Complex Numbers in Quantum Mechanics

[@RichBehiel]

Hi everyone, thanks for checking out this video. There are a couple caveats that I put in the video description, relating to the yin-yang metaphor and the connection between local U(1) symmetry and electromagnetism, so please check those out if you are interested. Also, I could use your advice about something. In this video, I added a bit of gray/black motion to the background, since this helps prevent KZbin's algo from adding compression artefacts to the video (moving color on a solid background would otherwise lead to a confetti-like appearance). The moving background also helps the video come to life a bit more, lets it breathe, you know? But I hope this effect does not come across as distracting or nauseating, so please let me know if in your opinion it was too much, and if I should make it more subtle or slow it down in future videos. Or, if you have another suggestion for how to add subtle motion to the background of a video without it being distracting, please let me know. By the way, if anyone has advice for how to speak more naturally into a microphone, I would love to hear it! I feel like there's a tradeoff between annunciation and flow, like if I try to say every word properly then I sound like a robot, but if I just talk conversationally then I find that I tend to mumble a bit. Maybe I just need more practice. But if anyone has any tips or tricks or vocal exercises, please let me know. And as always, if you have a question about anything presented in this video, just leave a comment and I, or another commenter, will get back to you soon. I highly encourage conversation around these topics, because odds are you're not the only one who has that question, so we can all learn together. That's really what this channel is all about :)

[@mechwarreir2]

Microphone quality sounds fine. Also the youtube animated background thing is annoying, but I've gotten use to it. Also it doesn't show up when full screen.

[@RichBehiel]

Awesome, thanks for your feedback. So next time I might try the same thing but with like 80% opacity on that layer to make it a bit less noticeable. Hopefully that would be enough to trick the compression algorithm without distracting from the video.

[@aprillomat]

I had to go back to try and spot the grey motion that you mentioned, and it's so subtle that it really isn't anything you should worry about. also an interesting tidbit that I didn't know about, and I am very deeply into compression so in my mind it would have had to be the opposite way around (adding motion would mean more bytes needed to compress that extra motion, and those then won't go into high quality foregrounds - but it could be that the newer codecs just work in mysterious ways, or that youtube will assign a lower compression target to videos with less visual complexity - anyway, it's interesting). As for @mechwarreir2's comment, I think they were talking about the new youtube feature which is called ambient mode, and which has nothing to do with your video in particular. Btw you can turn it off in the cog wheel menu of the video player :)

[@Ivarius321]

The way you're speaking is more than fine, nothing to worry about. Also, I didn't even notice the background was moving, lol

[@hyperduality2838]

Complex numbers are dual, real is dual to imaginary. Conjugate root theorem -- complex roots come in pairs or duals. Subgroups are dual to subfields -- the Galois correspondence. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! "Always two there are" -- Yoda. Splitting fields in group theory:- positive is dual to negative. Real number or the integers are self dual as they are their own conjugates:- kzbin.info/www/bejne/d6mzqJuAia2rick Elliptic curves are dual to modular forms. Electro is dual to magnetic -- Maxwell's equations. The inner product is dual to the cross product. Nothing wrong with duality once you understand it.

[@bowfuz]

we need more mathtube and sciencetube content where the speaker talks casually and laughs more, it's hard to pin down exactly why this makes it better but i conject that the usually neglected emotional aspect of videos like these is seriously improved with humor and the occasional fumble

[@villaratanaphom-sg3hg]

Probably because the technical fields have the stereotype of being unbearably dry and "unhuman"

[@nvanderhoff]

His voice also has good harmonics

[@kevon217]

yeah, seriously. The narration has that “tuck you in at night and read you a bedtime story” feel. The visuals are incredibly intuitive too.

[@SirFafer]

I think this video is perfectly fine with being straight to the point. You want to be entertained, he made this to educate.

[@DarkSchneider7]

I’ve been noticing a repetitive pattern of religious ads on scientific educational videos, and it’s starting to feel a bit off. The targeting seems somewhat mismatched with the content. It might be worth reviewing to ensure ad targeting remains relevant and respectful. Just to clarify, I’m not against religious institutions, but the constant repetition of these ads seems unusual and could benefit from a closer look. KZbin’s ad market should ensure that there are no aggressive intentions behind focusing specific ads on certain video types. Ads, especially religious ones, should ideally be broad-spectrum, particularly for educational content, since these videos attract a diverse audience. A more universal spread of ads would be more appropriate.

