Spinors for Beginners 2: Jones Vectors and Light Polarization
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- Опубликовано: 9 июл 2024
- Full spinors playlist: • Spinors for Beginners
Leave me a tip: ko-fi.com/eigenchris
Powerpoint slide files + Exercise answers: github.com/eigenchris/MathNot...
0:00 Introduction
1:09 Electromagnetic Waves
3:36 EM Wave Equations
5:52 Euler's Formula
8:21 Jones Vectors H and V
10:35 Jones Vectors D and A
12:34 Jones Vectors L and R
15:28 Sign convention
16:00 Conclusion + Exercise
![Spinors for Beginners 3: Polarizations and SU(2) Matrices [and O(3), SO(3), U(2)]](http://i.ytimg.com/vi/qICXIY5Dynk/mqdefault.jpg)
![Spinors for Beginners 3: Polarizations and SU(2) Matrices [and O(3), SO(3), U(2)]](/img/tr.png)
Thank you so much for making these. After I finished my physics undergrad degree and left physics, I think this is the only place to go to explain these concepts at a level above a physics undergraduate but without any more graduate training if I'm interested in learning more about physics.
You are incredibly gifted at teaching. I can follow every bit of this because you don’t hand-wave the math. You also give great visual insights and motivation. Please keep making these videos. I am waiting patiently for the next in this series. I think spinors are like the door that divides old quantum mechanics from new, meaning pre-group theory QM from group theory QM. Spinors are also an entre to field theory. They’re at a crux, so it’s important to understand them well.
Cool visualizations! I look forward to the next one.
Sometimes i think it's "Spinors for Beginors" just for the rhyme
This comment is a winor
Very nice coverage of polarization - one of the clearest I've seen.
Thanks Chris. Clear and illuminating as always
Very clear explanation. Thanks!
Great video. This was really well done. I could follow along easily, but at a fast enough pace that I didn't get bored. Thank you.
This video is crazy good! Hoping to see the next video soon :)
This series is immensely interesting.. Please continue..
You're definitely untangling the entanglement of math and physics. Looking forward to the next one!!
Great! Interesting how the pairs of complex numbers pop-out so naturally from the light wave description!
Also, it is very cool that by rotating a wave by Pi we come to the same polarization. So it seems kinda like 2pi turn for it!
Yup. That's exactly why we get spinor-like behaviour for wave polarizations.
@@eigenchris It is funny that we always working with C^2 instead of working with quaternions H directly. Not sure why is that this way? Maybe it is just easier?
In principle, there should be no difference. But probably quaternions could be associated with rotations more easily in some sense :)
All Spin n/2 reps (fermionic ones) of SU(2) are actually quaternionic in nature. That is why we have symplectic spinor metrics for them. And, I guess, can reinterpret any of those as a Sp groups of different signatures.
Waves are periodic, and the simplest periodic motion is going around a circle. Complex numbers are _the_ tool to "go spinny" in 2D. Quaternions are the most natural extension to complex numbers to go spinny in 3D. 3 planes/axes for rotation instead of just 1. Add 1 for the amount to _not_ rotate and you have 2 and 4 components respectively for complex numbers and quaternions.
Complex numbers actually have 2 much lesser known cousins, but neither of them goes spinny. The dual numbers, which basically just move linearly along a lone, and the split-complex numbers, which whoosh along a hyperbola. (Split-complex numbers are useful in special relativity since spacetime boosts act like hyperbolic rotations.) They also have extensions to 3D and beyond like the quaternions, and can even be mixed for 6 different quaternion-like 3D extensions.
@@angeldude101 As a random aside, dual numbers give you "Galilean boosts" in the same way split-complex numbers give you "Lorentz boosts". I'll briefly talk about them in future videos when we get to Clifford Algebras.
Me too I want to say Thank you. This video slammed all the things I've been studying together at once. So good!
Love this channel! 😃
Just in time! I'm doing a lab work on light polarization on friday, thanks a lot!
