Dirac Equation: Free Particle at Rest

[@sirwinston2368]

BSChE here. About ready to retire (13-14 mos.), move 500 miles, and go back to my alma mater (Aug 2025) to get a degree in Physics. This stuff is awesome. One of these days I am going to understand all of this. I swear. Your videos help. Thank you.

[@RichBehiel]

That’s awesome! It’s always inspirational to see people pursuing their passion for physics. And I’m glad these videos can help :)

[@sporefergieboy10]

🤨 I hope your work is paying for that

[@anth2]

I feel the same way friend. Started with a question and the MIT open courses lectures this Spring. It’s the first thing I’ve ever wanted to dedicate my life to learning about.

[@NolanFN.]

@@RichBehielhi

[@RichBehiel]

@Nolan_Rutledge hey Nolan!

[@cygnus_zealandia]

When I did Uni Maths and Physics around 45 years ago, there were no animations, just rapidly revolving blackboards moving so fast that it took most of my energy just to get the algebraic content down on paper. I'm so grateful for these beautiful visualisations and mathematical summaries. Many words, definitions and explanations are coming back to me all these years on, and much more of it is making a lot more sense. Thank you for presenting such content this way to re-view this content I remember of my youth, when I was a very visual student and struggled with the lack of visual content (and no animations). Visual content was part of my high school learning but it was not present much at University. Animations have made much of this material more enjoyable and comprehensible. I'm enjoying this immensely. Thanks again.

[@RichBehiel]

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

[@mariocesarsousa]

I am in the last year in physics. I feel I need 4 more years. This channel is great. I learn Physics and improve my English. ❤

[@sshkatula]

This is so much more interesting than back in the days at university! Thank you!

[@RichBehiel]

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

[@anth2]

This video series IS SO GOOD. Your visualizations are UNPARALLELED

[@RichBehiel]

Thanks, I’m glad you’re enjoying these videos! :)

[@arshraghuvanshi2039]

Quality teaching, as always, and a damn shame this doesn't get more views. But bro you gotta stop ending vids with cliffhangers😭😭 always have me looking if the next part has arrived...

[@RichBehiel]

Thanks! :) Honestly I try not to end with cliffhangers when starting to work on the video, but one concept always leads into another, and I have to end the video at some point 😅

[@armagetronfasttrack9808]

Great video! I hope in a future video you cover in detail the interpretation of the electron/positron components and how that relates to charge conservation through time. I did a final project a few years ago where I simulated (1D) Dirac electron wavepackets with momentum boosts and simple external potentials. I saw that you can initialize a wave packet with purely "electron" components (positive frequency components) and over time some of that electron component will transfer to "positron" components (negative frequency components) simply by initializing a wave packet with some spread in momentum space. I never fully resolved my confusion over how initial electron components can transfer to positron components and have charge conservation be maintained. I suspect it has to do with the electron/positron definitions not exactly corresponding to simply upper/lower components when momentum is non-zero, but I haven't seen this demonstrated clearly before. The Dirac sea argument also doesn't seem to be relevant for this issue since that argument is trying to resolve why electrons don't spontaneously decay (when interacting with the background EM field) into more and more negative energy states instead of discussing charge conservation during free-field dynamics.

[@RichBehiel]

Thanks for the thoughtful comment! :) It can be confusing, the way the electron and positron components smear into each other for nonzero momentum. In the nonrelativistic limit, this can basically be ignored, because as you said the electron and positron aren’t exactly defined to be on these components, and the small terms are so small that they don’t affect anything really. Same is true for your wavepacket example btw. But things get weirder at relativistic speeds. I’ll definitely have to do some videos on this topic.

[@armagetronfasttrack9808]

@@RichBehiel Looking forward to it. For my wave packets, I only saw the transfer of components when my momentum spread was relativisticly large (which is easy to do when you can reduce the value of c in your simulation :))

[@ripper5941]

The field manipulations in Dirac's equation are truly remarkable.

[@ansofficial709]

Absolutely outstanding! You are very gifted in your ability to explain physics. You could teach professors how they can be better teachers.

