Hi everyone, thanks for checking out this video, and I hope you had as much fun watching it as I had making it! :) Please let me know if you have any questions about any topics in this video, or suggestions for future videos, either for topics or format/editing suggestions. I’m trying to learn how to make these videos better, so constructive criticism is always well-received. Also, some of you have been asking about Patreon. Personally I wouldn’t want the pressure implied by paid subscriptions - these vids have to come from the heart. But if you enjoyed the video and want to leave a tip, there is the SuperThanks button under the video. That feels like a tip jar for work already done, which seems wholesome. All tips go toward textbooks and caffeine! :)
I use matplotlib for 2D stuff and plotly for 3D. Everything you see here is basically a plot evolving in a time loop. For example, the plot with the little blue dots that shows the probability distribution, I made a grid of about ten thousand points, gave each one a small random nudge so it wouldn’t look so grid like, then in each loop iteration generated a random number at each point while evaluating the probability density function, and if the random number is less than rho then that pixel comes to life and fades out over a few frames, otherwise it stays invisible. So that’s just plt.scatter(). I’m sure there’s an easier way to do this stuff, but that’s what I’m used to. I have looked into Blender but haven’t done anything with it yet. Going forward though I’d definitely like to get into that, because it seems incredibly powerful.
This video is excellent. Actually solving the entire SE from scratch would take absolutely forever and then some, so the walkthrough is more than adequate. Also, thank god for mathematicians, they really do make our jobs easier.
Student in nuclear physics, learning with books I sometimes don’t see behind math equations whereas with this video everything becomes more “alive” ! Thanks a lot for your time and I confess this is my new favorite video on RUclips. Should a video about interaction theory (photon or neutron on nucleus) be released, it would for sure become a killer video in the field for it does not exist on the net, at least not explained with your skills ! Cheers !
I got my BS in Nuclear Engineering, and one of my biggest regrets is not putting the "conceptual" with the equations. It's so hard to do problems when you have no idea what they're actually describing. I encourage you to keep watching videos like these..you'll understand the math work much better.
@@kidzbop38isstraightfire92 My theoretical mechanics teacher said that though math is paramont in physics, still more important is "physical intuition" about how things work. He forgave many sins for "the right idea". (I suspect that anyone who needs to say "righty-tighty" to themselves is lacking that intuition.)
@@mikemondano3624 yea, this is why I always hated the "shut up and calculate" idea that permeates modern physics...sure, there are certain cases where the equations manifested into novel concepts, but most of the time it's the reverse (where somebody has an idea first and then worked the equations later).
@@mikemondano3624bro could u give me some tips on improving my physics and maths I am 16 and trying to improve in physics. I live in India and here we have to give entrance exams to go for higher studies but the acceptance rate is less than 1% and 70 % of that one percent goes to schedule casts . So I wanted to know from an experienced guy where can I get some university courses in the field of physics on scholarship outside this shithole. I don’t study a lot tho
I wish I had this video around when I was taking quantum 50 years ago. I had to visualize all of this in my head, not knowing if I was right or wrong. Plus, imagine furiously writing down all these derivations from the blackboard while trying to understand the logic hoping my notes will make sense later on. Well done. ;-)
Thanks for the kind comment, and I’m glad you enjoyed the video! :) And I can relate to your experiences writing down derivations from the blackboard 😅 I hope these videos can make physics concepts more accessible.
This catapulted me back 35 years to my physics degree. We actually slogged through the derivation of all the important orthonormal functions (Legendre, Laguerre but also Hermite and Chebyshev etc.) in several three-hour maths lectures to equip us withe necessary tools, and then our QM lecturer showed how they were applicable to the hydrogen atom, electron-in-a-box and so on. It was a fairly painful process, and we would stagger out of the lectures afterwards with our heads reeling.
Bro if ur a physicist can u recommend me some tips for improving my critical thinking on physics and maths also I am 16 and also wanted to know if there are any university courses in quantum mechanics and computers where I can go after 18 .
@@ayanhassankango239 The sole piece of general advice I can give is practice, practice, practice problems in algebraic manipulation, calculus, trigonometry and linear algebra until you have the basics at your fingertips. If you're interested in physics pay particular attention to differential equations (for quantum mechanics and orbital dynamics etc.) and vector calculus (for electricity and magnetism).
@@ayanhassankango239 I recommend 3Blue1Brown playlist on Linear Algebra. It will give you an intuition of what's it's all about. But the rest is practice indeed... You can take any course on Coursera, they come with homework. It's free, you only have to pay to get a certificate of completion.
I am an engineer (retired) so the math and the particle physics is over my head but nevertheless this was a mesmerizing video to watch. The visuals and the animations help to make sense of it all. Please, dedicate additional 5-10% of your presentation to laymen like me. Give us a bit more of a 30,000 feet view. This will broaden your audience. It might be a bit boring or redundant for some of the specialists but it will definitely bring more people to your channel. or rather you can reach and enrich more people. Educating and bringing people up, what could be more noble than that! Thank you for the effort and the discipline to create, produce and post such videos.
Thanks for your thoughtful comment, and I’m glad you enjoyed the video! :) In future videos I’ll put more time into the 30,000 ft view. Even for those who are specialists, it’s always nice to take a step back sometimes and reflect on the context, and think about how the ideas are related to other things. Honestly that’s what I like most about physics, not getting stuck in the mathematical weeds but thinking about the big picture. But the math keeps one grounded, and there’s a beauty in its precision. I guess it’s a balance.
Hello, I'm also a Mechanical engineer and I took seriously and enjoyed learning Quantum Mechanics (QM) so be patient... And take a book and spend your time self enlightening you about this issues... And don't be afraid about math... Because without math it's impossible to understand truly what's going on.... I recommend a short and old RUclips on schrodinger equation derivation from an english man called Aphysics or something similar that can help you loose your fear for Mathematics.... Enjoy it!!! By the way, the vids from Richard Behiel are among the best videos on RUclips... But I recommend you to start maybe with Arvin Ash or Physics Asylum by Nick. Or ParthG. Those are less mathematically laden...
Also remember particles are not bals. They are more vibrations or oscillations and ways to move energy around... And we still don't know the real nature beneath everything.... What's an electrical charge???? Etc....
It is often difficult to see the forest (Physics) for the trees (Mathematics). Well done presentation. It has been more than 45 yers since I studied this closely in grad school. You refreshed my memory and I subscribed for more.
periodicity and quantization being highly related makes a lot of sense to me--to be periodic you have to have some kind of ultimately integral repetition, which lends itself to quantization. Interesting video!
my goodness it’s breathtaking that math is able to establish an intuition for the nature of reality. The phase change and switch visualization is astounding when you think of reality literally being that way - in and around us. And it all (or most of it anyway) can be explained by math.
Great video so far - again, really like not shying away from the math here. I think the first 20 minutes would have made more sense if we had the context from the slide 19:42 at the start. I think setting up why the spherical harmonic functions and radial functions are relevant is important prior to getting into the detail.
as someone who is essentially a humanities type-I have a decent grasp of common-or-garden calculus, and can dabble in linear algebra, multivariable calculus, etc, but not especially well-I am both absolutely fascinated by these videos and absolutely in awe of the people who figured all this stuff out in the first place. i feel like i grasp just enough of the maths to get a sense for how crushingly difficult doing something like figuring out solutions to the associated Legendre equation must be, and the fact that someone was able to do so is just ... wow. more generally, it's incredible to think that our ability to record, pass down and build on our collective knowledge means that as a species, it's taken us a few thousand years to go from our first steps in agriculture and engineering to working out exquisitely precise and detailed mathematical descriptions of the tiniest building blocks of reality itself. if we manage to avoid wiping ourselves out, who knows what we'll prove capable of?
Unfortunately, just as agriculture was humanity's biggest mistake, allowing the mass production of starch to power and enlarge a workforce (hunter-gatherers were taller, healthier, and longer-lived), private property, capitalism, slavery, and great inequalities, the human being is propelled ultimately by greed and knowledge is a tool of self-enrichment well beyond the necessary. We are not a sustainable species.
Finally found time to watch this. Really great work! I found your channel through your Hopf fibration video - I do physical optics simulations and wanted a way to convert electric field amplitude / Jones vectors to points on the Poincaré (Bloch) sphere for a visualization. I was amazed at how elegant & beautiful the mathematics behind that conversion is! My personal interest in these topics has been quite high recently, so I'm feeling very lucky that you're producing such nice videos on them at this point in time 😁
I had attended two semesters of lectures in quantum mechanics with Professor L. at the University of Stuttgart during my studies in electrical engineering. He only showed us the calculation of the ground state of the electron shell around a proton. You, on the other hand, even show the basics for calculating many orbitals of the hydrogen atom. Only 2 of 3 videos that easily go beyond the content of the lecture at that time. Many thanks also for the vivid visualisations of the solutions to the equations, where I could only do mental movies at the time.
I’m glad you enjoyed the video, and thanks for watching! :) Video has a lot of potential, compared to lectures. I think the future of education will be based a lot on videos, especially for physics since it often takes effort to make the equations come to life through imagination alone, but videos can provide a template for the imagination that helps kickstart the process of building intuition. Lecturing in person is an impressive skill though, something I probably couldn’t do very well. With video, it’s very helpful being able to edit everything before sharing it.
Thank you for striking the delicate balance between providing just enough mathematical details and the clarity of explanation. Much of modern physics is mired in mathematics that it is hard to find explanation that interprets the equations and solutions clearly. We need more such videos that is not afraid to tackle the mathematics and also provides sufficient physical interpretation. Please post videos expositing the way spin and magnetic quantum numbers(partly alluded to in this video with quantum number 'm') arising from SE.