Getting better all the time !!
[i tried to edit my comment to say] Your discussion here is worth referring to as an example to explain to people why engineers and scientists use complex numbers even for thinking about aspects of nature that *could* be accurately modeled with just real numbers and units of measure, but in relation to which complex numbers bring much convenience to the party.
Excellent video once again.
Very well explained. Great job!
Awesome video, thanks a lot!! I've studied math but never got into physics so hopefully I'm able to catch up but so far great explanation!! :)
Amazing work!!
Thanks for making this series. I am very interested in Penrose's Twistor Theory and I think these lessons on Spinors will lay important foundations for understanding the concepts needed to engage that subject matter eventually.
Just beautifully explained
Incredibly clear🎉
Cant wait for the next video!!!
Excellent! More please!
Excellent video
Thank you. These have helped me with understanding quantum computing better. The basis vectors just have different conventional names (e.g. |0>, |1>, |+>, |->).
Looking forward to the next video.
Excellent video 👍
Excellent!!!
Great Chris.
Sir you are a great teacher
Your the best..Thank You.👌
Fantastic!
Gracias por tu contenido. Es genial. Saludos desde Argentina. Agradezco la traducción.
This was awesome! If it's hard for anyone to follow the math, the easy way to visualize it is this: take a sine wave in the horizontal direction and a sine wave in the vertical direction. Their (vector) sum is a diagonal sine wave. Now take a sine wave in the horizontal direction and a *cosine* wave in the vertical direction -- i.e., a sine wave offset by pi/2 -- and add them. This will produce a circle.
thank you so much its interested one..
Very cool, waiting for new video
Thanks!
So far so good, starts like every lecture I ever attended. First few hours I'm like "boriiiiiiiing... easyyyyyy..." and then I get a punch, I lose consciousness, suddenly I'm sitting in the class with an exam in front of me and zero clue what's going on.
Let's begin the ride!
I'm curious: at what point in the video were you punched out of consciousness?
@@eigenchris I will sub to this thread! hah
@@eigenchris pesky vectors orthogonal to the plane of the screen, they punched him right in the face
@@eigenchris no no, I meant it will most likely come in the future videos :D
@Piotr Kołodziejski Ah, okay. I'll have an ambulance on standby.
Actually come back even you have not finished seems much more understandable. Thanks.
This is great, but I wish you'd mentioned how light can be elliptically polarized too, when the phase difference isn't a perfect multiple of quarter-turn.
Personally, I didn't know about it for a long time, since every description of light polarization I was exposed to was just "light can be linearly polarized in some direction, or circularly polarized with some handedness" with no mention of it.
It's cool that elliptical polarization shows up in the exercises; I just wish you also said it out loud or showed it in the video :)
Yeah, I tend to want to keep videos as short as possible. Elliptical polarization seemed like it wasn't necessary, so I skipped it. I can bring it up in the next video.
Very nice, polished video. One nit-pick on the complex representation of waves however - it is ultimately more elegant (and correct) to pair up each complex exponential with its conjugate. It keeps things real-valued, avoids the artificial sign conventions and (most importantly) correctly transforms under nonlinear interactions.
Can't wait for number 3.
Thanks! often times i am interested in a science topic but i can’t get past the cultural preamble and i end up having to learn more history and philosophy than science and math. For example, i know in the late 19th century, the second unification of physics came about by the work of Maxwell and his peers, but it took me forever for someone on youtube to show and explain the actual math and physics! I still don’t know how Planck derived his law / distribution during the great catastrophe and Im sorry to learn that bipolar Boltzmann was ostracized for a theory of statistical mechanics i still don’t know. I know all the liberal arts aspects of science without fully knowing the actual math or science! Knowing the actual Mathematical Sciences is super important So keep up the good work!
🎯 Key Takeaways for quick navigation:
00:00 🌟 The video introduces the staircase plan for explaining spinors, starting with basic examples in physics and focusing on two examples: light polarization and quantum spin states.