[@RichBehiel]

Thanks for the kind comment! That means a lot :)

[@karkunow]

That flag is just a representation of an SO3 element or a rotation in R^3. And on the deeper level it is a Hopf Fibration of 4D sphere: S1 -> S3 -> S2 Flag can't track down the spinor sign, which you can actually track in 4D. I guess the best way to go with spinors is understanding the SU2/SO3 and their irreps. Representation theory is a key to understanding spinors in some sense. But, of course, the more you know - the better.

[@Hampardo]

I've just been thinking about this. About how the bispinor may be represented as two 4pi invariant planes in 4D space. Very interested in your next video!

[@mjahore]

Love these videos! Thanks for making them and I hope you keep on. Good refresher for me, and lots of nuances that I never noticed as a student.

[@RichBehiel]

Thanks, I’m glad to hear that! :)

[@markloveless1001]

"Unsettling" is the best description of QM since Feynman. It is indeed. Superb.

[@Amb3rjack]

This stuff is dope and I'm not even sure what I'm looking at! Certainly don't know what any of it means. The graphics sort of give me an idea. Your videos are magnificent, Sir, and readind some of the comments, you are a life saver for some of these students undertaking degree and post degree studies. Bless you and thank you!

[@RichBehiel]

Thanks for the kind comment! :)

[@piotrwieckowski9607]

thank you, you made me understand something i thought was overcomplicated before this video

[@javierramos2915]

I've just come out of my Quantum Mechanics exam and this was one of the questions. Thank you so much and greetings from Spain!! ❤

[@RichBehiel]

Greetings from California! ❤️

[@xelaxander]

I love the brevity and non-chalance of your presentation.

[@Raspberry_aim]

Thank you so much, as always. These videos are superb!

[@RichBehiel]

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

[@skorrnnlol]

Brilliant and perfectly adds context to a lot of the Quant. Physics I've done in 2nd year, the whole quantum series has been really helpful and exciting, especially when I can begin to piece together the logic behind specific steps and where certain factors come from. Excellent addition on the end about the Stern-Garlech experiment too, did a brief cover of that at the end of the quantum component for my semester 1 module and its got me super interested in doing advanced quantum mechanics in 3rd year. Look forward to your next video!

[@RichBehiel]

Thanks for the kind comment, and I’m glad you’re enjoying the quantum series! :)

[@RyuGood0]

I just discovered your channel, and your videos are amazing. These are by far some of the best on KZbin. I suck at compliments, but thank you so much for making videos about such complex subjects and making them so great.

[@RichBehiel]

You’re great at compliments! Thanks for the kind comment :)

[@tune490]

Thank you Richard, great videos!

[@RichBehiel]

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

[@tomgraupner171]

wonderful! Thanks a lot and enjoy your holidays!

[@b.clarenc9517]

I wish I had found that kind of content on the 2013 KZbin, back when I was studying this.

[@nzuckman]

IT'S BISPINOR TIME BOIS 🥳

[@ceruleanbandit1692]

📣📣📣🗣️🗣️🗣️

[@digguscience]

Very difficult material explained in an easy way. Cool

[@Naman...123]

Thank you so much for such insights into Quantum physics , used to read in 2021 that's One of the best thing ever happened to me , Kind of nostalgia, Wanna Know What if you make Lecture series on the historical development of Quantum physics that's gonna be interesting if you have spare time from busy schedule will be appreciate ;⁠);⁠)

[@RichBehiel]

You’re welcome, and I’m glad you enjoyed the video! :) I’d like to get into the history of the subject someday, but for now I’m mostly focused on the ideas themselves, with some historical references included for context. In future videos I can include more references to historically important papers in the development of these ideas.

[@Naman...123]

Thanks for update, that's will be exciting 😁

[@aukir]

Can a particle ever actually be at rest? What does it even mean, a particle being its own observer by itself? That concept always gets me. The universe is all entangled with the big bang anyway.

[@monkerud2108]

i like your channel, you make good easy to follow content with good illustrations :).