As a person who has just started learning about quantum mechanics, this is a beautifully created video. The video was easy to follow with the animations while also gradually setting a rigorous base for solving the SE. I am thinking about persuing physics professionally and with such amazing explainers as these, I might just do it. PS: thanks for the reference articles as well; reading through them was really helpful.
Wow this series is amazing. I come from a more math background but have always been interested in quantum physics. IMO the video really strikes a wonderful balance between rigor and intuition. Also the visuals are so beautiful and has helped me depend my appreciation for topics I thought I was generally comfortable with. Really appreciate the experimental validations done to test the model of quantum mechanics, would love to see more of those!
Thanks for the kind comment, and I’m glad you enjoyed the video! :) Good idea about including more experimental validations. I think it often goes without saying that physics has been experimentally verified, but it‘s worthwhile to show how that was done. I’ll be sure to include more experimental validations in future videos.
I'm only in year 9 right now so I don't understand most of the math yet, however I've been really interested in particle physics as well as quantum physics and videos like this which give both a visual and mathematic/physics based explanation have allowed me to further help me understand the topic while being behind in the maths (but helping to understand the parts of the equation and what it all means). Thank you for making this series and you have no idea how much I want learn and hopefully fully understand all of the equations in this video.
Thanks for your kind comment, and it sounds like you’re on a good path with your education! :) A passion for knowledge is the most important thing. Eventually all the math will make sense.
Just found the channel; was a really well done video and yes, anticipating part three as well. The Dirac equation is fascinating. :) Was an excellent idea to place the functions themselves out front and explore their properties for a while, before diving into the nitty-gritty of equations. Thank you so much !
Wow. Thank you so very much. This is the most detail I've ever heard. Anyone watching should completely understand the many rabbit holes to which you often allude, but wisely avoid. No need to apologize. Ever.
Hi Richard, what a wonderful video! I had a course on quantum mechanics about 4 years ago at University, and I decided to refresh some of the old material before starting my PhD in about a month, and this video was an absolutely excellent refresher - great work.
Hi Richard, Many things, I don't understand as a chemist, but still I am trying! You presentation and voice clarity is excellent.This visualization of maths and it's working for the atom is amazing! Thank you for your nice efforts!
It is so wonderful that Bessel, Tchebychev, Legendre, Hermite, Laguerre, Jacobi and other had fun solving classes of ODE/PDE and gave us those polynômes in the XIXth century. Then Erwin Schrodinger could focus at the start of the XXth century on the physical interprétation of his equation and its solutions (building on the shoulders of its predecessors). Love maths
One equation models not just the hydrogen atom, but all elements in the periodic table. No wonder it has a kind of switchboard (the quantum numbers like n, l, and m): set them properly and voilà ! You've selected an element and one of its various configurations. It's a necessarily complex equation. But videos like this one lets us examine what is it all about. But, of course, this is not a basic quantum mechanics video, you'll have to look elsewhere for why energy is quantized (and how), why there is a complex (real and imaginary parts) field, and other related things.
18:13 It's important imo to also keep in mind, none of the energy eigenstates of Hydrogen want to sit near the proton. This is something a lot of people get wrong. The wavefunctions for the s-orbitals don't go to zero at the proton, but all of the radial distribution functions do (multiplying by r² guarantees this). That means there are no energy eigenstates in the Hydrogen atom where the electrons want to sit close to the nucleus. This is indeed a requirement of the uncertainty principle as you discussed in the previous video on why the electron doesn't just fall straight to the nuclei of atoms.
Omg the Hydrogen atom is so complicated :) It's astounding how all these pieces were discovered and put together, and agree with experiment with great precision.
Congratulations.... Again! : ) Your videos always bring bright insights from an educated physics who I assume loves what he is talking about and had spent long hours enjoying thinking about the physics and maths behind it. And I love the balance between raw physics math and equations that are usually preceded by animations and insights about the problem... In my case, I self study Quantum Mechanics with Griffiths book and also followed the MIT openware course on QM I and part of QM II and i use your videos to review things that I supposedly learnt, and see them visually and wonderfully expressed in your rigurously accurate animations.. But I always get great insights and a taste for the more complicated things that lay beneath QM... QM is amazing and hard to learn for a 52 year old mechanically engineer... But I love it... And I love your videos.... Great effort in the script... Great care in the animations and editing.... No shortcuts, no potholes.... Congratulations!!! And thanks a lot for sharing your thoughts on this hard QM matter!!! And I love also your next videos.. The Klein Gordon Dirac series.. I already viewed them twice, and now I will enjoy them again for the third time in preparation for the 3rd part of your hydrogen series... Those Dirac videos are the first time I understood completely the Dirac equation... Now I'm the quest to understand or at least have solid knowledge about spin mathematics and physics, and I use your videos and eigenchris and Noah_explains_physics...and other math resources and buff... And I'm anxious to review all of them again adding an additional onion layer or understanding!! I also have in the wish list your video on electromagnetism as a gauge theory... But I'm preparing for it as you ask for.... I think I'm probably the kind of public that can enjoy the most your videos... And I think there are thousands or hundreds of thousands of viewers like me... So your videos are wonderful and are helping lots of people.... So keep this way!!!! Congratulations!!! And many thanks!!!!! : )
Super duper excited for the third part! The pace of the content is perfect. PLease add the prerequisite videos soon. The pace of the delivery is just perfect.
Can't help but feel a bit nostalgic of my quantum mechanics courses 10 years ago (engineering physics). What an amazing subject! It's funny how 2 weeks ago when you released this video I was going back through my notes, trying to explain spherical harmonics and the hydrogen atom to my vetarian girlfriend (amazingly she is curious about this). I wish I had found your video ahah! I went back to quantum mechanics recently because I randomly asked myself, what the hell are spinors again? Oh boy I went down a rabbit hole down : weak hypercharge and isospin, higgs mechanism, majorana fermions, spontaneous symmetry breaking, etc. Now I feel like a toddler, but i'm happy. Anyway, thanks for your work and I can't wait for part 3!
Wow. I really love this channel alot, and all the videos 🙂❤️ Physics and maths and chemistry is very amazing! I love this soo muchh ✨️ Especially quantum physics 😌
Thank you for covering a lot my exam content from the what's going on side. This has been a huge help being able to visulise it and hopefully will help me to remember what physics to use where now that I have more of an understanding of what the maths represents. Now hoping part 3 comes out in the next couple weeks to save me haha
I just started learning about orbitals and their orientation in highschool and was a bit curious from where the shape of these orbitals come from. I think need to wait until university to understand how to solve these equations. But thanks for the excellent video, it is very clear and shows how to plot the function in different coordinate systems. I think i will understand this in comming 4-5 years and it will help me throughout my life.
This video is amazing! I can't imagine how much effort was put into animating the graphs and equations (BTW, what software do you use to do that), but I hope you know it's greatly appreciated, and I'm really looking forward to part 3!
Thanks, I’m glad you enjoyed it! :) I use python, with matplotlib. It’s kind of a tedious way of going about it, but it’s just what I had learned to use as an engineer. There’s probably better software out there for this kind of thing.
This is brilliant! I’m taking a course right now about hilbert spaces and P.D.E’s and the intuition of using the spherical harmonics as an orthonormal base is fascinating. Very well made video!
0:00: 📚 This video is about eigenstates, spherical harmonics, radial functions, and the absorption emission spectrum. 4:11: 🌐 The function plots complex numbers on a spherical plot and a two-dimensional plot, representing amplitude and phase. 8:13: 📚 The video discusses spherical harmonics and their properties. 12:15: 🌈 The plot shows a rainbow circle representing the complex phase and amplitude of a function. 16:10: 📐 The length scale of atoms can be calculated from Schrodinger's equation, with the Bohr radius being about one ten thousandth the wavelength of green light. 21:19: 🧪 The energy eigenvalues for each energy eigenstate in a hydrogen atom depend only on n and have a specific formula. 25:37: 🔬 The Schrodinger model of the hydrogen atom aligns well with experimental data and is precise and accurate. 29:23: 🔑 The essence of separation of variables is that both sides of the equation are constant, allowing us to separate the differential equation into two equations. 33:34: 📝 The first derivative of f with respect to Theta can be written as a derivative of f with respect to psi. 37:59: 🧩 The radial equation in quantum mechanics is influenced by the quantization of the angular momentum and the coupling constant. 42:12: 📚 The derivation shows that the atomic length scale can be determined from the Schrodinger equation. Recap by Tammy AI
12:29 I'm confused. In the circle, does each angle represent an input for theta, with the pattern in the circle changing with time at phi changes with time? If so, how are there multiple values for an angle?
Great question! The plot shows exp[i*m*phi], where I’m varying m in time to show the general form of the equation. You can see that most of the time, the curve doesn’t come back to itself, so the solutions which are smooth and continuous just have an integer m. The outer circle in the plot shows the magnitude (constant radius) and phase (color), with the flowery patterns representing the real and complex parts of the equation.
Thanks for watching! :) I’ll be doing some videos on the Dirac equation and electromagnetism as a gauge theory first, then will return to hydrogen for part 3, with a relativistic electron.
Thanks, I’m glad you enjoyed the video! :) I don’t have a Patreon, but I have superthanks enabled on the video 😉 Someday I might set up a Patreon, if/when I can commit more to putting RUclips videos out on a regular basis. For now RUclips is just a side thing, I love it, but I have to be able to take a break if something else comes up with family or work.