00:54 💡 Electromagnetic waves, including light, consist of electric and magnetic field vectors that vibrate perpendicular to their direction of travel. The polarization of light is determined by the orientation of its electric field.
03:36 🔄 Light polarization can be vertical (oscillating along the y-axis), horizontal (oscillating along the x-axis), or other polarizations obtained by combining these, with their characteristics represented using Jones Vectors.
09:39 🧮 Jones Vectors, 2x1 columns of complex numbers, represent polarization states of light. They consist of amplitudes and phases, containing all the information about polarization.
11:42 🔀 By changing the amplitudes and phases of Jones Vectors, various polarizations, such as diagonal and anti-diagonal, can be created by combining horizontal and vertical polarizations.
14:11 🌀 Circular polarizations (left-circular and right-circular) can be achieved by introducing phase shifts to vertical and horizontal polarizations, resulting in helical waveforms.
16:25 ⚛️ Jones Vectors representing wave polarizations are also spinors. In the next video, it will be shown how special matrices (SU(2)) can rotate these polarizations, revealing an angle-doubling relationship between physical and polarization spaces.
the complex component to an EM wave determines the DIRECTION of the wave momentum(vai Schrodinger).
Hello eigenchris!
I want to thank you so much for taking the time to make these videos. I have no idea how much I appreciate it! I have a question for you:
Do you have a rough plan or timeline for when you are planning on completing this series on spinors? I am currently writing my masters in physics and your idea of spinors as minimal ideals explains a lot of phenomenon that I am writing about! I am handing in my masters early in June. Perhaps a decent donation or two might expedite the process?
I saw your videos on spinors on the laroslav channel. I noticed that you explained the concept to an audience that already was familiar with the concepts, and thereby lost some of the audience on RUclips. I think making these videos on your own channel for the more general audience is a very good idea.
Things always take longer than I expect. I would guess I'd arrive at Clifford Algebras by Spring 2023 or so, but I can't guarantee anything. Donations won't help me work faster. I'm basically just going to release the videos when they're ready, whenever that happens to be.
If you're looking for a source to learn about spinors and Clifford algebras, you can download the thesis "An Interpretation of Relativistic Spin Entanglement Using Geometric Algebra" by Crystal-Ann McKenzie (just google the title + author and you should find it). In particular, Chapter 2 and Appendix B are probably worth reading for you. You can also leave comments on any of my videos if you have a specific question.
I'm curious what your thesis is on?
The convention to be used depends on the phase sign chosen :
Eᵧ(t,z) = A cos (ωt − kz ± φ) = A cos (kz − ωt ∓ φ)
= A sin (kz − ωt ∓ φ + π/2)
Thank you for putting all these, now it is clear that the introduction of complex number into equation is not about the imaginary part or any usefulness of its properties i^2 == -1 or introduce another dimension of properties. Simply because of using Euler’s equation to make the formula neat and clean. Why no one or textbook explicitly told me that up front?
idk why but I really want to watch you stream
like a game or streaming physics or something
At 14:34 It was first confusing to me because a left polarization traces a helix that seems to twist according to the right hand
VERY NICE NINJA!..YOU HAVE JUST DESCRIBE ALL THE POSSIBLE DIRECTIONS IN WHICH LIGHT COMES OUT OF A FREAKING LIGHT BULB..JAAAA.....I THINK AM GOING TO SUBSCRIBE TO YOUR CHANEL...
also interesting that two circularly polarized beams of light with a certain phase difference can restore vertical/horizontal polarization!
Yes, any 2 polarizations shown can recreate the others with the right coefficients. The space of polarizations is 2-dimensional (2-complex-dimensional, technically), so you only need 2 independent basis vectors to create any combination.
What books do you take reference from?
Hello, awesome job. Have a question 9:12. Why is it the waves we are super imposing have the same frequency and k? Is it choice?