[@nice3294]

I loved the motivation of complex numbers as extending the sense of "sign"/phase/direction from being discrete to continuous

[@MasterHigure]

Indeed. Much of the use that physicists have four complex numbers is precisely to make an up-and-down wave into a (virtual) rotational motion instead, because they are so much easier to work with. It is certainly a lot easier than having one wave representing the value at each point and another wave representing the rate of change at those same points (which would be the naive solution to the discussion that starts at 3:30). What physicists have instead is a single circular rotation at each point, and then they let the laws of physics take care of the rotational velocity. And complex numbers work so well that it seems like they were made for this.

[@giorgosg4032]

Finally, a very easy and comprehensive way to explain why complex numbers are so important for wave mechanics

[@ellepeterson9992]

WHY HAS NOONE EXPLAINED THIS TO ME LIKE THIS SO FAR this make so much sense

[@T0NYD1CK]

You can slao look at it this way: first, there was counting; that was closely followed by addition and multiplication. Everything was good until someone wondered about division. What happens if the there is no number to exactly show the result. That is why they invented fractions in between the numbers. Then they branched out into subtractions and all went well until they tried to take away more than they had to start with. Another invention required: negative numbers. The problem is that every time something new is invented you need to revisit all the old ideas to see if they still work. Everything was good until they looked at square roots. What to do with negative numbers. They already had forwards numbers and backwards numbers so they invented "sideways numbers". And the rest is history! EDIT: When it comes to waveforms, I think it is more intuitive to view a wave as a helix, like a spring or a corkscrew. You get the cosine part of the wave by looking at the side elevation or side view and the sine part by looking at the plan or top view. That fits exactly with imaginary numbers being sideways numbers. The real numbers go up and down while the imaginary numbers go in and out of the paper.

[@passingshots]

I struggled with complex numbers throughout all my education, I couldn't grasp the idea. The way you presented it makes complete sense because of the geometric representation. It's beautiful

[@YossiSirote]

Descartes … “that and Dualism” 😂😂😂😂 … and now you are one of my favorite people ❤

[@james-cucumber]

This was an incredible video. Leaving comment mostly for algorithm, but also to wish you the best of luck. This content deserves way more views

[@RichBehiel]

Thanks James! :)

[@aanchaallllllll]

0:03: 🧩 Quantum mechanics involves complex numbers, which initially seem confusing but are essential for understanding the subject. 3:12: 🌊 The concept of a wave and how numbers can capture its characteristics. 6:10: 📚 The imaginary and real parts of complex numbers are equally real, and representing waves as complex numbers allows for easier understanding of wave interference. 9:01: ✨ Complex numbers can be added and multiplied in the complex plane, with the product's magnitude depending on the magnitudes of the individual numbers and the phase angle depending on the sum of the phase angles. 12:05: 🔗 Complex numbers allow for the addition of waveforms in signal processing and Fourier analysis. 14:38: 🔍 Complex numbers in quantum mechanics are not about direction in physical space, but rather represent the two-dimensionality of a wave. 17:30: ✨ The amplitude squared of a complex number in quantum mechanics is often expressed as PSI star PSI, which represents the probability density relating to the wave function. Recap by Tammy AI

[@mat-hu5ys]

cool, but you could just do it yourself

[@0FAS1]

2 minutes in and you already blew my mind, way above my level of understanding in the later parts but somehow still coherent due to your wholistic approach, this channel is going to blow up in due time. Personally I find that I have the easiest time understanding when the purely abstract is intermingled with physical concepts and happenings and you were amazing at doing this. I think the same applies to many others as well. Thanks and I will definitely check out your other videos!

[@RichBehiel]

I’m so glad to hear that! Thanks for the kind comment :)

[@jippijip101]

Omg your visualization of a coherent state of the harmonic oscillator at 13:00 is FANTASTIC! Nice work!