[@enriquebalpstraffon]

Looking forward to SEE moving particles!

[@billfrug]

Fantastic:) Could you explain the reasoning that led Dirac to put the first order constraint you mentioned on this equation?

[@RichBehiel]

Yeah, it’s in my Klein-Gordon video :) The vid is 46 minutes long, but I put the timestamps for each section. Towards the latter half of the video I go into a few of the reasons why it should be first-order.

[@stauffap]

That an electron in a Stern Gerlach Experiment is either Spin-up or -down isn't mysterious at all if you interpret the spin as an angular momentum. That's how we expect angular momenta to behave in such magnetic fields. The mysterious part then is why the electron has just two spin states and why they have the specific value they have. One way to see that the direction of angular momenta must become directionally quantized is to understand to use the bar magnet model of a spinning electron and realise that it takes energy to change the direction of even a classical bar magnet in a magnetic field. It's as if the bar magnetic is in a potential that depends on the angle to the magnetic field lines. We've seen what quantum objects do in such situations. They get quantized. But since we don't know what exactly is spinning in an electron this seems to be useless to calculate anything. It just makes the directional quantisation in the Stern-Gerlach experiment less mysterious. PS: Of course If you imagine a more more realistic realistic spinning electron, you still get the directional quantisation, but you also get precession. Which adds yet another piece of understanding. Since we know that electrons are precessing from measurements, i find it usefull to think through these cases even though a lot of people keep saying that the electron isn't actually spinning.

[@KipIngram]

The best way to think about that "unsettling" thing is to regard the electron as only having one bit of spin information in it. It can't "quantify" more information that just that single bit. A bit has to be 0 or 1. Yes, an isolated (not entangled) electron DOES have a "spin axis" - a particular axis that when used as the measurement axis guarantees the result with 100% certainty. But you can't "get at" that axis - you can't measure it. Not from just one electron - it is only capable of giving you that one bit of information. If you measure around some other axis, say one perpendicular to the hidden "spin axis," then it will just randomly pick a one-bit result and give it to you. And then THAT becomes its new "hidden spin axis."

[@RichBehiel]

One bit of spin information - that’s a good way of putting it! I get it on a mathematical level, and can picture it as the flag aligning either parallel or antiparallel to the measurement axis. But I still have no idea how an electron can actually be that way! I don’t know how to draw a picture of an electron.

[@kmg3658]

Outstanding! 🍎

[@RichBehiel]

Thanks! :)

[@unrelentingawesomeness7501]

your videos are amazing

[@RichBehiel]

Thanks, I’m glad you’re enjoying them! :)

[@MostlyIC]

Richard, another great video !, (I'm rehashing now-->) as a mathematician I understand the concept of "operator" (a function maps a number(or number-ish thing) to another number(or number-ish thing), while an operator maps a function to another function), but I'm missing an intuition of "momentum operator" and "energy operator" and any other quantum mechanical "operator" there might be. this topic might make a good supplementary video ? ! (and yes, can't wait for the spinor supplementary video !)

[@stuff3219]

The momentum operator is found as one of the basic postulates of QM, that the classical momentum is replaced by the operator i/h * d/dx. What this amounts to is that, for example, a 1D particle with constant momentum is represented by a helix in position space (in other words, the complex phase just spins around and around as x is varied).

[@bogdanlevi]

It's a differential operator that maps a bispinor field on the Minkovsky space into another bispinor field. Or in the case of Schrödinger equation it's straight up p: C_n(R^4) -> C_{n-1}(R^4), where C_n is complex-valued n-differentiable.