Great question. I’m not sure if there’s a simple explanation, to be honest. The usual explanation is that we’re just lucky. Ultimately it has to do with the simplicity and symmetry of the coulomb potential of a point charge. But I don’t know how to say anything more insightful than that.
These visualizations are amazing, and the explanation is great! I could more-or-less follow it as a non-physics-person (but a sorta-math person). For the rainbow phase visualizations, I think it would be helpful consider using (Ok)LCH color space instead of HSV/HSL to plot the colors; human vision perceives 0,255,0 green noticeably brighter than 255,0,0 red or 0,0,255 blue. The LCH color space corrects for this (so e.g. all "80% luma" colors are actually equally bright, which is not true of 80% luminosity in HSL or 80% value in HSV), so there wouldn't be "banding" on the spherical harmonics. It might make the visualization smoother-looking and therefore maybe easier to interpret? Although it's hard to know for sure without trying it out; it's definitely a popular model for digital color design now.
I'm 17 years old I love maths and physics and I'm absolutely not able to deal with these crazy physics and maths concepts... But I really want to solve it one day!!!🤩🤩🤩🤩 This is how a hydrogen atom looks like!!! But unfortunately I cannot cope with it right now😅🤣
@RichBehiel Thank you sooo much!!! Within a few years I'm going to watch your videos so I can finally learn it!!! I'd like to study physics at university and now I'm in high school in Italy and it is a chemistry school (we mainly do chemistry) I'm studying maths on my own because I think that the more I understand maths the better I will understand physics one I get at university!!!🤩🤩🤩
@ryandias5153 Thanks!!! I want to start studying Newtonian mechanics this summer and I got from school 2 books about classical mechanics!!! Because I had to choose 2 books ('Cause I won a chemistry competition and I'm in a chemistry high school) and I chose these two... BTW they're in Italian and are the textbooks chosen by universities but I tend to read more in English rather than Italian!!!😅🤣🤣🤣
This understanding technics is very cool. This sir ,i will be learn morn some things for this video.i clear the quantam part of schidorgen equation which is very big😅😅😅
Thanks! :) I’d recommend Introduction to Elementary Particles, by Griffiths. It starts off prosaic and historical, then gradually transitions all the way into quantum field theory. Along the way, it covers most of the main concepts in particle physics and quantum mechanics, at a level which is mathematically rigorous but also general enough that you feel inspired to read papers about each topic. Honestly it’s one of my favorite books of all time.
I could never compare my work to Griffiths! But thanks for the compliment :) Video definitely has the potential to fundamentally change physics education. When looking at a page full of equations for the first time, the ideas don’t exactly leap off the page. But with video, they do, and they come to life. That has a much higher chance of inspiring a spark of insight in someone who’s trying to learn physics.
@@RichBehiel I believe it is a very interesting case for Physics Education Research. 10 years ago, they had a very active group in K-State. Not sure if they are still active there. As for the duality, I hypothesize a new model for the atom, but there is one small issue, which is the movement from continuous to discrete with functions. Until today I couldn’t find it away from you continuous to discreet the closest person. I’ve observed to solve this issue, however cheated, was Gregory Perelman on his approach to solve the Poincaré conjecture using Ricci Flow. However, he used the “surgical method” to remove some troubling singularities to be able to solve the conjecture. I theorized that going from continuous to discrete shape is like turning a sphere inside out without reaching a singularity
Love this video so much, thank you. I have a question about the emission spectrum with the experimental verification. Can the electron only occupy one energy eigenstate at one time? My understanding of these eigenfunctions was that they were like an orthonormal basis for the states the particle could occupy, and a particle occupies a superposition of them. In the theory (the one that's being experimentally tested around 22:00), what stops an electron from being in some linear combination (would this be a "superposition"?) of the eigenstates? Or if it can be in a combination of the eigenstates, then how is its energy determined? Clearly not by a combination of the energy eigenstates since then the bands wouldn't be discrete. I have a feeling that the energy quantization is the entire point but I'm missing where that's coming from in the theory
Is it that the state the electron is in has to be in the eigenspace of the Hamiltonian operator? So you can only combine states with the same energy eigenvalue?
I think the phase factor in spherical harmonics can be better represented as a height function on the surface of the sphere instead of colors. This will give a much better intuition of the "wavy" nature of the harmonics, like in this video: Spherical Harmonics (U2-05-05) by quantum vision.
I haven’t finished it yet, but am working on it :) I sort of went off on a tangent after Part 2, and am currently working on “Electromagnetism as a Gauge Theory”. That will lay out some of the concepts that we’ll see in Part 3.
"But it's actually a huge step forward, because partial derivatives are very painful, and ordinary derivatives are only an ordinary amount of pain." Between this, the elephant joke in the previous video, and the joke about the symbol for spherical harmonics being Y in this one, I have to say I absolutely love your sense of humor. One thing I would like explained, though, is why iħ∇ is the momentum operator.
Thanks! :) Good question. Honestly it’s hard to give a thorough answer in a comment. Feynman and Hibbs do a decent job of addressing it in their book Quantum Mechanics and Path Integrals. Long story short, wave mechanics is sort of a way of capturing the statistical behavior of a quantum system, but underneath the wave level of description there’s the principle of least action, acting on all possible paths. That’s really the level where you’d want to look at the nature of momentum, then from that, it follows that the momentum operator is related to the gradient of the wavefunction.
Thanks! :) Good question. Honestly it’s hard to give a thorough answer in a comment. Feynman and Hibbs do a decent job of addressing it in their book Quantum Mechanics and Path Integrals. Long story short, wave mechanics is sort of a way of capturing the statistical behavior of a quantum system, but underneath the wave level of description there’s the principle of least action, acting on all possible paths. That’s really the level where you’d want to look at the nature of momentum, then from that, it follows that the momentum operator is related to the gradient of the wavefunction.
@@RichBehiel Oh my god, you're right. I'd actually read about Noether's theorem and the relation between spatial translations, Lie groups, Lie algebras, and momentum as a generator of translations and completely forgot until now! It all came back when I was re-reading your comment for a second time. Thank you!
Fine video. I would appreciate a little more about non-energy eigenstates transitional States non-stationary states and their stability just perhaps qualitative. We seldom see pictures of those
due to the a-symetry of the spherical harmonics, the solution clearly cares about which direction is 'up'; ie which direction we choose to take our theta from, in real life by what metric is 'up' decided for the atom, is it just random? just some superposition of different axis choices?
Rich did not explain this, but the states he derived are eigenstates of three operators simultaneously: energy (denoted E), total angular momentum ( denoted L^2), and angular momentum along the z-axis (denoted Lz). Physically this means we can prepare a hydrogen atom to be in a state where we have knowledge of all three properties simultaneously. You are right that knowing the angular momentum along the z-axis is somewhat arbitrary - we could have also solved our equations and derived solutions where we would know E, L^2, and L_alpha (where L_alpha would be the angular momentum about an arbitrary axis). Rich didn't get into this at all, but while you can know the angular momentum about a particular axis (let's say Lz), you do not then get to know the angular momentum about orthogonal axes Lx and Ly. Mathematically, you can show that the solutions Rich derived all have the property that they are eigenstates of Lz, but are *not* eigenstates of Lx or Ly. I am guessing that we end up in this situation "in real life" when an interaction occurs (lets say a ground state hydrogen atom and a photon) and we know the photon has just the correct energy and z-axis angular momentum to excite the hydrogen atom into one of its excited states that then has the z-axis as a "special axis" (the asymmetrical axis of the spherical harmonics, as you are pointing out).
If you were to plug in negative n values and map r to -r for the Radial function, would this be the radial function for a positron? or is what im saying just nonsense
Good question! Actually a positron would just fly away from the proton, wouldn’t form a bound state since positive and positive repel. But if you had a positron and an anti-proton, then you’d have antihydrogen. It behaves just like hydrogen, same eigenstates and everything. The charges are just reversed.
@@RichBehiel I’m wondering if it is possible that the electron can temporarily go into a negative energy state, then be pushed back into a positive state by the proton. if so what would be difference in energy, and would a photon be emitted with that energy.
@beamshooter the electron is always in a negative energy state in hydrogen, zero being the unbound state when it’s infinitely far away. But there is a subtle effect, from quantum electrodynamics, in which the electron kinda sorta almost becomes positron-like when it’s very close to the nucleus. Google the “Darwin term”. It’s not as straightforward as the electron becoming a positron, but to contextualize it requires looking at the Dirac equation, where spin and antimatter play an essential role in the geometry of the situation.
@@RichBehielso if a photon adds enough energy to take E > 0, that electron no longer bound, but could it still spontaneously emit photon to bring it back into a negative energy bound state?