I'm assuming they have the same frequency.
@9:06, Why must the frequency and wave number be the same between each axis? Is this an inherent effect of a singular emitter to get the waves to overlap?
In theory, it doesn't have to be the same. any number of waves with different frequencies can be superimposed on top of each other. This is for the case of a single plane wave, where all the components have the same frequency.
There are two conventions of phase angle, wt - kz + phi or kz - wt + phi, as you mentioned. I've read in some reference (not clearly remember) that the physicists, especially in the quantum mechanics, prefer the convention of exp(-i(wt-kz+ph)). The phi in two phases of wt-kz and wt-kz+phi represents the phase shift of the two waves. The Schrodinger equation, momentum operators...all follows this convention as I know. Actually any conventions are just preference and so they do not make any difference in the real physics. But I think it would be a little more good to follow the conventions that many people prefer.
And one more thing. The definitions of the right-handed or left-handed polarizations can vary depending on the viewing from the receiver or source. (In Wikipedia of Jones Calculus, the basic convention is "kz-wt" used by Hecht, Circular polarization described under Jones' convention is called : "From the point of view of the receiver." In this convention, (1/root(2) (H + i V) is the Lef-Handed/Anti-Clockwise Circularly Polarized from the point of view of the receiver.). I think it would be better if such an explanation was added in your video. But overall your lectures are very very good!!!. I really continue to support you.
Thanks for the info. The way I define left/right circular polarizations includes the direction of wave propagation in my definition (the thump points in the direction of wave travel). So it doesn't matter from which perspective you view the wave.
Great video! At first I was like "I thought polarization vectors were just your run of the mill vectors" so I was on the edge of my seat trying to figure out when the vector-like behaviour of the electric field would be upgraded into a spinor-like one... Can't say I got it though... Maybe when we started ignoring the travelling wave? Or maybe right from the start when we considered this kind of "wave structure"? Or maybe that's a bad setting from which to look at it entirely. Anyway, eagerly looking forward to see what you have next!!
It's related to the fact that there are multiple Jones Vectors that can describe the same polarization. For example, [1, 0] and [-1, 0] describe the same polarization, just shifted by a half wave cycle. Polarization space can't tell the difference between [1, 0] and [-1, 0], so rotating from [1, 0] to [-1, 0] is viewed as a full turn. Rotating from [-1, 0] back to [1, 0] is a 2nd full turn. If you try the "homework" problem at the end, you'll see that there are only 3 total polarizations among the 6 Jones Vectors shown.
@@eigenchris Wow I see. So it's like the polarization is an entire plane instead of a single direction... which smells a lot like geometric/ Clifford algebra. Damn I can't wait for your next videos
Yeah, there's a kind of "projection" operation that can be done in Clifford algebras which makes it impossible to tell the difference between sets of clifford elements. While I'm familiar with the formal math behind this procedure (it's the "left minimal ideals" thing I mentioned in video #1), I'm not sure I understand the intuition behind it, or how it connects to the stuff in this video. Hopefully it will become clear to me by the time I get to those videos.
Damn, they didn’t mention how useful Jones vectors were for spinors in my optics class.
7:42 Can the fake imaginary part be interpreted as magnetic component of the EM wave?🙈
Please make the next video.
It is called phasor in EE😊
Wait, so when circular polarized, the magnitude of the electric force vector doesn't change, just direction?
So, then, frequency is like the rate of travel of the peek of the e force vector around a 2d circle, in circular polarized the surface of the 2d circle is facing the direction of travel like your diagram, in vertical polarized , then, it's like that 2D circle is rolling towards you in the direction of travel and we're ignoring the z component of the e field so we just see it go up and down to a circular function.
Okay then if I'm right so far, if we combine a vertically polarized wave with a circular polarized wave, do we get an elipse specificly or a different oval shape?
Yes, elliptical polarization is possible. A couple of the Jones vectors in the exercise at the end of the video denote elliptical polarizations.
