you have explained how when sine waves are integer multiple... they make orthogonal functions
believe it or not... this is the core idea behind OF..... 4G LTE technology ✌️
orthogonal is used to reduce noise generated by inter-symbol interference
Back in the 80's I did a lot of chemical spectroscopy. DSP was kind of new. But it was instructional to look at FFTs and time domain signals in almost real time. Writing code for FFT's or actual continuous integrals is also instructive.
@pbajnow i got my telecommunication education at the University of Illinois .... in the late 1980s.. thx for sharing...chears
@@AbuSous2000PR Lots of SDI money for signals work back then. What became of wavelet processing?
@pbajnow that is another exciting field... Wavelet...which i explore extensively years ago... there is an Indian professor from IIT...has an excellent course online on youtube... it is being used in JPEG encoding. Others use it model brain waves and earthquakes.... Multi-rate processing is another rabbit hole...chears
The content is extremely good, but I have to say I'm very happy with all the equations being perfectly formatted. The fact you used \dif x (or \dx iirc from the physics package) instead of just "dx" makes me very happy.
I've been struggling to "remember" the QM that I never really got comfortable with as an undergrad 30+ years ago. RUclips is a wonderful thing, and this video is a great example of why. About half way through, I had a light bulbs moment.
I've always been more comfortable understanding Fourier analysis in terms of acoustic or EM waves, but the probability waves of QM never felt natural. Your video builds to Fourier quite naturally but left me pondering, what is momentum in QM? I suddenly realized that it is directly analogous to frequency in sound or EM waves. This puts position in the role of (probability) amplitude. The ah ha was recognizing that momentum of an EM wave is proportional to frequency, and it all came together.
I had great instructors and any shortcomings were my own. Still, I keep realizing that in a formal physics classroom, they drowned you in the math and glossed over the big ideas. Either that or they presented the big ideas as a result of 10 blackboards of math with a statement like "so clearly ..." even thought the chalk was still smoldering and I was heads down trying to copy the last three boards. It's leave thinking the proof had probably said something really important, but I didn't quite catch it. I like the math, but appreciate the context as well.
Thanks for a great video.
This is because particles are not really particles, but waves. Also there may be nonlinearity in qm due to wavefunction collapse. Composition of waves should break at some point.
I've been waiting long for a video like this thank you very much
Amazingly detailed, clear, and fascinating explanation of how Fourier analysis relates to the quantum wave equation. I never really understood it very well until now.
One thing I would like to add is that the complex polynomial function P(x) = x^n when restricted to the unit disc, is given as P(e^{it}) = e^{nit}. That is, approximating a function on the unit circle with complex polynomials _exactly_ corresponds to the discrete Fourier analysis of the function.
What do you mean by that ? Please explain
I got your this channel as a follow-up of a good teacher's video. You too done as a classic.
The transformation from big Sigma to integral as you highlighted is some what partial because in very other situations one can derive similar relations with out following this formalism.
I am eager to see those in your forth coming and other topics.
Good things deserves good blessings.
Watching math like this all day, then coming back to the real world where you're reminded that many people still don't understand or fully comprehend mathematics. Immersing yourself in these types of videos makes you forget that many people still don't get these topics and how they apply to their everyday lives.
Anyways, thanks for sharing this.
Good to have you back Doc.
This is such an exceptional channel. I’m amazed it doesn’t have a few million subscribers yet.
Welcome back and thank you!
This was very interesting.
When you explained the Fourier series around 5:20, you _made_ the function periodic by demanding it repeat itself outside of the bounds we consider. In the example at the end you took a wave function that is zero everywhere but inside the box.
Are we still considering this to be periodic and repeating "behind infinity" or does the periodicity condition not apply when we take the infinite case?
It is also interesting to not that this works the other way around. You can get Schrödingers Equation from Hamiltonian mechanics by Identifying the Poisson bracket with Fourier conjugates.
Zap Physics did this in their last Let's Learn Physics stream on classical mechanics and it was really beautiful.
Thanks Narf! In the infinite space setup the wavefunction is not periodic, it's zero everywhere outside of that interval from -a to a. The area underneath |\psi(x)|^2 is fixed to one, remember.
@@PhysicswithElliot I'm not sure I understand. Perhaps I missed a step?
The construction of the Fourier _sum_ relied on the function being periodic, with periodicity related to R. If we take R to infinity to get the integral that perodicity becomes infinite as well, no? Is the limit of periodicity going to infinity just the same as a non periodic function? Can we make that statement in general?
Yes, because the space where you are working allows periodicity. Everywhere outside de box is 0. There’s no function outside
@@narfwhals7843 Yes the periodicity goes away in the limit. x ~ x + 2\pi R means that we identify each point x with every other point on the line that's shifted from it by 2\pi R. But when we send R to infinity, x isn't identified with any other points anymore. Then the discrete spectrum of Fourier waves becomes continuous, and all those waves interfere destructively outside of [-a, a] to make the wavefunction vanish instead of repeat.