[@RichBehiel]

Thanks, I’m glad you enjoyed the video! :) Good question about operators, and I probably should do a video clarifying that someday. A good place to start is to look at how to calculate the momentum expectation value for a particle in the context of the Schrödinger equation (Google “momentum expectation value particle in a box”). There you’ll see that the expected momentum for a particle is calculated by integrating over space, a quantity which looks like psi conjugate times psi (probability density), but sandwiched between those terms is the momentum operator. The operator acts on psi, and is then multiplied by psi conjugate. I like to imagine it as the momentum operator converts the wavefunction to a momentum field, then the psi star comes in to give the integral the flavor of a weighted sum, so the momentum expectation value is the net momentum. The energy operator can be explored in a similar context. In general, I like to think of a quantum operator as just something that acts on a wavefunction in some way, to give you some kind of field on the same domain, where the new field has something to do with the nature of the quantity that we’re using the operator to calculate. That’s a loose definition, but then the precision comes in from the definition of the particular operator.

[@MostlyIC]

@@stuff3219 I can get that level of definition from Wikipedia, what's missing in both cases is the intuition behind it.

[@MostlyIC]

@@bogdanlevi no cigar, this is a definition that is totally lacking in any intuition.

[@monkerud2108]

precession is essentially a two axis rotation, made possible by a constant torque on the top spinning, when the precession is damped but the "angular momentum" precessing is damped and driven at the same rate the angular momentum is constant but the orientations change in very non intuetive ways even if you have a good intuition for changing angular momentum and precession.

[@monkerud2108]

it does work like normal angular momentum, but on that scale the tops are very highly driven and damped tops, and the forces acting on the angular momentum are not always aligned so the effects are not obvious.

[@jedermann05]

The $4 kindle version of "Intro to Elementary Particles" by David Griffiths (recommended in the video) is poor quality. On an iPad it looks like a too-faint photocopy. On a kindle device the print is too small to read, even adjusting font size. You'll have to find a print copy.

[@RichBehiel]

Oh no, I’m sorry to hear that! In retrospect, it might have been better to refer to a book that’s free. Griffiths isn’t the cheapest book, and I wouldn’t want money to be a barrier stopping people from learning. It’s widely considered to be one of the best introductory texts for particle physics though.

[@godinhos7797]

awsomeeee!!

[@M-dv1yj]

Honest reflection/ question. Why is the electron spin being either up or down unsettling? I would imagine in a particle based conceptualization that could be the case. But in a field view… the fields probability weight balance “morphology” only allows those 2 states. If you see particle expression as the total probability weight balance as constructed by the internal field rebalancing from environment “boarder” interactions. I suppose in a way things like defined up or down spin help give us a glimpse into the “shape and dynamics” of the field and it’s particle expression. 😊

[@bogdanlevi]

If we now apply Lorenz transforms to the particle at rest, will we get all of the plane waves?

[@RichBehiel]

Yes! :) The space and time dependence transform like Klein-Gordon plane waves, and the spinors transform in a subtle way, such that one bispinor smears a little into the other.

[@thatguyfromw1rk983]

For the Stern-Gerlach experiment, is it that any spin-1/2 particle can have it's magnetic moment in any initial direction, but then under the influence of the B field it aligns and we get that 2 spot pattern?

[@RichBehiel]

Great question. Honestly I don’t have a good answer off the top of my head. But there have been other experiments to show that spin is quantized up or down when measured. It also comes from the consistency of the theory that spin should only be measurable in this way.

[@beamshooter]

Look up Stern-Gerlachs in series. Once spins are aligned to one apparatus, they maintain the orientation to the next. E.g. if you had another parallel Stern-Gerlach that only took in spin-UP particles from the one behind it, all particles would still be spin-UP.

[@thatguyfromw1rk983]

​@@beamshooterAh okay, so superposition of spin states, then once it's measured (taken out of superposition) it maintains that state until it's returned to it's initial superposition

[@beamshooter]

@@thatguyfromw1rk983 Kind of. A singlet state (UP/DOWN) in one orientation is still a superposition of orthogonal states with arbitrary phase. But yes, it would appear so long as we don't apply B field in another direction, spin state appears to stay the same (not considering uncontrollable external influences)

[@Nightmare-iq9tb]

NIce Vid!