Another fantastic video! I’m very curious, what do you do that would require you to know all this? Do you have a physics degree? This is all extremely cool and I’m curious what I could do in the future to learn more about this stuff
Thanks! :) Well I’ve always been obsessed with physics, since I was a kid. But when I went to college, I majored in mechanical engineering since I figured that way I could learn a decent amount of physics, while having a higher chance of paying the bills. But as an undergrad, I was drawn more towards physics rather than machines. So for my master’s degree I studied materials science, a bit aimlessly, dabbling in metallurgy but ultimately drifting toward condensed matter physics and falling in love with quantum mechanics. The class where we had to calculate bulk material properties from quantum ab initio simulations, was a turning point for me where I realized that QM was really existentially profound. After getting my master’s, rather than pursue a PhD in physics, I decided to go into industry instead, again due to considerations about paying bills. But I knew I had to do something physics-y. So I worked with micro-electromechanical systems for a few years, and that was a lot of fun. I got to work with a lot of tooling and techniques from the semiconductor industry, but applied it to weird futuristic devices. About a year and a half ago, I joined a startup that’s making superconducting tape for all kinds of scientific and industrial applications. It’s a kind of material that hasn’t really been commercialized at scale yet, so we’re trying to change that. So that’s been a lot of fun. Lots of quantum physics involved with characterizing this kind of material, diffraction using x-rays and electrons to study crystal structure, and also the material itself goes into an exotic state when it superconducts, pairing electrons together so they can collectively condense into the same quantum state and slide through the material literally like a ghost, an electrically charged ghost. And in a strong magnetic field, quantized flux vortices form, and we have to design our material to trap those vortices in an optimal way. It’s a whole world in there, with many analogues to deep concepts in fundamental physics. For example, flux vortices are a real version of what superfluid vacuum theorists dream about. Anyway, sorry for the wall of text here, but as for what you could do in the future, the possibilities are endless. Learning physics has probably the best return on investment, over any other subject. Well, depends on your circumstances I guess. But physics lets you engage with Nature on a whole other level. It’s spiritually satisfying, and often lucrative. Not always easy to learn though 😅
@@RichBehieldamn what an interesting life you had, I'm definitely envious lol. How's the industry/PhD choice for now? I'm thinking about going deep into academia but I have the same concern as yours, bills. In my field, it's hard to find the job that I'm passionate while still being enough for me to start a new family.
Personally I think I made the right choice with regards to the PhD, even though I really wish I had the luxury of pursuing it, I couldn’t justify it financially. Compared to working in industry after a master’s and saving aggressively, the difference in the two options is life changing. Compound interest is a powerful thing, and missing out on the opportunity to save in your twenties is a mistake that a lot of people make without realizing it. I have a few friends who stayed in academia and recently got their PhD. For the most part, they seem depressed and aimless, like they just finished working way too hard on something that wasn’t entirely in their interest, and now they’re not sure what to do. I might be a bit cynical, maybe this is just my own bias having not pursued a PhD, but at least in my friend group the PhDs tend not to be the happiest people. Maybe this will change as time goes on, and they presumably get access to high status jobs, but that’s not a sure bet. And they have some catching up to do, compared to those who have been working a few years already and have started to develop a sense of how the world works, which is something they don’t teach in school. Everything I just said would be totally irrelevant *if* academia were still the only way to learn things. If so, then I’d have a physics PhD because there’s just nothing I could do to stop myself from going down that path. But nowadays we have the internet and books and after a certain point you can teach yourself. Especially in physics, you can’t fool yourself because every step of the way you can just think “can I solve the exercises in this book and get the right answers?”. If yes then you know it, if not then you don’t. There don’t have to be gatekeepers involved in that learning process, you just follow what interests you and keep at it.
Good question! For the spherical harmonic plots, the radius is actually arbitrary since it’s just a function of phi and theta. The sphere is just a way of representing the 2D angular domain with its natural topography.
Yeah, I’d probably just mention that at the end of hydrogen part 3. The reason is fairly straightforward, the math just becomes super messy for a two-particle wavefunction where the electrons are still repealing each other while both are delocalized. Plus, the antisymmetry of the wavefunction and the exclusion principle.
@RichBehiel thanks! We didnt touch on helium much when i took QM besides the prof saying something like "things get non linear so its easier to just solve with a computer" but i really want to be able solve them as much as possible before resulting to solving them on the computer anyway.
Honestly, if i spend the whole day solving all those equations, i'd have less questions "what the heck is this variable doing?" So, now i have more questions than answers
I’m glad you enjoyed the videos! :) Part 3 is a work in progress, I’m going off on a detour into Lorentz transforms and the Dirac equation first, in order to set the stage for the ideas in that video. Part 3 will involve a lot of relativistic quantum mechanics, and while making it I realized that it’ll be helpful to do a couple videos first to cover a few of the prerequisite concepts.
Eventually 😅 I’ve gone off on kind of a tangent lately, currently working on some superconductor videos, then I might go into electroweak unification or might finally come back to H. Sorry for the delay.
I’m still working on it :) My next video will be Electromagnetism as a Gauge Theory, then probably the magnetic moment of the electron, then hydrogen part 3. Part 3 is going to be advanced, so I want to lay out the prerequisites first.
Thank you so much for the video Its a great help! The only part that has confused me about this is how the spherical part of the equations is solved. I know thay the Azimuthal solution is used but my brain won't accept that as a answer and I really would like to know how that solution was found. Does anyone know anywhere I can find a guide for that solution? I've tried looking but haven't found anything that helps but I probably haven't looked hard enough 😅.
I’m glad you enjoyed the video! :) For the azimuthal equation, are you asking about how we know the general solution is A*exp(im*phi) + B*exp(-im*phi), or how we were able to set B = 0 without loss of generality? If the former question, the solution arises from the famous property that e^x is its own derivative. So if you have the equation f’(x) = f(x) then you know right away that’s A*exp(x). Likewise if the equation is f’(x) = m*f(x) then by the chain rule, f(x) = A*exp(m*x). When the equation is f”(x) = -(m^2)f(x), then you just take the square root of the usual thought process, to get the solution. As for why we can set B = 0 without loss of generality, I can’t answer that sufficiently in a RUclips comment. But some partial answers can be found by considering that m can be negative, so there’s a bit of redundancy in the space of solutions when running through all m. Likewise because of the global U(1) symmetry of the wavefunction, the set of general solutions has a phase redundancy. And thinking in the context of the 2D angular equation, really we just need an orthonormal set of functions to span the solution to that equation. You would have to get into the weeds with the associated Legendre polynomial to really see why we can set B = 0, but I hope this comment points you in the right direction.
@@RichBehielOMG I forgot e^x is its own devrvaite Jesus I feel dumb 😅. But thank you yes it was the first comment you made about how Do we know what the solution to the equation is. The other comment you made about generality of B and bringing up the U(1) is something I will have to do more reasurch on. I see groups and U(1) group mentioned allot but never got my head around it. Thank you so much for the video can't wait for the third one!!!!!
![Noob To Max With DRAGON REWORK In Blox Fruits [FULL MOVIE]](http://i.ytimg.com/vi/LnBMOinoOvA/mqdefault.jpg)
Hi everyone, thanks for checking out this video, and I hope you had as much fun watching it as I had making it! :)
Please let me know if you have any questions about any topics in this video, or suggestions for future videos, either for topics or format/editing suggestions. I’m trying to learn how to make these videos better, so constructive criticism is always well-received.
Also, some of you have been asking about Patreon. Personally I wouldn’t want the pressure implied by paid subscriptions - these vids have to come from the heart. But if you enjoyed the video and want to leave a tip, there is the SuperThanks button under the video. That feels like a tip jar for work already done, which seems wholesome. All tips go toward textbooks and caffeine! :)
fr fr no cap. Thanks so much for making these videos - they're presented incredibly in an easy to understand way!
I’m very glad to hear that! Thanks for watching :)
@@RichBehiel Coded in Blender? Or maybe Manim? Could you do a tut or two about visualising these concepts in e.g. Blender?
I use matplotlib for 2D stuff and plotly for 3D. Everything you see here is basically a plot evolving in a time loop. For example, the plot with the little blue dots that shows the probability distribution, I made a grid of about ten thousand points, gave each one a small random nudge so it wouldn’t look so grid like, then in each loop iteration generated a random number at each point while evaluating the probability density function, and if the random number is less than rho then that pixel comes to life and fades out over a few frames, otherwise it stays invisible. So that’s just plt.scatter().
I’m sure there’s an easier way to do this stuff, but that’s what I’m used to. I have looked into Blender but haven’t done anything with it yet. Going forward though I’d definitely like to get into that, because it seems incredibly powerful.
Oh I came back for more brain melting 🫠
This video is excellent. Actually solving the entire SE from scratch would take absolutely forever and then some, so the walkthrough is more than adequate. Also, thank god for mathematicians, they really do make our jobs easier.
That makes me feel better, thanks! :) I was worried the video might not have been rigorous enough.
Well, not "forever", or even that long. It's at least in polynomial time. (P ≠ NP, by the way.)
IIRC it was like two weeks of lecture and homework in Pchem. And we still glossed over the P_l^m part
Best INDEPTH video on the Hydrogen Atom on RUclips. Look forward to Part 3 of 3
Student in nuclear physics, learning with books I sometimes don’t see behind math equations whereas with this video everything becomes more “alive” ! Thanks a lot for your time and I confess this is my new favorite video on RUclips.
Should a video about interaction theory (photon or neutron on nucleus) be released, it would for sure become a killer video in the field for it does not exist on the net, at least not explained with your skills ! Cheers !
Thanks for the wonderful comment! :) I’m glad you enjoyed the video. I’d love to do something on interaction theory in the future.
I got my BS in Nuclear Engineering, and one of my biggest regrets is not putting the "conceptual" with the equations. It's so hard to do problems when you have no idea what they're actually describing. I encourage you to keep watching videos like these..you'll understand the math work much better.
@@kidzbop38isstraightfire92 My theoretical mechanics teacher said that though math is paramont in physics, still more important is "physical intuition" about how things work. He forgave many sins for "the right idea". (I suspect that anyone who needs to say "righty-tighty" to themselves is lacking that intuition.)