6:55 I've personally find that just ignoring the imaginary part isn't the most intuitive explanation. I instead prefer to think that an actual electric field is always a mixture of a positive and negative frequency signals which have phases such that the imaginary parts cancel. Then it's sufficient to talk about the phase (and amplitude) of the positive frequency component because that uniquely determines the phase (and amplitude) of the negative frequency component when we require that the imaginary parts cancel.
I'm curious to learn why H, V, D, and A are considered "special" and we have spent no time one other linear combinations. I'm sure it has to do with SU(2) but I love to hear it explained. Regardless, this series is already invaluable.
There's nothing special about them. You can tilt your head (change your coordinate system), and A will become H, H will become D, D will become V. Any linear combinations are equally valid. But I felt the video was long enough at 17 minutes, so I didn't go into the details.
I believe it is related to the fact that the {D, A} basis is "the most different" basis to the {H, V} basis, in the sense that a measurement of a photon with D polarisation in the {H, V} basis will return either H or V with equal probability. Similarly, the {R, L} basis is just as different. I think the technical term is mutually unbiased bases.
The {R, L} basis is often regarded as the "most" natural because the coefficients are invariant under rotational symmetry (they don't depend on the chosen orientation for our x-y axes).
@@eigenchris Tangential fact - modern 3D cinema encodes stereo vision using the {R, L} basis because it is not affected by head tilt.
This should be required supplemental material.
When will you post your nexts vedios on spinors
Most likely in early January.
Great video! The diagram of the electro-magnetic wave at 1:44 is wrong. The E field and
B field are always 90 degrees out of phase. The diagram shows them in phase.
You mentioned that in the last video, but I don't believe this is true. Why do you think it is? You can try reading this: farside.ph.utexas.edu/teaching/em/lectures/node48.html
Hey man, what is your process? I want to learn how to condense so much information. Any advice will be appreciated, from anybody here.
For researching this, I started around January 2021 and just read every PDF and article on spinors I could find, and also some sections of a couple textbooks. A large portion of it was too hard for me to understand, so I made notes on the stuff I did understand and eventually presented it over a few discord calls to a math group I was in (you can still find that video on my channel from summer 2021). Having to explain it to others forced me to go over everything I knew and explain it in the best way possible. Now, that video is still pretty disorganized and has a few of mistakes. But over the next year I just kept trying to fill in the blanks in my understanding. Sometimes that would involve reading more, or drawing pictures, or talking it over with the math group on discord to hear their views.
Around Summer 2022 I felt I was finally ready to start this spinor series. I laid out a roughly 20-video plan on what I wanted to cover (although it will probably end up being longer--things always take more time than I think they will). Spinors are pretty abstract, and with abstract topics I usually take the "staircase" approach where I start with the simplest and most physical explanations possible, and then slowly work up to the more abstract definitions. After that I just go video-by-video, making the slides and graphics in powerpoint and doing the best I can to explain the topic in the way I would have wanted it explained to me 2-3 years ago when I started learning about it.
That was a long answer. Not sure if that answers your question!
@@eigenchris , so this series is two years in progress. Just color me impressed. I can't wait to binge through the series.😄
If the wave is moving in the z direction ala 5:12, shouldnt it be (kz-wt) then, and not (wt-kz) ala 4:32?
If we take cos(t-z) and look at t=0, z = pi/2, we get cos(0-pi/2) = 0. (This is the red point just in front of the origin where the wave crosses the z-axis.)
If we wait until t = pi/2 and look at z=pi/2 again, we get cos(pi/2 - pi/2) = 1, which is a peak. This shows the peak of the wave has moved forward from z=0 to z=pi/2.
@@eigenchris ah right, cos is an even func so in this case it doesn’t matter
👌🤝
1st of all these video series are awesome! And I still need to satisfy my curiosity.