[@RichBehiel]

Thanks! :)

[@wyager]

To take the "square root" of the KG equation and get the first-order dirac equation, you don't need a 4x4 matrix per se - that was only chosen because it's an easy-to-manipulate representation of the Clifford algebra Cl1,3, right? Let's call Cl1,3's elements T,X,Y,Z so I don't have to keep typing \gamma^0, \gamma^1, etc. We have TT=1, XX=YY=ZZ=-1, TX+XT=0, and so on. So the value of the 4 elements of a bispinor's vector representation are somewhat arbitrary, because they emerge only due to the (arbitrary?) choice of 4x4 matrices for T,X,Y,Z. What happens if we try to express the dirac equation *without* using this arbitrary 4-element vector representation of bispinors, but we just use the clifford algebra directly? Solving for TΨ=±Ψ, we get Ψ=(T+1)... for the electron case and Ψ=(T-1)... for the positron case. (This is because T(T+1) = 1+T and T(T-1) = (1-T) = -(T-1).) So it seems like (T+1) is analogous to [a,b,0,0] in the vector representation of bispinors, and (T-1) is analogous to [0,0,c,d]. Indeed, if you go back to the matrix representation, this checks out, as the matrix representation of T(T+1), which is Diag(2,2,0,0), picks out the first two elements of the bispinor vector and T(T-1), which is Diag(0,0,2,2), picks out the latter two. Anyway, this is a long-winded way of asking: can we just get rid of the matrices and vectors entirely, do everything purely algebraically directly on T,X,Y,Z (which have a matrix representation \gamma^{0,1,2,3}), and end up with a simpler or more elegant/intuitive result? For example, and I haven't worked this out farther than the contents of this comment, but I'm guessing that spin states would be represented as something like clifford algebra 2-blade values like XY, which seems very elegant, since it describes what the electron is doing (rotating like a unit vector going from Y to X). I'm having trouble finding much online about this - are you familiar with any writing along these lines, or papers working with the dirac equation without resorting to representation-theoretic matrices and vectors?

[@RichBehiel]

Brilliant thought process! :) That’s basically the idea behind Hestenes’s formulation of the Dirac equation in Spacetime Algebra. That formulation avoids these mysterious Dirac matrices in the way that you describe. When making these videos, I’ve thought about doing everything in Hestenes’s framework, or something similar. Ultimately I decided to use the more mainstream convention, to make the ideas more widely accessible, since the gamma matrices appear in like every particle physics textbook, and I didn’t want to add another layer of confusion to those are just getting into relativistic quantum physics.

[@wyager]

@@RichBehiel Great, thanks for the response. It looks like Hestenes has a lot of papers along these lines, some of which seem to be not easily accessible, and it also looks like people are saying there were some slight problems with his original approach. Do you have any suggestions for a recent/authoritative work by Hestenes or another author that does a good job covering the Dirac equation in STA?

[@erawanpencil]

So you need two complex numbers (4 real) to fully describe the wave function of an electron? So one Riemann sphere nested inside another?

[@RichBehiel]

Correct! In the standard Dirac formulation, each of the spinors is comprised of two complex numbers, four numbers total. And then there are two spinors, so four complex numbers, equivalent to eight real numbers. But in the nonrelativistic limit, assuming we’re dealing with only matter (or only antimatter) then it’s basically just a single spinor field. As for two Reimann spheres, that’s one way of looking at a spinor, although typically since we’re interested in unit spinors, visualizing these involves looking at a unit hypersphere (similar to U(1) on the unit circle in complex plane, but twice as many dimensions). That hypersphere picture can then be mapped to variety of different visualization schemes, most notably Penrose’s flag scheme, which lets us represent spinors in 3D, plus one axial rotation degree of freedom about the flagpole. That representation lends itself well to physical interpretation. But, it’s a 2-to-1 projection, so each flag corresponds to two points on the hypersphere. That, too, is physically meaningful, bizarre though it may seem. The next video is going to be all about exactly this topic :)

[@erawanpencil]

@@RichBehiel Oh wow I actually wanted to comment about Penrose's flags since I've seen that in his videos, but I've never understood it and it seems intimidating. If you could animate a twistor I think you'd be an internet hero!

