Complex Numbers in Quantum Mechanics
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- čas přidán 8. 06. 2024
- A brief introduction to the use of complex numbers in quantum mechanics. This video is intended mostly for people who are learning quantum mechanics and have some familiarity with things like the quantum harmonic oscillator, or the hydrogen atom, but might have some confusion around what all the complex numbers are all about. I hope this video provides you with an improved sense of familiarity with the complex numbers. These things are cool. They take a bit of getting used to, but they're cool.
My main goal in this video is to make the complex numbers feel as natural and accessible as possible, so I emphasize the perspective that the complex phase can be thought of as a generalization of positivity and negativity, and in particular that the phase oscillates between two poles (which I half-jokingly refer to as yin and yang). This approach, though real-part-biased, is motivated by the observation that the interference of two waves of the same frequency (constructive, destructive, and everything in between) provides a natural picture for one of the things the phase of a complex number might mean. I hope that helps to demystify how complex numbers are not an entirely absurd concept, because a stumbling block for many people, myself included for a while, is that the complex numbers seem too unrealistic for human intuition to sincerely glom on to. But, as I hope this video shows, the complex numbers can be made intuitive.
It should be noted, however, that the story does not end here. Once you are familiar with the complex numbers, you should stretch your mind out again by regarding the complex numbers as equipping a model with a circular degree of freedom. In particular, you can imagine a wavefunction as a section of a fiber bundle whose base is spacetime, whose fibers are circles of mysterious origin, and whose total space is some fragment of this thing we call reality. That should keep you up at night!
I should also add that the "U(1) Symmetry implies Electromagnetism" argument may well be completely backwards. It is true that, if one takes the Dirac field with minimal coupling to the photon field, and imposes local U(1) symmetry by fiat, then all the beauty of classical electromagnetism follows. But one can easily argue that such an imposition is contrived, and more indicative of a redundancy of our model than a genuine symmetry of physics. That argument is strengthened in light of Wigner's classification, pun proudly intended, since if we take the masslessness of the photon as our starting point, then the photon can only have helicity eigenvalues of +-1, not 0 (the photon has no rest frame), and therefore one must remove any physical contributions coming from longitudinal photon modes, since they cannot exist. This fictionalization of the longitudinal modes yields precisely the usual gauge symmetry of the four-potential (or so I am told... still need to work out for myself why this is true), and once you have the gauge symmetry of the four-potential, then your Dirac field better have local U(1) symmetry if you want to preserve minimal coupling!
Anyway, whichever direction of the argument is more true, it is still a beautiful idea that local U(1) symmetry of the Dirac field, and the usual gauge symmetry of A, and the masslessness of the photon are, for all intents and purposes, the same thing. It is still an open philosophical question as to whether all this symmetry and gauge freedom is a genuine reflection of natural symmetry, or of mere theoretical redundancy; that question boils down to whether the transformations involved are active or passive, respectively, and that quickly gets into some murky existential territory when you really think about it. Fiery debates are ongoing around these questions. But that's a topic for another time, and not one which is answerable within a CZcams video description.
Thanks for watching & reading :)
Chapters:
0:00 Introduction
1:00 Real vs. Complex Numbers
2:48 A Wavy Wave, Waving
4:33 Complex Representation of the Wave
7:48 Complex Addition, Multiplication, and Interference
12:10 Fourier Analysis & Superpositions
12:47 Examples: Harmonic Oscillator and Hydrogen
14:30 Plane Waves
16:49 Probability Density
18:07 U(1) Symmetry Implies Electromagnetism
#physics #quantum #math
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 CZcams'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 :)
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.
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.
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 :)
The way you're speaking is more than fine, nothing to worry about. Also, I didn't even notice the background was moving, lol
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:-
czcams.com/video/AxPwhJTHxSg/video.html
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.
I loved the motivation of complex numbers as extending the sense of "sign"/phase/direction from being discrete to continuous
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.