It was always curious to me what EM waves actually are? As far as I understand: the waves are oscillations of the environment. For example sound is an oscillations of an air. Or we can talk about water waves which are an oscillations of a water.
So following this logic am I right thinking that there is some special environment spread out through the space that is disturbed by charged particles and we can observe it like EM waves? And more over charge itself is a property of particle’s influence to the env?
EM waves are oscillations in the Electro-Magnetic field itself. This is a bit hard to describe since we can't directly see these fields. In the mathematical description, every point in space has an arrow for the E-field and an arrow for the B-field. You can "see" the B-field arrows/lines when you sprinkle iron filings around a magnet, but they are always there even if you can't see them.
Any time a charge accelerates, it will create a wave in the EM field. This is how antennas work: they accelerate electrons back and forth at a regular frequency, thus creating an EM wave of a given frequency.
Ok, so then I have to ask what is EM field? Does it exist by it's own?
@@piradian8367I'm not aware of a "simpler" explanation of the EM field. I just take it for granted that it exists.
Sorry, I'm not trying to find simple explanation. And I'm aware of Maxwel's equations which give us what you told above. Maxwell's interpretation requires introduction of the electric and magnetic fields. The reason why I ask is could we suppose that the properties of an electromagnetism are the properties of the space itself. Let's say like this the negative charge kind of "curves" the space, and the positive charge "curves" the space in opposite way.
@@piradian8367What you're describing sounds more like the way gravity is described in general relativity. All the formulations of E&M that I'm aware of require the existence of an E&M field, or something like it. It can't just be treated as a geometrical effect, as is the case with gravity.
At 14:37 you say the vector we have just calculated goes clockwise around the z axis according to the left hand rule but the red spiral goes in an anticlockwise direction.... (in both diagrams) and doesn't follow the fingers..surely it's a right circular polarisation you have drawn? Not a biggy but it messed with my head. It would be ok if the wave was travelling in the negative z axis direction?
Amazing videos though, thank you. I have watched them all, I am now going though them with pen and paper in hand...
14:56 oh, I'm really confused now. Shouldn't L = H - iV if e^ipi/2 = i = is a anticlockwise quarter turn around the z axis and we are going clockwise?
It's a bit confusing, but you have to imagine yourself frozen at the origin and looking at that the arrow in the XY plane does. The animation on the right has a black circle frozen at z=0 and you can see the arrow does travel in a clockwise direction.
If you look at the diagram on the left, here we've frozen time instead of freezing space, and in that case it does appear like the spiral moves in an anti-clockwise circle as z-increases. But I'm talking about the z=0 frozen case with t increasing.
@@eigenchris Ooo. Ok. I'll have another look. Thanks.
@@eigenchris Oooooo. The wave is travelling *through* the origin not a vector propagating forward from the origin. So the vector obeys the left hand rule but the spiral goes anti-sense as whole wave is moving forward.... I had the insight when opening a bottle of wine to try and calm myself down. The cork-screw spoke to me....
Wait, if the real part is the electric wave, I don't think you need to say we're "inventing" the sin part when we go to the vector form, it should be the magnetic field that we choose to ignore earlier, isn't it?
I'm not sure if that's something that can be done, but for the purposes of this video, we're just ignoring the magnetic wave altogether. The complex exponential notation is just a convenient way to write the electric field wave.
4:20. 4:50. 5:57. 6:39
will the series continue?
Hoping to upload the next one next week. Been taking a bit of a break over the holidays.
@@eigenchris Oh ok.. I'm waiting for it with great excitement. Thanks for these beautiful videos.
instalike!!!!!!!!
Yes, excellent presentation, very clear.
Is that possible for an real electromagnetic wave to have a phase shift as imaginary part? I mean phase is a complex number this way.
Not sure I understand the question. I'm ignoring the imaginary part of the EM wave for physical purposes. The complex number representation just makes it easier to write a phase shift using multiplication by e^i*(theta).