[@robotbugs]

At 6:43 you have an error with psi_3 repeated twice in the wavefunction on the left.

[@jameswarren6988]

Dude I get VERY little of this but want to understand more, and am understanding a very little bit more through the resources I can find. I had a question about the "stern gerlock" experiment you showed at the end of the video. This reminds me heavily of the double slit experiment. If I remember it correctly, (and Im going to have to go fact check myself bc im not remembering perfectly) when we obstruct light, either with something thin to split a laser or using the slit, because of light's wave properties, they end up creating a band of light with gaps of no light in between (instead of maybe splitting the beam into 2 dots or making a continuous band). Sorry for the horrible explanation, I just wanted to point out how and why this reminded me of that, and I wonder how or if this connects to Stern Gerlock experiment. If anyone feels like replying, I'd like to learn me some science lol.

[@beamshooter]

My theory for why spin states always collapse to the measurement basis is simply that the experimental apparatus itself localizes the spin states to its magnetic orientation.

[@RichBehiel]

That’s a great observation, and I don’t know off the top of my head why the experimental data wouldn’t support that perspective. So you’ve given me something to look into, thanks! :) Although, theoretically there’s an uncertainty relation involving knowing all the components of a particle’s angular momentum at once, which has implications for how the particle’s spin can be measured.

[@jakublizon6375]

But it's still probabilistic. You still have a superposition of states, so I'm not sure how you could explain that with spin states coupling to the magnetism of the measurement apparatus.

[@beamshooter]

@@jakublizon6375 The amplitudes for a single orientation are modified by the magnetic field. A |UP> + B |DOWN> approaches purely |UP> or purely |DOWN> They are most likely to be amplified such that the larger magnitude of A and B becomes mag 1 and the smaller 0 (hence keeping to the probabilities) Perhaps the unknown arbitrary complex phase plays a role somehow.

[@ryangraffius1474]

If (at 6 mins) electrons/positrons live on separate ‘half’s’ of the bispinner, in separate ‘channels’ could it be appropriate to assume that in the special case scenario a Mid/Side transformation of the Dirac impulse can occur? As in doubling and or halving of the magnitude of the impulse? Sum and Difference? As above, so below, if indeed the Dirac is an analogue to the Kronecher Delta?

[@ryangraffius1474]

As an aspect of statistical uncertainty under a bell curve (facilitated by a M/S transformation) to address the Möbius transformation of the polar coordinates. We may not know where they are, but in the special case we know the highest and lowest (asymmetric balancing) of the magnitudes with respect to alignment with the angular momentum.

[@jeremyelser8957]

What would have happened with a different number of spatial dimensions? We can only make 4 gamma matrices out of combinations of 1 and i. Is the number of dimensions fundamental to why we have electrons, or vice versa?

[@fable4315]

Idk if anyone can answer this here 12:50 regarding to this experiment. Couldn’t it be that the interactions with the magnetic field „force“ the electron to have either spin up or spind down? Because we put it inside a strong enough magnetic field and if it has „slightly“ spin up then the magnetic field „forces“ it up and the other way around for down? I am not a physicist and I know this question might sound dumb or naive, it is just an intuitive question I had seeing this picture.

[@OBGynKenobi]

I'm just a physics enthusiast and there's one thing I don't get, does "at Rest" mean completely motionless, ie, 0 degrees Kelvin (which is not physically possible)? If not, then is that term even correct?

[@RichBehiel]

Great question! :) In this context, “at rest” means the particle’s momentum is precisely known, and precisely zero. Because the momentum is precisely known, the position is totally unknown (equal probability everywhere). This is an idealization, but it lets us build up the space of solutions to the Dirac equation. Because once we have the eigenstate for zero-momentum, we can Lorentz boost it to get all the momentum eigenstates, and then we’ll have a basis which we can use to construct more general solutions.

[@therevivalists2187]

Science is great in knowing the reality of matter & energy. But the real question is Who has put these systems in the Universe, Who has formed these laws to work. Is it done automatically without anyone intervention or there is someone behind this fantastic universe. Therefore, excelling in Science & Technology is essential to progress while knowing and having belief in the Creator of entire universe and ourselves is important over all other things.