[@RichBehiel]

Thanks! :)

[@xanterrx9741]

18:31 an hour i think you've said three , YOU ARE AMAZING Richard you create amazing content that is packed in suck a way that anyone could realy understand it . I'm learning QM and QFT on my own and your videos are just way to learn and relax for an hour or more or less but that time is greatly spended , again i would like to say that you are just amazing

[@RichBehiel]

Thanks for the very kind comment, that means a lot! :) I’m glad you’re enjoying the videos.

[@mmer1687]

This is one of the most beautiful math videos i've seen. I hope you will continue doing them.

[@RichBehiel]

Thanks! :)

[@ЯСуперСтар]

Сразу понял что ты русский)

[@mmer1687]

@@ЯСуперСтар как?

[@ЯСуперСтар]

@@mmer1687 Native скажет i've seen so far или i've ever seen, а вместо continue doing them скажет keep making it. Самое заметное, что сразу бросается в глаза. А вообще, я рад, что такие видео смотрят и у нас. Не имею ввиду ничего плохого

[@zacwarnest-knowles9139]

wow I’ve just found your channel and this is crazy quality stuff and a really great intuitive perspective that helped me see the complex plane in a different light. I was a bit shocked when I saw your subscriber count I expected you to have atleast in the 10k to 100k range. You will surely blow up soon making things to this standard.

[@RichBehiel]

Thanks, I’m glad you enjoyed the video! :) It’s funny you say that, since posting this video a couple days ago my subscriber count has almost doubled 😮

[@spacecowx3116]

This hits wayyy different than those low quality ear grating lectures i'm accustomed to finding on youtube. Its also way different than those documentary style videos that seem to only scratch the surface. Keep up the good work

[@RichBehiel]

Thanks for the kind comment, I’m glad to hear you enjoyed the video! :)

[@Marc-tm4xh]

This is stuff that I've been thinking and wondering about (as a layman) for literally years. Your videos are so fantastic at giving me insight into all these ideas. I can't imagine how long it took to make all those beautiful mindblowing visualizations. Truly amazing work, thank you!

[@RichBehiel]

Thanks, I’m glad you’re enjoying the videos! :) It does take quite a lot of time, but it’s very satisfying.

[@michaelguidera1876]

50 yrs ago, one of my elective classes in Mechanical Engineering at Cal Berkeley was about complex numbers and their application to airfoil design. The professor showed us how to set up equations used to calculate the lift in different wing designs on aircraft. It was fascinating.

[@kraamesh]

Thank you for uploading the fantastic video with discernible animations, explaining the significance of complex numbers in understanding quantum mechanics. It has been a while since I attempted to create my first KZbin video about the complex number from a physicist's perspective. However, due to a lack of coding tools and experience, I haven't been able to proceed. You have shown my first and final steps, but there are two more steps in my idea: i -> LCR -> Fourier -> QM... I am not sure i will be able to proceed but your video gives motivation...

[@tedsheridan8725]

Another outstanding video! Though I kept on waiting for the wave (3:00-6:00) to pop out and be depicted as a rotating helix, with a continuum of complex planes perpendicular to the axis of wave propagation. That's how I've always pictured them (at least to the extent that I've dabbled in QM), but it's rarely shown that way. It seems like such an obvious way to illustrate how the 'zero' point can still have a magnitude, and explain the sinusoid as a rotation through complex space.

[@RichBehiel]

Dang, I should have done that! 😩 That would have been so cool.

[@anth2]

you are an artist. And you’ve found your portal into the realm of art via pure math, and it’s really stunning. I’ve never encountered anything like this. I am humbly taking the first steps of a long journey towards understanding math and physics now, and I can intuitively confirm your sentiment “it’s one of the most wholesome things you can do”. Really grateful for these videos. You are helping me find the applied science hidden in plain sight in the work I’ve devoted my life to doing (which is teach music to children)

[@davidwright8432]

Clear, excellent, charming, informative, reassuring (of sanity) and fun! This is the way complex numbers should be introduced. As intriguing, an invitation to thought; not as an affront to reason. I wish my high school math teacher had been as eloquent and persuasive.