@@PhysicswithElliot Thank you, that helped!
Cannot wait to watch. Really interesting topic, that pops up quite often:
In different lectures and an old submarine movie.
Hopefully I get a step closer to understanding how to apply it.
I don't know how to thank you sir. I really appreciate the effort you put to make this video, notes and that useful website.
So good!!! I have some fundamental understanding of quantum mechanics. Then I watch this video, Fourier Transform is awesome!
I am studying quantum mechanics these days, and I really love your explanation about fourier transform by the system of particle on a ring. Thank you for making great video.
Fantastic Lesson. Crystal clear explanation. Watching this lecture and reading Griffiths QM 2nd, section 2.4. Perfect combo!
this video was amazing! would be very happy to see one explaining the state vector in different bases
You are so kind to make these amazing videos and publish them for free, thank you so much
Always thought of the idea that all quantum mechanics courses should first teach the Fourier series and Fourier transform and all that stuff about signals and frequencies and distributions, and only after this they can hit the students with all that unintuitive facts about the double-slit experiment, the uncertainty relations, measurement processes, operators ...etc. And You assured the rightness of that idea.
Agreed. First time I learned about quantum mechanics was in pchem2 course and only because of my professor did I understand the basics. I don't recall the book (Atkins 6E) going into details of Fourier series and transforms. That professor covered some of it. It wasn't until I took a math elective on PDEs and variations calculus that I got into the nitty gritty as an undergrad. By the time I got into grad school, the math became so much easier lol
This is so cool, actually seeing WHY uncertainty principle is true, this is amazing, thanks.
Thanks Qi!
Another superb video Elliot 👍. Keep up the good work. Proud to be one of your patreons
Thanks Bart!
Thank you so much for explaining these wonderful subjects in such a simple and intuitive way!
I would love it if you could try making a video on some of the less intuitive parts of thermodynamics/statistical mechanics :)
Glad you liked it Yoad!
Thanks for making clear that momentum and position are two bases for the same vector space. Physicists often talk of “momentum space” and “position space” representations for a particle. But these terms should be replaced by the two above “bases” since they span the same Hilbert space.
من أروع ما شاهدت
فشكرا جزيلا على الشرح والعرض
ولن ابرحك ابدا حتى استفيد من فيض معارفك
Instant sub, this is the kind of crystal clear and intuitive but yet mathematically rigourous and detailed content that I'm looking for when it comes to explaining physics.
You are right up there with Physics Explained as far as I'm concerned !
And of course 3B1B for the mathematical aspect.
Bless the internet.
Thank you very much for your work, I look very much forward to your next videos ! (before that, I have some catch up to do with your previous videos haha)
it is not mathematical rigorous. The eigenfunctions of the momentum operator are not square integrable. The operator is ill defined, thus the concluding statements are also not rigorous. The momentum operator like this has no eigenfunctions which are related to quantum physics. Operators in quantum mechanics are densely defined one the space of square integrale functions such that they have a complete set of generalized eigenfunctions.
@@youtubesucks1885 I should have said "more rigorous than the average". This video is supposed to popularize scientific ideas so it is a given it won't be 100% mathematically rigorous. I think the author is clearly aware of this but intentionally did not mention those deviations in ordre to not get too much in the way of the main ideas.
I only had few classes regarding quantum physics so quantum operators are still untamed beasts for me :)
Either I wasn't aware of this condition on the eigenfunctions or I forgot about it. Either way, thanks for the clarification.
But I remember a lot of physics (not just quantum physics) is based on square integrable functions which is a rather nice condition in general compared to other purely mathematical problems where conditions can quickly and easily become more strict.
I’m sad because I never went beyond Algebra 2, and I’ve forgotten much of that, even. I love learning about the concepts of quantum physics, things like sterile neutrinos and penrose diagrams and Electroweak Symmetry Breaking, but this was just utterly beyond me to follow. Still, I didn’t get *nothing* out of it, and your style and delivery are delightful. Thank you for sharing.
oh your graphics are BRILLIANT, I love the use of emojis too. great lesson!
OK I am a newbie in understanding ..but 2:14 seems like a derivative of where I turn out to be according to all the places I could have been? A bit like Humphry Bogart in Casablanca? but it has a dynamical component somewhere?
Amazing video! This is a great complement to 3blue1brown's videos on the Fourier transform where he looks at it from a math perspective.
Thanks John!
🎯 Key Takeaways for quick navigation:
00:00 🌌 Quantum Mechanics and Fourier Transform Introduction
- Overview of the video's focus on the connection between the Fourier transform and quantum mechanics.
03:16 🔄 Particle on a Circle and Fourier Series
- Describing a quantum mechanical particle living on a circle of radius r.