[@monkerud2108]

intuitions from rough analogs in the macroscopic world like spinning up a large magnet doesn't work, because adding in the right kinds of damping and driving makes no sense until you understand what it is like for an electron.

[@monkerud2108]

if you havent you should check out real stern gerlach results, there is always a smear there, the peculiar thing is the split into two orientations, it looks like a kind of a arc on one side and a strange spiked line on the other, this has to do with the shape of the magnets, if you could easily produce a non uniform field, that doesn't have this shape, it would be more like two slightly thinkened lines. anyway, the effects of the stuff these fields and effects emerge from are not that intuitive and easy either but it is fun to think about.

[@advikdutta]

Make another lesson but on bosons instead of spinors or how the Dirac Lagrangian correlates with the Dirac equation

[@CM-bq9fp]

On min 3:00, aren't the pi operators? But then you set them to zero. This would be ok here,if psi_i are their eigenfunctions with eigenvalue zero. But you should not assume it.

[@RichBehiel]

You’re right. Typically the free particle at rest is assumed to be the four-momentum eigenstate with 3 momentum [0,0,0], because that’s the state where we know with 100% certainty that it’s at rest. But, suppose for example we had a Gaussian wavepacket with net zero expectation value of momentum, but still a distribution so it disperses over time and in principle can be measured to have nonzero momentum. Such a thing could arguably also be called a free particle at rest, but in that case we would have to account for the distribution by integrating a Gaussian over the eigenstates.

[@enotdetcelfer]

YES woopwoop

[@davidrandell2224]

Electrons are expanding and atoms too at different rates: I.e. never at ‘rest.’ “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for proper physics.

[@monkerud2108]

you need to really master the classical account of angular momentum and precession to get a glimpse of what is going on, and then imagine that the vacuum can carry angular momentum in some really strange ways, and certain configurations corresponds loosely to configurations of magnetic fields and so on. hard to provide equations that capture it faithfully but probably a good idea to review normal classical precession, play with damping and driving in all sorts of ways, get some intuition there and come back to it.

[@vansf3433]

Nor any single form of matter is ever at rest, but human wild imagination keeps coming with all sorts of fictions

[@RichBehiel]

True, the particle at rest eigenstate is only an idealization. But we can generalize this solution to eigenstates for particles with nonzero momentum, and from there we can sum over that basis in various ways to address more realistic scenarios.

[@monkerud2108]

what is going on with a macroscopic rotating magnet not oriented in the field direction:) that is just as strange, we have some torque on a spinning top from empty space causing it to precess , therefore in the macroscopic stern gerlach experiement you get a combination of up and down and the result is some spread depending on the distribution of initial orientations, and the process of going from constant angular momentum to precessing. naively we can just say a bunch of little fundamental moments went down and some went up and the result is some orientation for the big magnet, sounds very strange. this is just heuristic ofc but still, for a small tiny little electron, it goes mostly one way or the other, there is always some spread ofc but lets not talk about that, but anyway, because electrons are so small they are really some field configuration emergent from a unified field i dont want to go into the details of too much, but in a strong magnetic field the microscopic state of the electron can be oriented mostly in line or opposite the magnetic field because those configurations are lower in gradient energy than the intermediate orientations. that isnt so satisfying without seeing the unified construction, but it is because of some story like that, the electron is like a little top being torqued, but it sort of very quickly decays into an orientation by a driven and dissipative process that conserves angular momentum, because of the topological nature of the microstate of an electron, as basically an arrangement of angular momentum in the underlying unified gunk making up the fields. lets not go into details, it would be too long a comment, but it depends on some gravitational dynamics for maintaining the topological defect to to speak along with minimizing the gradient energy of stuff spinning and carrying angular momentum in the vacuum, which would take too long to go into properly, and a hand wavy half way there explanation would still take long and not be very informative. anyway it i sort of analogous to a top being torqued that increases its rate of precession but is stuck at a point it cant balance so it never reaches a rate of precession where the change in angle form the magnetic field line stops, until it reaches some value close to aligned or anti aligned, then it stops, it is basically a sort of friction acting on the rate of precession and a driving force resoring the lost angular momentum traded into precession as you change the rate of precession, it is a quite complicated process that happens fast, and can happen to either anti align or align with the field direction. handwavy, short and gives no details of how it happens, but an intuetive analogy is decent enough think :) it is a mechanism that allows the magnetic field to turn the electron almost all the way to aligned and only when the friction from the precession becomes small does the electron stop changing its rate of precession a lot, so ultimately it is kind of a weird ass thing, the way i stated it isnt quite right, missing some details, but it serves as a decent intuition of the kind of thing going on, to give you these plus or minus spins.