[@RichBehiel]

And I wish all KZbin comments were as kind and flattering! Thanks for watching the video, and I’m glad you enjoyed it :)

[@BlueGiant69202]

@@RichBehiel Please consider the idea of making an intriguing, invitation to thought video about Spacetime Algebra and its relationship to complex numbers in Geometric Algebra. Spacetime Algebra as a Powerful Tool for Electromagnetism by Justin Dressel, Konstantin Y. Bliokh, and Franco Nori. "Abstract" "We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field." "Keywords: spacetime algebra, electromagnetism, dual symmetry, Riemann-Silberstein vector, Clifford algebra" Dressel, J., Bliokh, K.Y., Nori, F., 2015. Spacetime algebra as a powerful tool for electromagnetism. Physics Reports 589, 1-71. doi:10.1016/j.physrep.2015.06.001 digitalcommons.chapman.edu/cgi/viewcontent.cgi?article=1373&context=scs_articles

[@hydropage2855]

The instant you showed the animation for the Fourier square wave generator I had to drop a like. I’ve manually computed those myself and it is one of the most beautiful mathematical concepts I’ve ever taught myself

[@stuartriley]

Richard thank you for the presentation and your insight on complex numbers. I have studied the works of many particle physics and few had noted the world that exists in the quantum field theory of the impact of complex numbers, and their conjugates. What we sense is not what our reality is; we cannot see it but it (complexity) is there. Thank you once again for this presentaton.

[@philipm3173]

So lucid and comprehensible, tremendous job!

[@RichBehiel]

Thanks! :)

[@2ndPortal]

Beautiful explanation! I was waiting for an intuitive understanding of the imaginary numbers! Greatly appreciated🍀

[@markawbolton]

Great timbre and natural Narration. Very pleasant and easy to follow.

[@michaellara695]

Wow this video is incredible! It's just a matter of time before this channel becomes huge, amazing content!

[@RichBehiel]

Thanks, I’m glad you enjoyed the video! :)

[@ToneHelix97]

You are gonna be a star, I am looking forward to the upcoming contents from this channel!

[@RichBehiel]

Thanks Vikash! :)

[@jaw0449]

Thank you for this!! I've always struggled to 'visualize' this part of QM. By the way, the tangent at 6:15 is spot on lol...also, those equations at the end are some of my favorite!!

[@MusicEngineeer]

Viewing the bidirectional real number line as two unidirectional number rays with a binary second coordinate to pick on which side we are is a very interesting way to see it. I have never thought about it this way - but it does make a lot of sense indeed. We observe, that the left half of the number line is obtained from the right by a reflection (a discrete geometric transformation) and furthermore that a reflection can also be expressed as a rotation (a continuous geometric transformation) by 180° and then we just allow all angles instead of just 0° and 180°. I think, when we think about complex numbers that way, we kind of directly and naturally arrive at the polar form rather than first thinking about their cartesian form. We kind of "bypass" the idea of the cartesian form and immediately think in terms of length and angle.

[@RichBehiel]

It’s a bit of a quirky perspective, but I think it makes the complex numbers feel more natural, or at least shows one of the ways we can get into the complex numbers without starting too far from what we already know.

[@samuelthecamel]

You are an amazing presenter. This channel deserves way more subs.

[@RichBehiel]

Thanks! :)

[@SolveEtCoagula93]

It has been, literally, years since I've found something so fascinating as this video. OK, I'm getting on a bot and I no longer tend to look at this type of stuff but, boy, does this wake the brain up and say, WOW! Thank you so much for the style of your presentation with the occasional interjection of humour, and the clarity with which you explain such a difficult subject. For the first time, I just begin to glimpse the beauty of these things. Hope I can follow the rest of the series! (I just wish my brain would stop making some of the animations pop into 3D instead of being in a 2D plane!)

[@moralboundaries1]

12:50 What a beautiful and profound animation. Really captures the essence of superposition, doesn't it!

[@AdrianBoyko]

This is the first video I’ve seen that provides any insight into how position and momentum necessarily combine in a wave function. All other instructional material seems to just state that position and momentum probabilities can be derived from a wave function, as if that’s some sort of axiom. Until now, the wave function has always looked like position information, to me, with momentum information being buried in there in some mysterious, imperceivable way.

[@karsonio3543]

I know you’ve already gotten a lot of positive comments, but that won’t stop me from doing the same! Great video :) happy to have found this channel before it blows up!

