To Understand the Fourier Transform, Start From Quantum Mechanics

Develop a deep understanding of the Fourier transform by appreciating the critical role it plays in quantum mechanics! Get the notes for free here: courses.physicswithelliot.com...
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The Fourier transform has a million applications across all sorts of fields in science and math. But one of the very deepest arises in quantum mechanics, where it provides a map between two parallel descriptions of a quantum particle: one in terms of the position space wavefunction, and a dual description in terms of the momentum space wavefunction. Understanding this connection is also one of the best ways of learning what the Fourier transform really means.
We'll start by thinking about the quantum mechanics of a particle on a circle, which requires that the wavefunction be periodic. That lets us expand it in a Fourier series---a superposition of many sine and cosine functions, or equivalently complex exponential functions. We'll see that these individual Fourier waves are the eigenfunctions of the quantum momentum operator, and the corresponding eigenvalues are the numbers we can get when we go to measure the momentum of the particle. The coefficients of the Fourier series tell us the probabilities of which value we'll get.
Then, by taking the limit where the radius of this circular space goes to infinity, we'll return to the quantum mechanics of a particle on an infinite line. And what we'll discover is that the full-fledged Fourier transform emerges directly from the Fourier series in this limit, and that gives us a powerful intuition for understanding what the Fourier transform means. We'll look at an example that shows that when the position space wavefunction is a narrow spike, so that we have a good idea of where the particle is in space, the momentum space wavefunction will be spread out across a huge range. By knowing the position of the particle precisely, we don't have a clue what the momentum will be, and vice-versa! This is the Heisenberg uncertainty principle in action.
0:00 Introduction
2:56 The Fourier series
16:08 The Fourier transform
25:37 An example
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About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com.

[@MaxxTosh]

You seriously give 3B1B a run for his money, this is fantastic. I can’t wait to see more!

[@PhysicswithElliot]

Thanks Maxx!

[@leif1075]

@@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?

[@aloysiusdevadanderabercrombie8]

@@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.

[@leif1075]

@@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?

[@aloysiusdevadanderabercrombie8]

​@@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.

[@ki-ka]

This is almost incomprehensibly beautiful on a number of levels.

[@alejrandom6592]

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 ♡

[@DudeWhoSaysDeez]

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.

[@mintakan003]

Outstanding use of visuals, along with how the transitions happen, when the quantities are varied.

[@wdavis6814]

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!

[@lewiswinner6634]

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

[@apophenic_]

The amount of intuition this video provided was remarkable. Thank you.

[@johnhamilton7762]

Incredible description, incredible graphics,incredible explanation. Simply beautiful. So much work must have gone into this labour of love. Thank you.

[@kenleybrown7224]

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!👍

[@Guido_XL]

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!

[@EnchantedGardenGnome]

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. 🙏🏽

[@OmnipotentO]

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.

[@magellan500]

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.

[@lewiswinner6634]

Amazing to get a deeper understanding of the Fourier transform, very thorough as always. THX!

[@PhysicswithElliot]

Glad you liked it!

[@ViTaLC0D3R]

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.

[@krustykrewe]

this video was amazing! would be very happy to see one explaining the state vector in different bases

[@bishwajitbhattacharjee-xm6xp]

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.

[@pelegsap]

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.

[@maryamgholinasab4531]

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.

[@ResetToZero3210]

Fantastic Lesson. Crystal clear explanation. Watching this lecture and reading Griffiths QM 2nd, section 2.4. Perfect combo!

[@paulg444]

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.

[@tappetmanifolds7024]

Simplicity from complexity? I would never have believed it. Very well done !

[@antman7673]

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.

[@Valeriy7D0]

that's brilliant! Many thanks for sharing such a great tutorial!

[@chicopolar7666]

Great video Elliot the connection of FS and wave function gives me new purpose in learning Physics.

[@joelcurtis562]

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.

[@PhysicswithElliot]

Thanks Joel!!

[@AbuSous2000PR]

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

[@sambhavgupta4653]

amazing best. and thanks for the notes

[@insertname1184]

This video was amazing!!! Thank you for sharing!

[@jeengbe]

This was beautiful, thank you for making this

[@Jacked_R_Us]

I've been waiting long for a video like this thank you very much

[@kevconn441]

Good to have you back Doc.

[@bengineer_the]

Thank you for such a wonderful explanation. Your wording is very illuminating, thank you. :)

[@jamesangel9512]

A delightfully satisfying presentation. Thank you!

[@frankreashore]

Another totally amazing video. Huge thanks.

[@marverickbin]

Mind blown! It is really satisfying to see everything connecting. Is kinda frustrating not being able to tell my friends this experience.

[@quercus6856]

Excellent video, I had a few mind blown moments, thanks!

[@MuralidharKamidi]

This is an excellent video. Thank you for doing this.

[@mathjitsuteacher]

Awesome video! Congratulations.

[@ricardodelzealandia6290]

Earned a subscribe. Thank you.

[@LydellAaron]

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.

[@hqs9585]

amazing video, thanks!

[@manla8397]

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.