The imaginary part will be ignored for getting real polarizations and magnitude of jones vectors. However photon is a quantum particle and have complex wavefunction before making a measurement. May be that's why you used the Euler analogy. @@eigenchris
It looks like a Jones vector is a vector of phasors only. There is something else?
That's basically it.
tip: vectors look better when the top arrow doesn't cover the subscript. This is \vec{e}_x instead of \vec{e_x}
Probably true. I've been writing them the 2nd way for 5 years now, though... I'll most likely just keep doing it this way.
Great job, man! but you messed up a bit visualizing right-handed polarization while talking about left-handed😬
I think it's correct. You have to imagine a point in the xy-plane travelling in a circle around the original as the wave passes the xy-plane. It's not what you would guess at first looking at a still image of the wave.
😀
Look at it classically. A charge is moving. With Fourier transforms we can split any movement into a sum of circular movements, so let's say it's moving in a circle. Its movement creates an electromagnetic field, and the radiation part of it is only from movement perpendicular to the direction it travels.
So if you look at the circularly-moving charge from anywhere in the plane it's moving, you get a linear sine wave. That's it's polarity. Look at it perpendicular and you get a circular sine wave. At any other angle you see an ellipse.
That's all there is to it. The math just describes that. Nothing more.
Great video, but where are the solutions to the exercises? At least some Jones vectors lead to non-normalized solutions.
Answers should be linked in the description. You can have non-normalized Jones vectors, but the polarizations don't care about the normalization. If you scale a jones vector by 5, the polarization doesn't change. So for polarizations we can just choose a "representative" Jones vector that is normalized to 1.
@@eigenchris I've found it on the github, thank you.
Spinors for Beginnors
There are 3 directions that are 'up' and can be polaraized. There's up/down, left,right, and even forward-backward, which is harder to detect, but present none-the-less.
How can you polarise a wave forward if the forward component is zero?
you can't polarize the wave forward-backward if that moves with a speed of light into that direction, right?
@@karkunow @Iaroslav Karkunov that rotates at the speed of light; although interestingly doesn't actually get to rotate because it's moving at the speed of light. One side of the spin is 0 and static, the other side is the speed of light (actually I guess it has to be 2c for a instant on the opposing side). If your axis can be sideways, and get a wave horizontal/vertical, then in that direction part of the wave is stationary. Sort of goes back to Newtons first law with a extension; a thing moving in a direction will keep moving that way unless an external force; but also a thing spinning around an axis will keep spinning around that axis unless a force changes it. At the macro scale we have at least 7D to describe things and their physics; why wouldn't the same dimensions apply all the way down?
There is also helically polarized light; though I think that's more about the emission in time and is really mutliple photons. There was just a video that showed that circle too.. (but the circle of the wave wasn't also moving with the wave)
@@karkunow ruclips.net/video/a6z1eqekX44/видео.html I think
Hey eigenchris, nice video showing the correspondence between spinors and Jones vectors.
Your video inspired me to do a lecture on optics (ruclips.net/video/vTWslacroLQ/видео.html). Here, I use conic sections to show why polarizations are particular shapes. Additionally, I discussed the Jones vectors and how they give rise to different polarizations. I guess the novelty of my video is that I show you can superpose Jones vectors to yield new polarizations. The other beautiful thing is that I show the correspondence between differential operators and physical variables.
I thought I was one of the first people to talk about the connection between the Jones vectors and the Bloch sphere, but it seems that you did it in a subsequent video.
Once again, thanks for the inspiration!
Hello, I couldn't find your email contact so please allow me to make inquiries here. I am an assistant professor at UCR, where I teach linear algebra, electromagnetics, among other courses. I find some of your animations and pictures particularly illustrative, so I was wondering if I could have your permission to use part of them in my classes (I will make clear of your credits if I do so). Thank you very much!
Sure, go ahead. Thanks. If you ever want to try making something similar, I made them with 3D models from paint3D, and animated them in Powerpoint with the "morph" option in the "Transitions" tab.