[@RichBehiel]

Thanks Karson! :)

[@Toaster278]

God finally this is has left me with a really intuitive way of understanding what the real and complex parts of a wave actually imply in an intuitive sense

[@everyotherodd]

This deserves 1M+ views - the question that was answered in this video brought a lot of existential satisfaction 👏

[@johnlard]

Can I just say that I love your casual yet knowledgeable tone. It makes it so much easier to follow what you're talking about!

[@arthurbehiel4632]

Fantastic video! I’ve watched it several times. One point of clarification. I think the reason complex numbers are two-dimensional is that the waves they represent have two components. Waves oscillate between components, like electric and magnetic fields, current and voltage, or kinetic and potential energies. The two dimensions of complex numbers allow us to express both components with one value. To your point, both components are equally real. (I dropped out of HS math when my teacher could not tell me why we had to learn about imaginary numbers. I thought he was just wasting my time. 😂)

[@RoyMustang.]

Thank you sir ! This will be very useful for my PhD thesis !

[@stephenlashley6313]

Finally!! I have been using it this way for eons without ever seen it written this way. To curve fit some of nature's favorites there must be vibration and probability waves as described in this channel. Excellent!

[@lepidoptera9337]

You can't tell the difference between curve fitting and geometry? That's pretty lame. ;-)

[@Χριζαϊων_Ζηνόβῐος]

Funny how I never liked math until after i finished my math credits in college. Now that I can learn stuff how I want to I can see how interesting a lot of fields of math and physics are to me. Great explanation and video, even if some of the notation was lost on my inexperience.

[@stephendaedalus7841]

Excited for the gauge symmetry video. I took two semesters of QFT as an undergrad so I think I'm kinda maybe following lol great work!

[@cademcmanus2865]

Never heard the complex numbers described as a generalization of binary directionality. Really cool stuff.

[@kennybeach342]

Hey man, thank you for making these videos, I'm taking modern physics currently and have been struggling to wrap my head around a lot of it, your videos help to clarify a lot of my confusion

[@RichBehiel]

I’m glad to hear that! :)

[@ocerams1826]

the visual aids in complex representation of the wave are sooooo good

[@RichBehiel]

Thanks, I’m glad you enjoyed them! :)

[@alanmiessler8174]

This music goes beautifully with the graphics and narration. Beautifully edited 👌

[@RichBehiel]

Thanks! :)

[@WindmillEntertainmentGames]

Everything you make is an instant watch for me, I love your videos:)

[@RichBehiel]

Thanks, that means a lot! Glad you’re enjoying the videos :)

[@tune490]

Thank you Richard this was an awesome video. I really can't get enough of physics.

[@ClemoVernandez]

Great video! Really cool animations and clear explanations. Keep up the great work! :)

[@RichBehiel]

Thanks! :)

[@EigenRovak]

I defo wish I saw an animation like the one at 13:40 when I first learned the QHO in undergrad. Would've helped prevent the "ok, now what?" moment after calculating the energystates.

[@younesaitelhadi8135]

Finally! The question that all my physics professors never answered have been answered clearly 🙏

[@hannibalbirca2]

Best explanation of complex numbers ever !!!

[@StephanBuchin]

So well done. Clear and informative video 🙂

[@LucaFanciullini]

Fantastic work, I hope to see yout next videos soon. Good luck!

[@naturegirl1999]

I like your videos, I also like that you explain what the variables mean, I like that the equations are large print, I like that you talk through what each part of an equation means

[@unnikrishnanvr186]

Also amazing video :) hope your channel blows up soon . You truly are a hidden gem of youtube

[@RichBehiel]

Thanks for the kind comment! :)

[@DanielL143]

Ok so this was the absolute best video on the internet (I've watched them all) for explaining the connection between complex numbers and QM. Please please do one on Hilbert space and linear algebra and gauge symmetries. Thank-you sir! -your new Subscriber

[@RichBehiel]

Thanks for the very kind comment! :) I’ll definitely be getting into gauge symmetries and linear algebra, most likely Hilbert space too. So many topics to cover, so little time! 😅 Thanks for subscribing.