@@mikemondano3624 yea, this is why I always hated the "shut up and calculate" idea that permeates modern physics...sure, there are certain cases where the equations manifested into novel concepts, but most of the time it's the reverse (where somebody has an idea first and then worked the equations later).
@@mikemondano3624bro could u give me some tips on improving my physics and maths I am 16 and trying to improve in physics. I live in India and here we have to give entrance exams to go for higher studies but the acceptance rate is less than 1% and 70 % of that one percent goes to schedule casts . So I wanted to know from an experienced guy where can I get some university courses in the field of physics on scholarship outside this shithole. I don’t study a lot tho
I wish I had this video around when I was taking quantum 50 years ago. I had to visualize all of this in my head, not knowing if I was right or wrong.
Plus, imagine furiously writing down all these derivations from the blackboard while trying to understand the logic hoping my notes will make sense later on.
Well done.
;-)
Thanks for the kind comment, and I’m glad you enjoyed the video! :) And I can relate to your experiences writing down derivations from the blackboard 😅 I hope these videos can make physics concepts more accessible.
This catapulted me back 35 years to my physics degree. We actually slogged through the derivation of all the important orthonormal functions (Legendre, Laguerre but also Hermite and Chebyshev etc.) in several three-hour maths lectures to equip us withe necessary tools, and then our QM lecturer showed how they were applicable to the hydrogen atom, electron-in-a-box and so on. It was a fairly painful process, and we would stagger out of the lectures afterwards with our heads reeling.
Bro if ur a physicist can u recommend me some tips for improving my critical thinking on physics and maths also I am 16 and also wanted to know if there are any university courses in quantum mechanics and computers where I can go after 18 .
@@ayanhassankango239 The sole piece of general advice I can give is practice, practice, practice problems in algebraic manipulation, calculus, trigonometry and linear algebra until you have the basics at your fingertips. If you're interested in physics pay particular attention to differential equations (for quantum mechanics and orbital dynamics etc.) and vector calculus (for electricity and magnetism).
@@ayanhassankango239 I recommend 3Blue1Brown playlist on Linear Algebra. It will give you an intuition of what's it's all about. But the rest is practice indeed... You can take any course on Coursera, they come with homework. It's free, you only have to pay to get a certificate of completion.
I am an engineer (retired) so the math and the particle physics is over my head but nevertheless this was a mesmerizing video to watch. The visuals and the animations help to make sense of it all. Please, dedicate additional 5-10% of your presentation to laymen like me. Give us a bit more of a 30,000 feet view. This will broaden your audience. It might be a bit boring or redundant for some of the specialists but it will definitely bring more people to your channel. or rather you can reach and enrich more people. Educating and bringing people up, what could be more noble than that! Thank you for the effort and the discipline to create, produce and post such videos.
Thanks for your thoughtful comment, and I’m glad you enjoyed the video! :) In future videos I’ll put more time into the 30,000 ft view. Even for those who are specialists, it’s always nice to take a step back sometimes and reflect on the context, and think about how the ideas are related to other things. Honestly that’s what I like most about physics, not getting stuck in the mathematical weeds but thinking about the big picture. But the math keeps one grounded, and there’s a beauty in its precision. I guess it’s a balance.
Hello, I'm also a Mechanical engineer and I took seriously and enjoyed learning Quantum Mechanics (QM) so be patient... And take a book and spend your time self enlightening you about this issues... And don't be afraid about math... Because without math it's impossible to understand truly what's going on.... I recommend a short and old RUclips on schrodinger equation derivation from an english man called Aphysics or something similar that can help you loose your fear for Mathematics.... Enjoy it!!! By the way, the vids from Richard Behiel are among the best videos on RUclips... But I recommend you to start maybe with Arvin Ash or Physics Asylum by Nick. Or ParthG. Those are less mathematically laden...
Also remember particles are not bals. They are more vibrations or oscillations and ways to move energy around... And we still don't know the real nature beneath everything.... What's an electrical charge???? Etc....
It is often difficult to see the forest (Physics) for the trees (Mathematics). Well done presentation. It has been more than 45 yers since I studied this closely in grad school. You refreshed my memory and I subscribed for more.
a friend made me discover your channel, it was a pleasure to watch please continue !
periodicity and quantization being highly related makes a lot of sense to me--to be periodic you have to have some kind of ultimately integral repetition, which lends itself to quantization. Interesting video!
"integral" is an interesting word here. Indeed, you inteGRATE something and you get an inteGER as a result! 🙂
my goodness it’s breathtaking that math is able to establish an intuition for the nature of reality. The phase change and switch visualization is astounding when you think of reality literally being that way - in and around us. And it all (or most of it anyway) can be explained by math.
Great video so far - again, really like not shying away from the math here.
I think the first 20 minutes would have made more sense if we had the context from the slide 19:42 at the start. I think setting up why the spherical harmonic functions and radial functions are relevant is important prior to getting into the detail.
You’re totally right! Man I should start running these videos by you before posting them 😅
as someone who is essentially a humanities type-I have a decent grasp of common-or-garden calculus, and can dabble in linear algebra, multivariable calculus, etc, but not especially well-I am both absolutely fascinated by these videos and absolutely in awe of the people who figured all this stuff out in the first place. i feel like i grasp just enough of the maths to get a sense for how crushingly difficult doing something like figuring out solutions to the associated Legendre equation must be, and the fact that someone was able to do so is just ... wow.
more generally, it's incredible to think that our ability to record, pass down and build on our collective knowledge means that as a species, it's taken us a few thousand years to go from our first steps in agriculture and engineering to working out exquisitely precise and detailed mathematical descriptions of the tiniest building blocks of reality itself. if we manage to avoid wiping ourselves out, who knows what we'll prove capable of?
Unfortunately, just as agriculture was humanity's biggest mistake, allowing the mass production of starch to power and enlarge a workforce (hunter-gatherers were taller, healthier, and longer-lived), private property, capitalism, slavery, and great inequalities, the human being is propelled ultimately by greed and knowledge is a tool of self-enrichment well beyond the necessary. We are not a sustainable species.
Finally found time to watch this. Really great work! I found your channel through your Hopf fibration video - I do physical optics simulations and wanted a way to convert electric field amplitude / Jones vectors to points on the Poincaré (Bloch) sphere for a visualization. I was amazed at how elegant & beautiful the mathematics behind that conversion is!
My personal interest in these topics has been quite high recently, so I'm feeling very lucky that you're producing such nice videos on them at this point in time 😁
Thanks, I’ve glad you’re enjoying the videos! :) What kind of optics?
@@RichBehiel Polarization gratings, mostly!
I had attended two semesters of lectures in quantum mechanics with Professor L. at the University of Stuttgart during my studies in electrical engineering. He only showed us the calculation of the ground state of the electron shell around a proton. You, on the other hand, even show the basics for calculating many orbitals of the hydrogen atom. Only 2 of 3 videos that easily go beyond the content of the lecture at that time. Many thanks also for the vivid visualisations of the solutions to the equations, where I could only do mental movies at the time.
I’m glad you enjoyed the video, and thanks for watching! :) Video has a lot of potential, compared to lectures. I think the future of education will be based a lot on videos, especially for physics since it often takes effort to make the equations come to life through imagination alone, but videos can provide a template for the imagination that helps kickstart the process of building intuition.
Lecturing in person is an impressive skill though, something I probably couldn’t do very well. With video, it’s very helpful being able to edit everything before sharing it.
This is the best, most well formatted description of something that is wrong I've ever seen!
Thank you for striking the delicate balance between providing just enough mathematical details and the clarity of explanation. Much of modern physics is mired in mathematics that it is hard to find explanation that interprets the equations and solutions clearly. We need more such videos that is not afraid to tackle the mathematics and also provides sufficient physical interpretation. Please post videos expositing the way spin and magnetic quantum numbers(partly alluded to in this video with quantum number 'm') arising from SE.
As a person who has just started learning about quantum mechanics, this is a beautifully created video. The video was easy to follow with the animations while also gradually setting a rigorous base for solving the SE. I am thinking about persuing physics professionally and with such amazing explainers as these, I might just do it.
PS: thanks for the reference articles as well; reading through them was really helpful.
Wow this series is amazing. I come from a more math background but have always been interested in quantum physics. IMO the video really strikes a wonderful balance between rigor and intuition. Also the visuals are so beautiful and has helped me depend my appreciation for topics I thought I was generally comfortable with. Really appreciate the experimental validations done to test the model of quantum mechanics, would love to see more of those!
Thanks for the kind comment, and I’m glad you enjoyed the video! :) Good idea about including more experimental validations. I think it often goes without saying that physics has been experimentally verified, but it‘s worthwhile to show how that was done. I’ll be sure to include more experimental validations in future videos.
I'm only in year 9 right now so I don't understand most of the math yet, however I've been really interested in particle physics as well as quantum physics and videos like this which give both a visual and mathematic/physics based explanation have allowed me to further help me understand the topic while being behind in the maths (but helping to understand the parts of the equation and what it all means). Thank you for making this series and you have no idea how much I want learn and hopefully fully understand all of the equations in this video.
Thanks for your kind comment, and it sounds like you’re on a good path with your education! :) A passion for knowledge is the most important thing. Eventually all the math will make sense.
Something you can study on your own right now that will be a real boon to you when learning quantum mechanics is linear algebra.
@@debrachambers1304 I’ll start having a look into that, thanks for the tips!