@@eigenchris Many thanks!
I think we'll be in the dark ages for as long as the statistically average person believes that one (1) is a number.
Nifty.
spinors for beginors
This really should be the presentation method long before the math itself is introduced
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EM Waves are 90 degrees out of phase. Current (electrostatic) presence is ahead of voltage(electromagnetic). They do not occur at the same time... (1:47 animation)
"Current (electrostatic)"??? Electrostatics concerns stationary charges not moving ones lol
@@kepe7323 you mean you're not also going to say voltage is a static thing and doesn't move? EM both the charge and magnetic field are in motion; otherwise you'd have a M wave not an EM wave propagating.
@@3zdayz I'm saying current is an electrodynamic phenomenon, not an electrostatic one
Suppose E(t, z) = A cos(omega t - k z), with A, t and k three fixed real numbers.
If you take the four Maxwell equations (in vacuum, so just the four with E and B) and plug this E in there, you get that B is described by B(t) = A/c cos(omega t - k z), where now c is the speed of light.
In particular, B is orthogonal to E (as a vector).
So yes, they occur precisely at the same time. That is, for a free wave in a vacuum, E and B are **in phase**.
If you want a mathematical intuition about why that is, first notice that: the derivative of a cosine function is pi/2 out of phase with it, and therefore the integral of a cosine function is pi/2 out of phase with it as well.
Taking the second Maxwell equation we have that the curl of E (which is a derivative) is equal to the derivative in time of B. Therefore, the derivative in time of B is pi/2 out of phase with E, which means that B itself must be in phase with E. (or, at most, they will be pi out of phase. This, however, doesn't change a thing because it just means they are in phase with the negative of each other, it just flips one of them)
Holy hell, I got 95% of that...
I code computer games / games engines so I know basic vectors and matrices. Plus, I do neural nets, same thing but in higher dimensions. What I don't know is Euler's formula which you presumed knowledge of but still, you say you horseshoed the imaginary component in. Plus, with imaginary numbers, it's just a function (computer sense) where you return a negative number for negative multiplication and obviously that results in rotation. Honestly, I don't know why you bothered with it when a simple rotation matrix would do the job but ok.
The basic rotation transform. No problem whatsoever. I could code that in ten mins. It's just bog standard rotation matrices, nothing fancy at all.
So you're just saying that the rotation over time is a two turn to return to the same overall state. Much like - on the very macro scale - the solar cycle is 22 years but 11 years in each polarisation. That's self-evident, a geometric inevitability.
Ok, I appreciate this was the easy step. I absolutely could code this.
Except... It gives no explanation whatsoever as to why nature behaves in this way. All you're doing is modelling it.
"It gives no explanation whatsoever as to why nature behaves in this way. All you're doing is modelling it" is pretty much what physics is, when you get down to it. I have no idea why spin-1/2 particles exist in nature, and I don't think anyone does. But people have managed to figure out the mathematics of how they behave.
@@eigenchris Huh. So it's a tautology of sorts, that's unsatisfying. "Why does the bird fly? Because it's in the sky" sort of thing. Well, I suppose in fairness that's not modelling anything but still.
Some things are fully explained in physics though. Galileo's Principle of Relativity - ok, it's a bit axiomatic but minimalistically so and it's complete. Newton's 1st follows from it; do nothing, nothing happens. No need to programme that one in a sim.
S = k log W makes sense, on a fundamental level I mean; no further explanation required. Complex systems result in dispersal / arrow of time.
Point is that physics often gives complete explanations on other things, why not this? There must be more to it. I mean, if matrices work, then clearly it's geometric.
Very interesting but why the unnatural voice ? It's very distractive.
I'm have a pretty monotone voice.
Unfortunate choice of sign convention, as in almost all cases, it's the other way around.
I came across both sign conventions when doing research for this video.
When will you post your next vedios on spinors
Will aim to post by this weekend.