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
Probably because the technical fields have the stereotype of being unbearably dry and "unhuman"
His voice also has good harmonics
yeah, seriously. The narration has that “tuck you in at night and read you a bedtime story” feel. The visuals are incredibly intuitive too.
Finally, a very easy and comprehensive way to explain why complex numbers are so important for wave mechanics
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
WHY HAS NOONE EXPLAINED THIS TO ME LIKE THIS SO FAR this make so much sense
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.
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
Descartes … “that and Dualism” 😂😂😂😂 … and now you are one of my favorite people ❤
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
Thanks James! :)
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!
I’m so glad to hear that! Thanks for the kind comment :)
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.
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 😮
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.
Finally! The question that all my physics professors never answered have been answered clearly 🙏
Wow this video is incredible! It's just a matter of time before this channel becomes huge, amazing content!
Thanks, I’m glad you enjoyed the video! :)
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!
This is one of the most beautiful math videos i've seen. I hope you will continue doing them.
Thanks! :)
Сразу понял что ты русский)
@@user-vm1hi7bo5s как?
@@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
Beautiful explanation! I was waiting for an intuitive understanding of the imaginary numbers! Greatly appreciated🍀
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
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.
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
This deserves 1M+ views - the question that was answered in this video brought a lot of existential satisfaction 👏
Great timbre and natural Narration. Very pleasant and easy to follow.
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. 😂)
So lucid and comprehensible, tremendous job!
Thanks! :)
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!
Thanks, I’m glad you’re enjoying the videos! :) It does take quite a lot of time, but it’s very satisfying.
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
Thanks for the kind comment, I’m glad to hear you enjoyed the video! :)
I am looking forward to you unpacking how U(1) symmetry implies E.M. awesome video as always!
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! :)
@@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.
this is am amazing program of visual education! Can’t wait to get more!
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!
Thanks Karson! :)
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)
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 CZcams 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...
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.
Dang, I should have done that! 😩 That would have been so cool.
Omg your visualization of a coherent state of the harmonic oscillator at 13:00 is FANTASTIC! Nice work!
Thanks! :)
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.
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.
You are gonna be a star, I am looking forward to the upcoming contents from this channel!
Thanks Vikash! :)
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!)
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.
thank you so much for this! **rewatches until my brain melts**
You are an amazing presenter. This channel deserves way more subs.
Thanks! :)
Never heard the complex numbers described as a generalization of binary directionality. Really cool stuff.
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!!
The way you said “i dunno” is golden man..
Thank you sir ! This will be very useful for my PhD thesis !
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
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.
And I wish all CZcams comments were as kind and flattering! Thanks for watching the video, and I’m glad you enjoyed it :)
Can you please tell me in few words what you understood about the use of complex numbers in QM from that video?
@@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
Everything you make is an instant watch for me, I love your videos:)
Thanks, that means a lot! Glad you’re enjoying the videos :)
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.
Great video! Really cool animations and clear explanations. Keep up the great work! :)
Thanks! :)
So well done. Clear and informative video 🙂
Thank you Richard this was an awesome video. I really can't get enough of physics.
i am so grateful for the visual understanding !
This music goes beautifully with the graphics and narration. Beautifully edited 👌
Thanks! :)
12:50 What a beautiful and profound animation. Really captures the essence of superposition, doesn't it!
Always get excited when it leads back magnetism, power of the cosmos.
Fantastic work, I hope to see yout next videos soon. Good luck!
Best explanation of complex numbers ever !!!
In before you blow up! Great video quality and concise meaningful explanations :).
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!
As someone who is starting their undergraduate physics degree this fall this video was at times both scary and very exciting
The way physics is supposed to be! :)
I really look forward to your upcoming video!!
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.
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
I’m glad to hear that! :)
This videos are amazing! Keep up the good work, I'm looking forward to seeing more videos!
Thanks! :)
So great🙌 Imaginary numbers need a better name.
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
Fantastic video! This channel has some serious potential. Keep it up! 😁
Thanks! :)
Wait, you're the tungsten cube reviewer! What a coincidence. And great video btw!