Just found the channel; was a really well done video and yes, anticipating part three as well. The Dirac equation is fascinating. :)
Was an excellent idea to place the functions themselves out front and explore their properties for a while, before diving into the nitty-gritty of equations.
Thank you so much !
Thanks, I’m glad you enjoyed the video! :)
Wow. Thank you so very much. This is the most detail I've ever heard. Anyone watching should completely understand the many rabbit holes to which you often allude, but wisely avoid. No need to apologize. Ever.
Wow… your a really good teacher. Deeply knowledgable.
Thanks! :)
Danke!
Thank you!! :)
Hi Richard, what a wonderful video! I had a course on quantum mechanics about 4 years ago at University, and I decided to refresh some of the old material before starting my PhD in about a month, and this video was an absolutely excellent refresher - great work.
Thanks, I’m glad you found the video helpful. Best of luck with your PhD! :)
Hi Richard,
Many things, I don't understand as a chemist, but still I am trying!
You presentation and voice clarity is excellent.This visualization of maths and it's working for the atom is amazing!
Thank you for your nice efforts!
This is exactly what I'm learning in my modern physics class... thank you for explaining it so well this actually helped me.
Awesome, I’m glad to hear that! :)
The quality of these videos is astonishing and the explanation is very clear and intuitive. Thank you for this incredible content 🔥
Thanks, I’m glad you’re enjoying the videos! :)
Superbly done. I am very much enjoying this series. There is so much beauty in the mathematics and you have shown that so well. Thank you
Thanks, I’m glad you’re enjoying these videos! :)
It is so wonderful that Bessel, Tchebychev, Legendre, Hermite, Laguerre, Jacobi and other had fun solving classes of ODE/PDE and gave us those polynômes in the XIXth century. Then Erwin Schrodinger could focus at the start of the XXth century on the physical interprétation of his equation and its solutions (building on the shoulders of its predecessors). Love maths
wow, this is way too advanced for me, but I can't wait to understand all that next year !
Don't hold your breath.
3 years later I still don't know shit 😅
One equation models not just the hydrogen atom, but all elements in the periodic table. No wonder it has a kind of switchboard (the quantum numbers like n, l, and m): set them properly and voilà ! You've selected an element and one of its various configurations. It's a necessarily complex equation. But videos like this one lets us examine what is it all about. But, of course, this is not a basic quantum mechanics video, you'll have to look elsewhere for why energy is quantized (and how), why there is a complex (real and imaginary parts) field, and other related things.
@@wafikiri_ It is an insoluble differential equation for all but n=1 to n=3.
18:13 It's important imo to also keep in mind, none of the energy eigenstates of Hydrogen want to sit near the proton. This is something a lot of people get wrong.
The wavefunctions for the s-orbitals don't go to zero at the proton, but all of the radial distribution functions do (multiplying by r² guarantees this). That means there are no energy eigenstates in the Hydrogen atom where the electrons want to sit close to the nucleus. This is indeed a requirement of the uncertainty principle as you discussed in the previous video on why the electron doesn't just fall straight to the nuclei of atoms.
Omg the Hydrogen atom is so complicated :) It's astounding how all these pieces were discovered and put together, and agree with experiment with great precision.
Great video. The last time I saw the H solution was when I was an undergrad physics major; your presentation was much more interesting and enjoyable.
Congratulations.... Again! : ) Your videos always bring bright insights from an educated physics who I assume loves what he is talking about and had spent long hours enjoying thinking about the physics and maths behind it. And I love the balance between raw physics math and equations that are usually preceded by animations and insights about the problem... In my case, I self study Quantum Mechanics with Griffiths book and also followed the MIT openware course on QM I and part of QM II and i use your videos to review things that I supposedly learnt, and see them visually and wonderfully expressed in your rigurously accurate animations.. But I always get great insights and a taste for the more complicated things that lay beneath QM... QM is amazing and hard to learn for a 52 year old mechanically engineer... But I love it... And I love your videos.... Great effort in the script... Great care in the animations and editing.... No shortcuts, no potholes.... Congratulations!!! And thanks a lot for sharing your thoughts on this hard QM matter!!! And I love also your next videos.. The Klein Gordon Dirac series.. I already viewed them twice, and now I will enjoy them again for the third time in preparation for the 3rd part of your hydrogen series... Those Dirac videos are the first time I understood completely the Dirac equation... Now I'm the quest to understand or at least have solid knowledge about spin mathematics and physics, and I use your videos and eigenchris and Noah_explains_physics...and other math resources and buff... And I'm anxious to review all of them again adding an additional onion layer or understanding!! I also have in the wish list your video on electromagnetism as a gauge theory... But I'm preparing for it as you ask for.... I think I'm probably the kind of public that can enjoy the most your videos... And I think there are thousands or hundreds of thousands of viewers like me... So your videos are wonderful and are helping lots of people.... So keep this way!!!! Congratulations!!! And many thanks!!!!! : )
Thank you, I took my modern physics class earlier, but I need a refresher for my quantum class later
Superb video. I am an engineer just wanting to master spherical harmonics, but this makes me want to go further. Thank you.
Thanks, I’m glad you enjoyed the video! :)
litteraly can't wait for part 3
You have such a great channel. Thanks for all the work you do Mr. Behiel.
Thanks Haresh, I’m glad you’re enjoying the videos! :)
Super duper excited for the third part! The pace of the content is perfect. PLease add the prerequisite videos soon. The pace of the delivery is just perfect.
Great set of videos. You don't really need to be as apologetic as you were there at the end - you laid out a great road map.
Can't help but feel a bit nostalgic of my quantum mechanics courses 10 years ago (engineering physics). What an amazing subject! It's funny how 2 weeks ago when you released this video I was going back through my notes, trying to explain spherical harmonics and the hydrogen atom to my vetarian girlfriend (amazingly she is curious about this). I wish I had found your video ahah!
I went back to quantum mechanics recently because I randomly asked myself, what the hell are spinors again? Oh boy I went down a rabbit hole down : weak hypercharge and isospin, higgs mechanism, majorana fermions, spontaneous symmetry breaking, etc. Now I feel like a toddler, but i'm happy. Anyway, thanks for your work and I can't wait for part 3!
Good luck. Stay positive and remember that there's other fish in the sea. You'll find somebody else (and try people also who are not from Vetaria).
I’m following (I think) so far. I’m an amateur - grossly untrained and unintuitive - and yet I followed along. Awesome stuff, thank you, keep going.
Thanks, and I’m glad to hear that! :)
Wow. I really love this channel alot, and all the videos 🙂❤️ Physics and maths and chemistry is very amazing! I love this soo muchh ✨️
Especially quantum physics 😌
Thanks for the kind comment! :)
You're welcome ^^ ❤️
Thank you for covering a lot my exam content from the what's going on side. This has been a huge help being able to visulise it and hopefully will help me to remember what physics to use where now that I have more of an understanding of what the maths represents. Now hoping part 3 comes out in the next couple weeks to save me haha
I just started learning about orbitals and their orientation in highschool and was a bit curious from where the shape of these orbitals come from.
I think need to wait until university to understand how to solve these equations.
But thanks for the excellent video, it is very clear and shows how to plot the function in different coordinate systems.
I think i will understand this in comming 4-5 years and it will help me throughout my life.
Your approach to the topic is incredible! Also I think the jokes are very funny =) Keep up the good work, sir!
Thanks! I’m glad you enjoyed the video :)
This video is amazing! I can't imagine how much effort was put into animating the graphs and equations (BTW, what software do you use to do that), but I hope you know it's greatly appreciated, and I'm really looking forward to part 3!
Thanks, I’m glad you enjoyed it! :) I use python, with matplotlib. It’s kind of a tedious way of going about it, but it’s just what I had learned to use as an engineer. There’s probably better software out there for this kind of thing.
谢谢！
感謝您觀看我的視頻 :)
This is brilliant! I’m taking a course right now about hilbert spaces and P.D.E’s and the intuition of using the spherical harmonics as an orthonormal base is fascinating. Very well made video!
Thanks, I’m glad you enjoyed the video! :)
Totally awesome! I had hoped to learn how you calculated the animation of the orbitals involving the randomness.
0:00: 📚 This video is about eigenstates, spherical harmonics, radial functions, and the absorption emission spectrum.
4:11: 🌐 The function plots complex numbers on a spherical plot and a two-dimensional plot, representing amplitude and phase.
8:13: 📚 The video discusses spherical harmonics and their properties.
12:15: 🌈 The plot shows a rainbow circle representing the complex phase and amplitude of a function.
16:10: 📐 The length scale of atoms can be calculated from Schrodinger's equation, with the Bohr radius being about one ten thousandth the wavelength of green light.
21:19: 🧪 The energy eigenvalues for each energy eigenstate in a hydrogen atom depend only on n and have a specific formula.
25:37: 🔬 The Schrodinger model of the hydrogen atom aligns well with experimental data and is precise and accurate.
29:23: 🔑 The essence of separation of variables is that both sides of the equation are constant, allowing us to separate the differential equation into two equations.
33:34: 📝 The first derivative of f with respect to Theta can be written as a derivative of f with respect to psi.
37:59: 🧩 The radial equation in quantum mechanics is influenced by the quantization of the angular momentum and the coupling constant.
42:12: 📚 The derivation shows that the atomic length scale can be determined from the Schrodinger equation.
Recap by Tammy AI
12:29 I'm confused. In the circle, does each angle represent an input for theta, with the pattern in the circle changing with time at phi changes with time? If so, how are there multiple values for an angle?