[@SampleroftheMultiverse]

13:14 As a science enthusiast my ability excides my knowledge of course. I observed a mechanical effect when I was very young and felt it was interesting to play with. I knew enough about frequencies, amplitudes to know it model something. When I found that I could make structural panels out of it, I thought I could make money fabricating light weight panels using the process. A profession told me it looks like the energy levels of the wave function at 13:14 in your video. The effect touches on several point in QM that it would be worth someone knowledge to study for themself. Also, at just before 13:14 in your video you talk about square or u shape waves. My process shows this just before it jumps to the next energy level. I, not knowing if it relates, think it showing how all possible wave functions are being consider before the next transition. Just my inane thoughts. Your video is very well done. wish I could make a better video on my studies.

[@oversquare6625]

TBH, I dont understand why people dont just say that complex numbers are merely an index into a twist or turn. They are a convenient substitute for angle or radians.

[@filipelqj]

We sort of do, If I understand you correctly. It's called Euler's notation.

[@Daryoushatami]

Me neither❤

[@kbostr]

torsional numbers

[@bernardomarques4306]

This videos are amazing! Keep up the good work, I'm looking forward to seeing more videos!

[@RichBehiel]

Thanks! :)

[@luisabril9692]

Fantastic video! This channel has some serious potential. Keep it up! 😁

[@RichBehiel]

Thanks! :)

[@necrosudomi420thecuratorof4]

thanks i watched lots of complex number video and that kind of stuff and your explanation is A1! good job.

[@RichBehiel]

Thanks, glad you enjoyed the video! :)

[@jamesgray3312]

In before you blow up! Great video quality and concise meaningful explanations :).

[@andytroo]

19:30 the links between Guage theory and Noethers Theorum are amazing - U(1) symmetry Implies Electromagnetism, but invariance of the laws of physics under translation implies U(1) symmetry, and the conservation of charge..

[@omarelzeki_]

amazing content! I cant wait for this channel to grow up.

[@RichBehiel]

Thanks! :)

[@TheWyrdSmythe]

I’ve read that Gauss wanted to call them the “lateral” numbers rather than “imaginary” which makes a lot of sense. The complex plane also makes it clear why +1 x +1 = -1 x -1 = 1, which I’ve always thought was kinda cool. It also makes it clear why sqrt(-1) = i - halfway between +1 and -1.

[@RichBehiel]

“Lateral” would be a much better name! I might start calling them lateral numbers, in hopes that it catches on 😂

[@AdrianBoyko]

To maximize confusion, I vote that real/imaginary terms should be replaced with one of the following: • up/down • charm/strange • top/bottom

[@RichBehiel]

Let’s define three versions of the complex numbers, which differ only in scale 😈

[@cleon_teunissen]

My preference would be to execute the following two renamings: Rename 'complex number', to 'composite number', which I feel sounds less daunting, and rename 'imaginary number' to 'internal number'. The metaphor is then to have the internal component of a composite number as something of an internal degree of freedom of each number on the real number line. There would also be an association with the cyclic property of the internal number 'i'. There is a 4-cycle: i*i*i*i=i (Maybe even rename 'complex number' to 'cyclic number')

[@AdrianBoyko]

@@cleon_teunissen “Binions” with “first” and “second” parts

[@alpirtyx]

Incredible video, you deserve many more subs

[@RichBehiel]

Thanks Alberto! :)

[@richardjowsey]

Well done! I'm currently writing a paper on a novel complex exponential formulation of Special and General Relativity, which is all about complex numbers and phase angles. Also the Poincaré group and U(1) symmetries, so it quite naturally unifies with EM. If you're interested in the exp(iφ) math, I'd be happy to share.

[@richardjowsey]

@@jcd-k2s I've published a couple papers in Fundamental Physics, but this exp() formulation of GR is still being written. I've got all the math done, I'm just wrapping it up in discussion and implications. Yeah, in general, everything is interesting!

[@RichBehiel]

Sounds like an interesting paper! I’d love to read it :)

[@laziz193]

I am looking forward to you unpacking how U(1) symmetry implies E.M. awesome video as always!