[@Dominoes0]

Maybe a silly question. Definitely outside the scope of the video, as described in the first section, but... Does anyone know of an intuitive, visual, or numerical explanation of how to calculate the Dirac spinor for a simple (eg Hydrogen) atom? The links I find in a search are pure analytical, and leave a lot to the imagination! I understand that this is complicated due to it being a Diff Eq with many degrees of freedom. Let me know what you think! I'm mainly interested in trying to solve for arbitrary potentials, but being able to solve Hydrogen is a good step! For example, do the individual 4-component Psi components look like variants on the Schrodinger equation solution for hydrogen? Are they all 0 except for one etc? Thank you!

[@RichBehiel]

Great question! It’s a very complicated answer, unfortunately. But that’s exactly the question I plan on addressing in Hydrogen Part 3 :)

[@BiswajitBhattacharjee-up8vv]

Working with ELECTRON SPIN RESONANCE SPECTROSCOPY, which is a microwave SPECTROSCOPY for molecules with single electron or paramagnetic moments these population inversion in very dilute and low power is a fact. Various quenching mechanism is used to explain. Your question on spinors in presences of positron ,in one state is a good quest . Why quantum mechanics make spinors an mathematical object found suitable for electron. Not Dirac .

[@billfrug]

spinor = two complex numbers = two planes = two joysticks = game controller ;)

[@MinMax-kc8uj]

"We have to be careful." That is getting overused. It's in everyone's videos at least a dozen times per video.

[@adama8570]

Very well presented! But please use the english pronunciation "ZED" instead of "ZEE" for Z which invites mistakes!

[@worldhaseverything]

First

[@h.n.4060]

Second ;P

[@RichBehiel]

@@h.n.4060third!

[@aydscenter]

fourth :D

[@Handelsbilanzdefizit]

You're such a nerd 🤓

[@RichBehiel]

😂

[@alanthayer8797]

Dude CELEBRATING Holidays CANT B THAT SMART! And learn GEOMETRIC ALGEBRA & Geometric calculus aka Non Newtonian maths instead of Dirac and Otha MAINSTREAM mathematics bcuz there LESS forms of GA! Goto SUDGYLACMOE Channel & Bi-Vector Channel ta catch up!

[@RichBehiel]

Nice comment 😂 I’m actually a huge fan of Hestenes’s work, and would love to explore it on the channel someday. But the mainstream Dirac framework is more accessible and in many ways easier to work with, even though it’s also less elegant. So I’ll stick with Dirac until hydrogen part 3 at least.

[@RichardHicks-bp8yc]

Particles r never still promise u r playing with light that is literally from the core of our planet or are morning star it's electric promise it's not capable of not vibrating

[@pghislain]

'The particule is not moving.'. May be you could say, "The observer's referencial is in the referencial of the particule and he measure no movement between the 2 referencials." Without the observer and the referencials,, there is no quantum mecanics neither special relativity. Speed is aways relative to and is always a measurement. It is constant only in the very specific case mentionned before, Just a comment.

[@RichBehiel]

True! The particle is at rest with respect to us, but could be moving with respect to another reference frame.

[@psiwavee]

Your videos are amazing

[@RichBehiel]

Thanks, I’m glad you’re enjoying them! :)