[@well321321]

Спасибо за работу! Прекрасная визуализация! THX!

[@bartpastoor1028]

Another superb video Elliot 👍. Keep up the good work. Proud to be one of your patreons

[@PhysicswithElliot]

Thanks Bart!

[@odins_claw]

Wow, this is a brilliant explanation. Subscribed 🎉

[@trelosyiaellinika]

Excellent explanation! Thanks.

[@nomOfyourBusiness]

God tier explanations! never stop :D

[@faldhanizanov]

come on upload again everything about quantum mechanics or relativity.. I'm so excited to be waiting for

[@corsaircaruso471]

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.

[@elmehdiben895]

Fantastic. Carry on 👏🙏. I am very interested in quantum computing, I hope you do a series on this subject 🫶

[@hrperformance]

1st class video as always. Thank you so much

[@redared9389]

من أروع ما شاهدت فشكرا جزيلا على الشرح والعرض ولن ابرحك ابدا حتى استفيد من فيض معارفك

[@Duskull666]

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 :)

[@PhysicswithElliot]

Glad you liked it Yoad!

[@bobbyking8976]

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.

[@PhysicswithElliot]

Thanks Bobby! I did the bulk of the animation in Keynote

[@david203]

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.

[@Johnny.Fedora]

@@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.

[@commanderthorkilj.amundsen3426]

“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.

[@lilblackduc7312]

Teach them all academics and NOT ideologies, Bobby...🇺🇸 😎👍☕

[@manitro337]

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.

[@alainhamel7163]

fantasticly explained ,great !

[@VolkGreg]

Superb job, Elliot

[@qibingsia9212]

This is so cool, actually seeing WHY uncertainty principle is true, this is amazing, thanks.

[@PhysicswithElliot]

Thanks Qi!

[@rockyrococo3175]

Very nice job.

[@jml_53]

I've been struggling to "remember" the QM that I never really got comfortable with as an undergrad 30+ years ago. KZbin 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.

[@frun]

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.

[@karamashi520]

high quality amazing video you are so underrated mate

[@raxirex6443]

Effective, clean visualisation

[@maciejmulak8285]

It's really brilliant, many thanks!

[@MrFtriana]

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.

[@crh8803]

It is just an excellent class. Amazingly didactic. Congratulations.

[@radwizard]

Thank you! I'm digging deep into Fourier Transforms currently.

[@PhysicswithElliot]

Good timing!

[@davidmarshall8628]

Does your toolbox include Lebesgue measure and integration theory?

[@omargaber3122]

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.

[@AJ-et3vf]

Great video. Thank you

[@haiyunsky]

great video, thank you very much.

[@apdj94]

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

[@stephanecouvreur1377]

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.

[@bryanfuentes1452]

thank you.... i wish you included the part why the uncertainty product is greater than hbar/2

[@cerioscha]

Gold dust, thanks for making these videos !

[@urnext6874]

This video is great thx ...plz upload videos more frequently sir

[@deepakjanardhanan7394]

Back again... Thank you.

[@martinstyblo6355]

Wish to have seen this some 30 years ago during my basic course!

[@mohammedpatel3051]

Excellent explanation Great 👍

[@ayhamhalalsheh221]

miss you , keep the good work up please

[@elamvaluthis7268]

Very nice explanation 🙏🙏🙏.

[@alikarimi-langroodi5402]

Excellant. Thank you

[@mtate405]

This is a brilliant video

[@lexinwonderland5741]

oh your graphics are BRILLIANT, I love the use of emojis too. great lesson!

[@spencerwenzel7381]

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.

[@mariocesarsousa]

This is ❤perfect ❤. I am in the last year of physics and things unclear are much better.

[@AnyVideo999]

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.

[@Krish_202]

What do you mean by that ? Please explain

[@lewiswinner6634]

Literally what we’re doing rn at uni! My last homework was about the Fourier Transformation 😅

[@PhysicswithElliot]

Perfect!

[@kyrilo1993]

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) ?

[@paulbalduin]

Such a good video!

[@ankidokolo]

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.

[@masterludovicus802]

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

[@realcygnus]

Superb stuff

[@cane988]

Each day I stand closer to understanding your magic, magic man

[@odysseus231]

I NEVER THOUGHT YOU COULD GET THE FOURIER TRANSFORM AS A LIMIT OF FOURIER SERIES THAT'S SO COOOOOL I gotta run tell someone.

[@johnchessant3012]

Amazing video! This is a great complement to 3blue1brown's videos on the Fourier transform where he looks at it from a math perspective.

[@PhysicswithElliot]

Thanks John!

[@dillbourne]

I'd love to hear you explain how Papov Ghosts and Goldstone Bosons work in QFT.

[@alexandreolle2340]

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)

[@youtubesucks1885]

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.

[@alexandreolle2340]

@@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.

[@Micro-Moo]

Very nice. The very root of things in quantum mechanics in half an hour.

[@marcolanucara4551]

In minute 04:00, why is the condition imposed on psi and not on its modulus square?

[@notmyname7698]

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.

[@MathPhysicsFunwithGus]

Great video!