- Introduction to periodicity, ensuring the wave function repeats on the circle.
- Exploring Fourier series as a sum of Sines and Cosines for periodic functions.
06:40 🔍 Representation of Fourier Series in Exponentials
- Introduction to expressing the Fourier series using exponentials.
- Euler's identity as a key tool in transitioning between Sines/Cosines and exponentials.
- Explanation of coefficients PSI sub n and the significance of wave number K.
09:43 🌊 Fourier Waves as Eigenfunctions of Momentum
- Fourier waves (e to the ikx) as eigenfunctions of the momentum operator.
- Significance of H-bar times the wave number (k) as the eigenvalue.
- Explanation of quantized momentum values due to periodicity on the circle.
13:19 ➡️ Calculating Fourier Coefficients
- Introducing the method to compute Fourier coefficients using integration.
- Demonstrating the integral process to find the coefficients PSI sub n.
- Linking the coefficients to the contribution of individual Fourier waves.
16:29 🔄 Transition to Infinite Space and Momentum Continuity
- Discussing the transition from a particle on a circle to an infinite x-axis.
- Elimination of momentum quantization as radius (R) goes to infinity.
- Acknowledging the continuity in momentum values.
18:56 🔀 Fourier Series in the Infinite Radius Limit
- Presenting the Fourier series formula in preparation for the infinite radius limit.
- Introducing modifications in notation for the limit.
- Simplifying the coefficients for the Fourier series in infinite space.
21:14 🌌 Fourier Transform Basics
- The Fourier transform emerges from the Fourier series for a wave function on a circle in the limit where the circle becomes infinitely large.
- Each Fourier wave represents a state of definite momentum, and the transform is a superposition of these basic waves.
- The transition from discrete Fourier series to continuous integral is explained, highlighting the underlying concept's consistency.
23:06 🔄 Position and Momentum in Quantum Mechanics
- The probability of finding the momentum in a window is related to the squared proportion of the Fourier wave, mirroring the position probability.
- Position space wave function (ψ of x) and momentum space wave function (ψ-hat of K) are two representations of the same quantum state vector.
- The relationship between position and momentum probabilities sets the stage for the uncertainty principle.
25:37 📊 Example: Fourier Transform of a Wave Function
- Illustration of the Fourier transform for a particle localized in a position space interval.
- Momentum space wave function (ψ-hat of K) graphically explained, showing how different Fourier waves contribute.
- Introduction to the uncertainty principle by analyzing the width of position and momentum space wave functions in different scenarios.
Made with HARPA AI
thank you.... i wish you included the part why the uncertainty product is greater than hbar/2
This video was amazing!!! Thank you for sharing!
Wow, very neatly explained.
I've found this video very helpful in my journey to Quantum mechanics...
Love Euler's Identities!
Thanks alot!
5:25. the wave function DOES allow for some probability outside the box (exponentially decaying), NOT ZERO!
This video made understanding the topic so much easier!
Another totally amazing video. Huge thanks.
Love your video. It explains Fourier transform in such a simple way. Prof instead of using a square position function, is it possible to show us how to get the same result by a more general function.
This is very well made. I have been waiting a long time for a good video explaining this at sufficient depth. Appreciate your hard work.
If I can make one request: while describing equations, can you use what the variable represents instead of the letter? For example instead of saying "p = h bar over i" say "momentum equals Planck's constant divided by root of -1 which we denote as i" or something like that. I usually pause the video and think this through myself but occasionally I am not sure what a variable represents and am stuck. The repetition of what variables represent will promote conceptual understanding.
This is not a criticism, just a humble request. Your videos are great and I make time in my schedule to ensure I can watch them.
Great explanation of why the Fourier transform and its inverse look the way they do!
Wow!! What tools do you use to create these videos?
Thank you! I'm digging deep into Fourier Transforms currently.
Good timing!
Does your toolbox include Lebesgue measure and integration theory?
high quality amazing video you are so underrated mate
that's brilliant! Many thanks for sharing such a great tutorial!
This was beautiful, thank you for making this
It's too detailed explanation ✨️
ഒരുപാട് നന്ദി🙏
Спасибо за работу! Прекрасная визуализация! THX!
Cool!
There are more things that should be said about the Fourier transform and the theory of representations in QM, but it will be more fun with the Dirac's Bra - Ket notation.
Will you make a video on how you can modify the schrodinger equation to make it work on curved space-time?
The Schrödinger equation cannot be modified to work on curved spacetime. Instead, you need to look into Lagrangian densities that account for curvature in spacetime, and derive the quantum field theories from there.