Great question! The plot shows exp[i*m*phi], where I’m varying m in time to show the general form of the equation. You can see that most of the time, the curve doesn’t come back to itself, so the solutions which are smooth and continuous just have an integer m. The outer circle in the plot shows the magnitude (constant radius) and phase (color), with the flowery patterns representing the real and complex parts of the equation.
instant watch every time 👍👍 great work as always
Thanks, I’m glad you enjoyed the video! :)
Thank you for posting!
Need a 3rd part... but ur the goat
What a wonderful video!!! Thank you for this enlightening content
Thanks for this very informative video ( with a high pitch voice because of all the hydrogen you managed to make me inhale ).
Atoms are smaller than 17:03 visible light!
Q. Is there way that I can solve this in book or excel or Python?
Thank you so much sir.
When will you upload the third part?
Thanks for watching! :) I’ll be doing some videos on the Dirac equation and electromagnetism as a gauge theory first, then will return to hydrogen for part 3, with a relativistic electron.
Wow!! what a great video! I'm super excited for part 3! Do you have patreon?
Thanks, I’m glad you enjoyed the video! :) I don’t have a Patreon, but I have superthanks enabled on the video 😉
Someday I might set up a Patreon, if/when I can commit more to putting RUclips videos out on a regular basis. For now RUclips is just a side thing, I love it, but I have to be able to take a break if something else comes up with family or work.
Hi thanks for the wonderful video. Is there a simple explanation on why we can assume the solution to be separable in r, \theta, \phi?
Great question. I’m not sure if there’s a simple explanation, to be honest. The usual explanation is that we’re just lucky. Ultimately it has to do with the simplicity and symmetry of the coulomb potential of a point charge. But I don’t know how to say anything more insightful than that.
These visualizations are amazing, and the explanation is great! I could more-or-less follow it as a non-physics-person (but a sorta-math person).
For the rainbow phase visualizations, I think it would be helpful consider using (Ok)LCH color space instead of HSV/HSL to plot the colors; human vision perceives 0,255,0 green noticeably brighter than 255,0,0 red or 0,0,255 blue. The LCH color space corrects for this (so e.g. all "80% luma" colors are actually equally bright, which is not true of 80% luminosity in HSL or 80% value in HSV), so there wouldn't be "banding" on the spherical harmonics. It might make the visualization smoother-looking and therefore maybe easier to interpret? Although it's hard to know for sure without trying it out; it's definitely a popular model for digital color design now.
Great idea! :) I’ll give that a try for sure, I agree that it would be nice if the blue didn’t look so dark.
genuinely great account
Thanks! :)
I'm 17 years old I love maths and physics and I'm absolutely not able to deal with these crazy physics and maths concepts... But I really want to solve it one day!!!🤩🤩🤩🤩 This is how a hydrogen atom looks like!!! But unfortunately I cannot cope with it right now😅🤣
Keep studying and you’ll become an expert someday! :)
@RichBehiel Thank you sooo much!!! Within a few years I'm going to watch your videos so I can finally learn it!!! I'd like to study physics at university and now I'm in high school in Italy and it is a chemistry school (we mainly do chemistry) I'm studying maths on my own because I think that the more I understand maths the better I will understand physics one I get at university!!!🤩🤩🤩
@@JonnyMath any and every doubts you come up to have in the future about basics and advanced physics
, talk to me
@ryandias5153 Thanks!!! I want to start studying Newtonian mechanics this summer and I got from school 2 books about classical mechanics!!! Because I had to choose 2 books ('Cause I won a chemistry competition and I'm in a chemistry high school) and I chose these two... BTW they're in Italian and are the textbooks chosen by universities but I tend to read more in English rather than Italian!!!😅🤣🤣🤣
This understanding technics is very cool. This sir ,i will be learn morn some things for this video.i clear the quantam part of schidorgen equation which is very big😅😅😅
Love your content, keep it up man! 👍
Thanks! :)
Freaking awesome course, I've never heard anyone explain it so well! What book is that by chance, I'd like to buy it!
Thanks! :)
I’d recommend Introduction to Elementary Particles, by Griffiths. It starts off prosaic and historical, then gradually transitions all the way into quantum field theory. Along the way, it covers most of the main concepts in particle physics and quantum mechanics, at a level which is mathematically rigorous but also general enough that you feel inspired to read papers about each topic. Honestly it’s one of my favorite books of all time.
Thanks a lot for the video, although I didnt understand anything, it was interesting to see how complex physics can be. :-)
Holy that is a lot of math lol... very clear instruction thank you so much, cant wait for the 3rd video if you're still "cooking"!
thank you professor, that helps a lot
Beautiful derivation!
Thanks, I’m glad you enjoyed it! :)
man i love this!! It's gonna sound evil and selfish. but don't die before uploading the 3rd part.
i need it.
I’ll try my best not to! 😅
So in two videos, you summarized Griffiths’s book.
One video would be equal to a whole course for undergraduate physics student lol
I could never compare my work to Griffiths! But thanks for the compliment :)
Video definitely has the potential to fundamentally change physics education. When looking at a page full of equations for the first time, the ideas don’t exactly leap off the page. But with video, they do, and they come to life. That has a much higher chance of inspiring a spark of insight in someone who’s trying to learn physics.
@@RichBehiel I believe it is a very interesting case for Physics Education Research. 10 years ago, they had a very active group in K-State. Not sure if they are still active there.
As for the duality, I hypothesize a new model for the atom, but there is one small issue, which is the movement from continuous to discrete with functions. Until today I couldn’t find it away from you continuous to discreet the closest person. I’ve observed to solve this issue, however cheated, was Gregory Perelman on his approach to solve the Poincaré conjecture using Ricci Flow. However, he used the “surgical method” to remove some troubling singularities to be able to solve the conjecture.
I theorized that going from continuous to discrete shape is like turning a sphere inside out without reaching a singularity
Great video and definitely inspiring
Love this video so much, thank you. I have a question about the emission spectrum with the experimental verification. Can the electron only occupy one energy eigenstate at one time? My understanding of these eigenfunctions was that they were like an orthonormal basis for the states the particle could occupy, and a particle occupies a superposition of them. In the theory (the one that's being experimentally tested around 22:00), what stops an electron from being in some linear combination (would this be a "superposition"?) of the eigenstates? Or if it can be in a combination of the eigenstates, then how is its energy determined? Clearly not by a combination of the energy eigenstates since then the bands wouldn't be discrete. I have a feeling that the energy quantization is the entire point but I'm missing where that's coming from in the theory
Is it that the state the electron is in has to be in the eigenspace of the Hamiltonian operator? So you can only combine states with the same energy eigenvalue?
I think the phase factor in spherical harmonics can be better represented as a height function on the surface of the sphere instead of colors. This will give a much better intuition of the "wavy" nature of the harmonics, like in this video: Spherical Harmonics (U2-05-05) by quantum vision.
Dear Richard, I've only just discovered your channel. Has part 3 of the hydrogen atom been released yet please? Or has it been deleted?
I haven’t finished it yet, but am working on it :) I sort of went off on a tangent after Part 2, and am currently working on “Electromagnetism as a Gauge Theory”. That will lay out some of the concepts that we’ll see in Part 3.
@@RichBehiel Thank you. I shall look forward to it.
Waiting for the part 3 of H atom ⚛️,
"But it's actually a huge step forward, because partial derivatives are very painful, and ordinary derivatives are only an ordinary amount of pain."
Between this, the elephant joke in the previous video, and the joke about the symbol for spherical harmonics being Y in this one, I have to say I absolutely love your sense of humor.
One thing I would like explained, though, is why iħ∇ is the momentum operator.
Thanks! :)
Good question. Honestly it’s hard to give a thorough answer in a comment. Feynman and Hibbs do a decent job of addressing it in their book Quantum Mechanics and Path Integrals. Long story short, wave mechanics is sort of a way of capturing the statistical behavior of a quantum system, but underneath the wave level of description there’s the principle of least action, acting on all possible paths. That’s really the level where you’d want to look at the nature of momentum, then from that, it follows that the momentum operator is related to the gradient of the wavefunction.
Thanks! :)
Good question. Honestly it’s hard to give a thorough answer in a comment. Feynman and Hibbs do a decent job of addressing it in their book Quantum Mechanics and Path Integrals. Long story short, wave mechanics is sort of a way of capturing the statistical behavior of a quantum system, but underneath the wave level of description there’s the principle of least action, acting on all possible paths. That’s really the level where you’d want to look at the nature of momentum, then from that, it follows that the momentum operator is related to the gradient of the wavefunction.
@@RichBehiel Oh my god, you're right. I'd actually read about Noether's theorem and the relation between spatial translations, Lie groups, Lie algebras, and momentum as a generator of translations and completely forgot until now!
It all came back when I was re-reading your comment for a second time. Thank you!
Fine video. I would appreciate a little more about non-energy eigenstates transitional States non-stationary states and their stability just perhaps qualitative. We seldom see pictures of those
Thanks for this vide it was verry informative
Thanks for watching! :)
Great video! thanks😁
due to the a-symetry of the spherical harmonics, the solution clearly cares about which direction is 'up'; ie which direction we choose to take our theta from, in real life by what metric is 'up' decided for the atom, is it just random? just some superposition of different axis choices?