[@RichBehiel]

Thanks, glad you enjoyed the video! :) I’m looking forward to it as well, it’s a wonderful concept but it’ll take some building up to. First I’m planning on doing a hydrogen atom Schrodinger video, using that to introduce the Dirac equation, then Dirac plane waves, then Poincare group and Wigner’s classification, then I think the U(1) -> electromagnetism video will make a lot more sense. Actually I should probably do one on the four potential too, like showing how it relates to voltage and the Lorentz transform. Lots of good stuff coming up! :)

[@zacwarnest-knowles9139]

@@RichBehiel that sounds awesome in terms of a build up towards getting a true understanding of how maths and abstract theory leads to the familiar ideas of electromagnetism and chemistry.

[@sdsa007]

this is am amazing program of visual education! Can’t wait to get more!

[@physics_enthusiast_Soorya]

I'm glad I understood/can imagine most of the stuff you said in this video except the Local U(1) symmetry at the end or whatever... and the Quantum Harmonic oscillator in between. Great video. I love you and your channel 👍✨️ Keep it up, you're like a god of mathematics for me 😅 And your Teaching and understanding skills are on the roof!

[@lepidoptera9337]

Unitarity follows from the fact that quantum mechanics is an ensemble theory. The total number of ensemble members does not change during the evolution of the ensemble, i.e. it can be described with a unit vector of constant length one after we apply a normalization. The only (continuous) operations that we can perform on such a vector are (generalized) rotations. For the physical case of a free particle that reduces to a time dependent phase factor.

[@MertowVA]

Incredibly underrated content.

[@chem7553]

I really look forward to your upcoming video!!

[@nopenope3024]

Wow what an amazing video in its subject and especially composition!

[@RichBehiel]

Thanks! :)

[@MinusMedley]

Always get excited when it leads back magnetism, power of the cosmos.

[@brendawilliams8062]

At 18:19 the motion is mesmerizing. Thankyou

[@sdsa007]

i am so grateful for the visual understanding !

[@Arithryka]

thank you so much for this! **rewatches until my brain melts**

[@eventyonwhite686]

you just blew my mind with those tasty intuitions man

[@nicholassullivan6105]

Thank you for the great video! I'm interested to hear what you say about U(1) and electromagnetism. As far as I understand, in the standard model the Dirac field is not really a wavefunction like in QM, as you say in the description. So the U(1) symmetry should really be a field symmetry rather than a wavefunction symmetry, but I suspect the distinction can be a little subtle when representation theory comes into play. There are respectable textbooks such as Cohen-Tannoudji's QM that derive the electromagnetic gauge field from a local U(1) phase symmetry in a single-particle wavefunction, but I'm still a bit iffy on this idea!

[@RichBehiel]

That’s a great point, and now that you’ve got me thinking about it, I’ll have to read up more on the nuances of this idea when it comes to multi-electron wavefunctions. I’ve always heard the idea presented in the context of a single electron interacting with the EM field. In that case, its wavefunction is a bispinor comprised of four complex numbers. I’m under the impression that U(1) symmetry means you can swing the phase of all four of those components by the same amount, at least that’s how it looks based on the Lagrangian. But now you’ve got me second guessing myself 😅

[@fernandohadad]

At 07:00, where you prove Euler's formula using Taylor series, I wanted to share an elegant alternative proof that reveals the deep geometric meaning behind this formula. By defining f(θ) = e^(i·θ) and examining its derivative f'(θ) = i·f(θ), we can write f(θ) = u(θ) + i·v(θ) where u and v are real functions. This leads to a simple system of differential equations: u'(θ) = -v(θ) and v'(θ) = u(θ). Combined with the initial condition f(0) = 1, these equations show that u(θ) = cos(θ) and v(θ) = sin(θ), proving that e^(i·θ) = cos(θ) + i·sin(θ). What makes this proof particularly illuminating is that f'(θ) = i·f(θ) shows us that at each point, the rate of change is perpendicular to our position in the complex plane (since multiplying by i rotates by 90 degrees) - precisely the behavior we expect in circular motion. This deep connection between complex numbers and rotational motion helps explain their natural emergence in quantum mechanics, where wave functions and phase relationships are fundamental to describing quantum states.

[@shabnamalizadeh2525]

your videos are amazing thanks a lot ,your clarity in explaining, the background music ,animations are perfect. I can not find the 3th video about Hydrogen atom ,would you please guide me?