@@angelmendez-rivera351 im pretty sure you can, like for example instead of using the normal laplacian you use the laplace-beltrami operator (which is basically the laplacian but generalized to (pseudo) riemannian manifolds)
but im not sure of the details
31 minutes of absolute brilliance
As I saw the time running out on the video I honestly got worried that we weren't gonna get to Heisenberg in time 😅 but we got a nice taste of the principle in this video
just now, after ten years of studying quantum mechanics, I can
say that I understand what the wave equation means. Please any one tell Fayman about it.
, and if anyone is related to Roger Penrose, send him this video, he was experiencing the same problem .
Thank you so much.
This was one of the most beautiful things I’ve ever seen 😭💯
I like the old background better (the white one with a box that has a grid, and has equations outside of it). I like the new fonts that you are using, though.
This channel is phenomenal, absolutely top-class explanations of such deep concepts. It’s a real treat to see such an elegant presentation of mathematics and physics where the derivations and results follow so seamlessly. I hope y’all continue this channel for years to come, these videos are a gift to the world. This nuclear engineer sincerely thanks y’all for the content, keep it up!👍
Great video Elliot the connection of FS and wave function gives me new purpose in learning Physics.
I feel like this video was sent to me from the gods. The proportion of people who know a lot about rigorous QM and don't know a lot about Fourier transforms has got to minuscule. This has been an absolute treat
Can Somebody please explain, why we replaced psi_k with 1/√2πR • psî(k). 19:01
Thinking of it, psi_k is a constant whereas psî(k) is a function. How we can exchange them?
What am I missing here?
Also, 1/R is a sensitive variable here. (Going to be ‰ in further steps) So why (is it allowed to do so) include it into the replacement term??
Hi! You've made an outstanding video which not only explains the Fourier series and integrals in an excellent matter, but also shows the maths behind it. However, I do have a question. At 16:30, how are we able have k under the summation sign and not n=-inf to inf like we had earlier? Thanks!
amazing best. and thanks for the notes
Literally what we’re doing rn at uni! My last homework was about the Fourier Transformation 😅
Perfect!
Brilliant. Excellent. Spectacular.
Awesome video! Congratulations.
Could you please tell us the specific tools that you use in order to make these amazing animations?
21:21 isn't this a bit hand wavy? If k depends on R, and R is in the denominator of the exp argument, shouldn't the exponential go to 1? (exp(1/inf)) why does the k just stays there?
Back again... Thank you.
Mind blown!
It is really satisfying to see everything connecting.
Is kinda frustrating not being able to tell my friends this experience.
In minute 04:00, why is the condition imposed on psi and not on its modulus square?
I think something that's worth clarifying here is that the form of the momentum operator can be inferred by matching the generator of translations to the de Broglie Hypothesis. You *need* the de Broglie Hypothesis to build quantum mechanics "from the ground up" or otherwise assume uncertainty or the momentum operator axiomatically. I think this is important to note because the de Broglie Hypothesis is essentially an intuitive guess that ended up being instrumental for the success of quantum mechanics.
De Broglie's theory tried to justify a local view of QM, to answer some of Einstein's feelings about QM. Bell's theorem shows that any local view of QM must be wrong on basic principles. Entanglement also shows that a local view must be wrong. If you want to find a more sensible interpretation of QM, look into David Bohm's paper of 1952, which describes QM as deterministic and nonlocal.
@@david203 That's not the point I'm making here. I'm saying that the form of the momentum operator is either taken itself to be axiomatic or derived from the De Broglie Hypothesis along with considering the translation operator's generator. This is not a statement about interpretations of quantum mechanics.
come on upload again everything about quantum mechanics or relativity.. I'm so excited to be waiting for
This is an excellent video. Thank you for doing this.
So this is settled, you have the best Physics youtube channel. For sure...!
I NEVER THOUGHT YOU COULD GET THE FOURIER TRANSFORM AS A LIMIT OF FOURIER SERIES THAT'S SO COOOOOL
I gotta run tell someone.
Great explanation as always ! I just had a small question. Is the continous model using the fourrier transform consistent with the observtions of quantized momentum ? How can k be continous if momentum is quantized (discrete) ?
Awesome! Welcome back. This is a great topic.
Thanks James!
miss you , keep the good work up please
25:20 wouldn't the position space wave function use the momentum basis and vice versa?
Of course, it is
Wish to have seen this some 30 years ago during my basic course!
Amazing video as always! I'm curious, what software do you use to create your videos?
Thanks Petaro! Keynote, Motion, Final Cut mainly
Wait, shouldnt the dx be between the integral and the function it is integrating?
That's just the convention used in higher level physics, f(x)dx ~ dx•f(x). Works either way, but I agree the former looks better notation wise
The strong force is said to act over a short range. Does this imply that quarks have a finite range of probability where they could be found and thus potentially have discrete momentum states rather than continuous momentum states?
2:01 why is it the square of the wave function * the width (but not the function itself?
The probability density curve is the modulus squared, ie absolute value squared, partly because we hope to measure actual quantities, the wave function is complex