Incredibly underrated content.
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.
Also amazing video :) hope your channel blows up soon . You truly are a hidden gem of youtube
Thanks for the kind comment! :)
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)
At 18:19 the motion is mesmerizing. Thankyou
your videos motivate me so much thank you my friend
Thank you! :) This is really beautiful and inspiring!
the visual aids in complex representation of the wave are sooooo good
Thanks, I’m glad you enjoyed them! :)
Wow what an amazing video in its subject and especially composition!
Thanks! :)
thanks i watched lots of complex number video and that kind of stuff and your explanation is A1! good job.
Thanks, glad you enjoyed the video! :)
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
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.
amazing content! I cant wait for this channel to grow up.
Thanks! :)
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.
“Lateral” would be a much better name! I might start calling them lateral numbers, in hopes that it catches on 😂
To maximize confusion, I vote that real/imaginary terms should be replaced with one of the following:
• up/down
• charm/strange
• top/bottom
Let’s define three versions of the complex numbers, which differ only in scale 😈
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')
@@cleon_teunissen “Binions” with “first” and “second” parts
From 14:36 to get a grasp of the environment of the electron: two ways to interpret SU2.
1. A grid plus marbles
An infinite chessboard grid where every crossing is not a point, but a marble.
The location of an electron is the combination of the grid point, and as well the spot on the marble at this grid point. The position on the marble determines the spin.
2. Onion plus grids
An infinite onion, where every spot on an onion layer is not a point but an infinite chessboard. The location of the electron on the onion shell determines the spin.
Now that we know SO3 is just our interpretation of what the electron is doing, it may be wise to pick the model the least resembling SO3. What is: Onion plus grids.
According to this model our 3D space is a layering of grids sown together by spin.
Absolutely brilliant!
Incredible video, you deserve many more subs
Thanks Alberto! :)
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..
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!
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 😅
09:07 I want this animation as my computer desktop wallpaper.💯
im mesmerized by your animation.😉
god i love your style so muchhhh
This is so well done!
Thanks! :)
I was just listening to 2pac and this randomly popped up next in my recommendation. Did NOT disappoint.
Great work with the animations.
Thanks! :)
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.
I don't know much about complex formulation of GR, but complex formulation of SR has already been done a while ago. Einstein stated it was useless.
@@richardjowsey Do you have a website or papers so I can make my own idea about whether it's original, interesting, cranky or revolutionary? Not that I am an expert, but I'd like to know what you mean. In general everything is interesting.
@@benoitavril4806 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!
Sounds like an interesting paper! I’d love to read it :)
what a scary title, yet this might be the Best classroom introduction of complex numbers!
that part of the video, anyway :D
quantum phys final in a few days… realizing my foundations of this subject were not quite accurate 😅
Super thankful for this video tho as I am a visual concept learner.
Cute, love it. Time. Convergence. Cool. Field theory, still one dimensional. Time regression, kind of. How to achieve wave function, still up in the air for one dimensional algebraic schools of thought.
Thanks for the Descartes dualism remark instant sub (although the rest of the content itself is sufficient for an instant sub)
Great video, thank you 👍
You’re welcome, thanks for watching! :)
Very nice!
Excellent presentation. Since 2021 a new interpretation of quantum mechanics will reinforce this video. It deals with the known idea that the universe is composed of "stuff in a media." This interpretation says that the "stuff" are the elementary particles and the "media" is the quantum space in oscillation. In such a way that the space oscillates between the observable 3D and the 4th dimension. The particle will be randomly at 3D meanwhile its space is. The total energy, momentum, charge, etc are contained in this 4th dimension that makes essential the management of complex numbers. The real ones deal with some physical parameters and the imaginary with the others. The presence of these physical parameters is out of phase as described by Heisenberg's Uncertainty Principle. You can read more in the book titled "Can relativity and quantum mechanics go together?" hope you like it and get inspired regards.
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.
I like being reminded that doing physics is super wholesome. 😊