[@RichBehiel]

Thanks! :) I’m still working on part 3.

[@unnikrishnanvr186]

Complex numbers are just amazing , but complex algebra is just... Beyond Traumatizing. Geometry,vectors, functions, and stuff thats applicable for complex nos alone... Its a whole pack! No other topics in mathematics has ever reached its level of greatness in my pov(other than calculus)

[@hodysensei3438]

The way you said “i dunno” is golden man..

[@ThomasGutierrez]

Fantastic video. I will definitely be referring my students to this for its clarity, accuracy, and accessibility. The visualization of local gauge invariance video you are working on will be a great contribution to the scientific communication community. If you could consider crafting a visualization of Dirac spinors and visualization of how chirality and spin and particle/antiparticle-ness interrelate in that context, that would be wonderful.

[@crytp0crux]

For the topic at hand, this video presents an educational strategy and teaching result too great to even measure. It makes us wonder how those of us struggling to genuinely understand difficult abstract concepts were being taught by pretentious beings still struggling to comprehend their own subject material.

[@lepidoptera9337]

There is nothing difficult about complex numbers. The magnitude of a complex number represents a scaling operation and the phase (aka argument) is a rotation angle. The reason why complex numbers come in handy in physics is because we are dealing with a lot of rotations. Quantum mechanics in particular is basically a theory of high dimensional rotations.

[@OnePieceFan4765]

As someone who is starting their undergraduate physics degree this fall this video was at times both scary and very exciting

[@RichBehiel]

The way physics is supposed to be! :)

[@TriangularCosmos]

So great🙌 Imaginary numbers need a better name.

[@connormcmk]

Don't forget that Descartes also thought that the heart beats because when there's no blood in it it's cold so it stretches out and when it's hot it compresses (so he thought the heart was a perpetual motion machine). But yeah, also dualism

[@wargreymon2024]

Probably the best explanation on complex numbers and physics.

[@Sphyrch]

Wait, you're the tungsten cube reviewer! What a coincidence. And great video btw!

[@scottgreen3807]

I can share this with you about complex numbers. I was taught complex ac circuit analysis at the age of 21. Ten years latter and with much practical professional application, I began to completely understand. Electronics uses the “j” operator to avoid the term imaginary number because it’s not. Its answer is impossible but physics and math together handle the situation brilliantly. Wave functions oscillate and require trigonometric function to analyze them meaning we need an answer to the square root on -1. At sixty five years of age, i now find it natural like observing wave in a lake. And I used to think I could explain it. It’s about resolving the electrical reaction to having both capacitance and inductance in an alternating current electronic circuit. Every circuit has natural “parasitic” properties of both reactive components and analysis also introduces resistance as the second “part” of your “number”. See it? You mentioned j operator addition and multiplication, you add in polar and multiply in angular form. Conversions are complicated in the middle of equations when it takes many as in parallel and series circuit calculations.

[@鑰恩曨]

your videos motivate me so much thank you my friend

[@stephenfoster6940]

At 3:36 the point is zero and going up, not down, and at 4:17 the point is zero and going down, not up, as the wave propagates to the right. This doesn’t change the substance of this excellent video though! 😊

[@andreizelenco4164]

Thank you! :) This is really beautiful and inspiring!

[@GamerDS76]

in the start I could follow it but my brain broke in the middle and just kept on watching. Anyway great video, will visit back when I understand more

[@Technoyote]

Love your videos. Here's a nice bit of etymology. The name "complex numbers" refers to the fact there are multiple numbers in one, not that they're complicated. They're complex in the sense of an apartment complex.

[@RichBehiel]

I’m glad you’re enjoying the videos, and thanks for teaching me something new! :)

[@marciliocarneiro]

Congratulations!A very good explanation,but perhaps it would be interesting to search Schrödinger own words explaining how complex numbers entered in the world of physics

[@amoghk.m.6769]

That would indeed be interesting. Where can I read his explanation?

[@Verrisin]

what a scary title, yet this might be the Best classroom introduction of complex numbers!

[@Verrisin]

that part of the video, anyway :D

[@elindauer]

Thank you. This scratched a major itch for me

[@RichBehiel]

I’m glad to hear you’ve been enjoying these videos! :)