Rich did not explain this, but the states he derived are eigenstates of three operators simultaneously: energy (denoted E), total angular momentum ( denoted L^2), and angular momentum along the z-axis (denoted Lz). Physically this means we can prepare a hydrogen atom to be in a state where we have knowledge of all three properties simultaneously. You are right that knowing the angular momentum along the z-axis is somewhat arbitrary - we could have also solved our equations and derived solutions where we would know E, L^2, and L_alpha (where L_alpha would be the angular momentum about an arbitrary axis). Rich didn't get into this at all, but while you can know the angular momentum about a particular axis (let's say Lz), you do not then get to know the angular momentum about orthogonal axes Lx and Ly. Mathematically, you can show that the solutions Rich derived all have the property that they are eigenstates of Lz, but are *not* eigenstates of Lx or Ly.
I am guessing that we end up in this situation "in real life" when an interaction occurs (lets say a ground state hydrogen atom and a photon) and we know the photon has just the correct energy and z-axis angular momentum to excite the hydrogen atom into one of its excited states that then has the z-axis as a "special axis" (the asymmetrical axis of the spherical harmonics, as you are pointing out).
Where is part 3?
Amazing video!
Thanks, I’m glad you enjoyed it! :)
If you were to plug in negative n values and map r to -r for the Radial function, would this be the radial function for a positron? or is what im saying just nonsense
Good question! Actually a positron would just fly away from the proton, wouldn’t form a bound state since positive and positive repel.
But if you had a positron and an anti-proton, then you’d have antihydrogen. It behaves just like hydrogen, same eigenstates and everything. The charges are just reversed.
@@RichBehiel I’m wondering if it is possible that the electron can temporarily go into a negative energy state, then be pushed back into a positive state by the proton. if so what would be difference in energy, and would a photon be emitted with that energy.
@beamshooter the electron is always in a negative energy state in hydrogen, zero being the unbound state when it’s infinitely far away.
But there is a subtle effect, from quantum electrodynamics, in which the electron kinda sorta almost becomes positron-like when it’s very close to the nucleus. Google the “Darwin term”. It’s not as straightforward as the electron becoming a positron, but to contextualize it requires looking at the Dirac equation, where spin and antimatter play an essential role in the geometry of the situation.
@@RichBehielso if a photon adds enough energy to take E > 0, that electron no longer bound, but could it still spontaneously emit photon to bring it back into a negative energy bound state?
Another fantastic video! I’m very curious, what do you do that would require you to know all this? Do you have a physics degree? This is all extremely cool and I’m curious what I could do in the future to learn more about this stuff
Thanks! :)
Well I’ve always been obsessed with physics, since I was a kid. But when I went to college, I majored in mechanical engineering since I figured that way I could learn a decent amount of physics, while having a higher chance of paying the bills. But as an undergrad, I was drawn more towards physics rather than machines. So for my master’s degree I studied materials science, a bit aimlessly, dabbling in metallurgy but ultimately drifting toward condensed matter physics and falling in love with quantum mechanics. The class where we had to calculate bulk material properties from quantum ab initio simulations, was a turning point for me where I realized that QM was really existentially profound.
After getting my master’s, rather than pursue a PhD in physics, I decided to go into industry instead, again due to considerations about paying bills. But I knew I had to do something physics-y. So I worked with micro-electromechanical systems for a few years, and that was a lot of fun. I got to work with a lot of tooling and techniques from the semiconductor industry, but applied it to weird futuristic devices.
About a year and a half ago, I joined a startup that’s making superconducting tape for all kinds of scientific and industrial applications. It’s a kind of material that hasn’t really been commercialized at scale yet, so we’re trying to change that. So that’s been a lot of fun. Lots of quantum physics involved with characterizing this kind of material, diffraction using x-rays and electrons to study crystal structure, and also the material itself goes into an exotic state when it superconducts, pairing electrons together so they can collectively condense into the same quantum state and slide through the material literally like a ghost, an electrically charged ghost. And in a strong magnetic field, quantized flux vortices form, and we have to design our material to trap those vortices in an optimal way. It’s a whole world in there, with many analogues to deep concepts in fundamental physics. For example, flux vortices are a real version of what superfluid vacuum theorists dream about.
Anyway, sorry for the wall of text here, but as for what you could do in the future, the possibilities are endless. Learning physics has probably the best return on investment, over any other subject. Well, depends on your circumstances I guess. But physics lets you engage with Nature on a whole other level. It’s spiritually satisfying, and often lucrative. Not always easy to learn though 😅
@@RichBehiel
ruclips.net/video/tF-1dF0eBts/видео.html
Universe in whole & it's function
@@RichBehieldamn what an interesting life you had, I'm definitely envious lol. How's the industry/PhD choice for now? I'm thinking about going deep into academia but I have the same concern as yours, bills. In my field, it's hard to find the job that I'm passionate while still being enough for me to start a new family.
Personally I think I made the right choice with regards to the PhD, even though I really wish I had the luxury of pursuing it, I couldn’t justify it financially. Compared to working in industry after a master’s and saving aggressively, the difference in the two options is life changing. Compound interest is a powerful thing, and missing out on the opportunity to save in your twenties is a mistake that a lot of people make without realizing it.
I have a few friends who stayed in academia and recently got their PhD. For the most part, they seem depressed and aimless, like they just finished working way too hard on something that wasn’t entirely in their interest, and now they’re not sure what to do. I might be a bit cynical, maybe this is just my own bias having not pursued a PhD, but at least in my friend group the PhDs tend not to be the happiest people. Maybe this will change as time goes on, and they presumably get access to high status jobs, but that’s not a sure bet. And they have some catching up to do, compared to those who have been working a few years already and have started to develop a sense of how the world works, which is something they don’t teach in school.
Everything I just said would be totally irrelevant *if* academia were still the only way to learn things. If so, then I’d have a physics PhD because there’s just nothing I could do to stop myself from going down that path. But nowadays we have the internet and books and after a certain point you can teach yourself. Especially in physics, you can’t fool yourself because every step of the way you can just think “can I solve the exercises in this book and get the right answers?”. If yes then you know it, if not then you don’t. There don’t have to be gatekeepers involved in that learning process, you just follow what interests you and keep at it.
Great video!!
Thanks, I’m glad you enjoyed it! :)
GREAT VIDEO!!!
Excellent. Is the radius in the first section on the angular dependency approx a0, thanks
Good question! For the spherical harmonic plots, the radius is actually arbitrary since it’s just a function of phi and theta. The sphere is just a way of representing the 2D angular domain with its natural topography.
After you finish with Hydrogen will you touch on the schodinger equation amd Helium or at least explore why heavier atoms lack exact equations?
Yeah, I’d probably just mention that at the end of hydrogen part 3. The reason is fairly straightforward, the math just becomes super messy for a two-particle wavefunction where the electrons are still repealing each other while both are delocalized. Plus, the antisymmetry of the wavefunction and the exclusion principle.
@RichBehiel
thanks! We didnt touch on helium much when i took QM besides the prof saying something like "things get non linear so its easier to just solve with a computer" but i really want to be able solve them as much as possible before resulting to solving them on the computer anyway.
Honestly, if i spend the whole day solving all those equations, i'd have less questions "what the heck is this variable doing?"
So, now i have more questions than answers
@RichBehiel where is part 3 of 3? These vids are really good please finish the series
I’m glad you enjoyed the videos! :) Part 3 is a work in progress, I’m going off on a detour into Lorentz transforms and the Dirac equation first, in order to set the stage for the ideas in that video. Part 3 will involve a lot of relativistic quantum mechanics, and while making it I realized that it’ll be helpful to do a couple videos first to cover a few of the prerequisite concepts.
A bit lost in spherical harmonic related to m n l; any computer program to generate ….
Great video indeed...
Do you happen to know what book the linked paper comes from? I would like to find the full book. I appreciate the way it is written
Is part 3 comming out?
Eventually 😅 I’ve gone off on kind of a tangent lately, currently working on some superconductor videos, then I might go into electroweak unification or might finally come back to H. Sorry for the delay.
@@RichBehiel That's alright, keep up the good work!
@@RichBehielGo wherever your hearts leads you! Work made out of passion is greater than work made out of duty ❤
How do you create these simulations graphics, is it in c++?
Python, matplotlib for 2D stuff and plotly for 3D.
My exams are in 10 days and these two lectures made me understand a hell lot of things..but where's the part 3!🥺
I’m still working on it :) My next video will be Electromagnetism as a Gauge Theory, then probably the magnetic moment of the electron, then hydrogen part 3. Part 3 is going to be advanced, so I want to lay out the prerequisites first.
Thank you so much for the video Its a great help! The only part that has confused me about this is how the spherical part of the equations is solved. I know thay the Azimuthal solution is used but my brain won't accept that as a answer and I really would like to know how that solution was found. Does anyone know anywhere I can find a guide for that solution? I've tried looking but haven't found anything that helps but I probably haven't looked hard enough 😅.
I’m glad you enjoyed the video! :)
For the azimuthal equation, are you asking about how we know the general solution is A*exp(im*phi) + B*exp(-im*phi), or how we were able to set B = 0 without loss of generality?
If the former question, the solution arises from the famous property that e^x is its own derivative. So if you have the equation f’(x) = f(x) then you know right away that’s A*exp(x). Likewise if the equation is f’(x) = m*f(x) then by the chain rule, f(x) = A*exp(m*x). When the equation is f”(x) = -(m^2)f(x), then you just take the square root of the usual thought process, to get the solution.
As for why we can set B = 0 without loss of generality, I can’t answer that sufficiently in a RUclips comment. But some partial answers can be found by considering that m can be negative, so there’s a bit of redundancy in the space of solutions when running through all m. Likewise because of the global U(1) symmetry of the wavefunction, the set of general solutions has a phase redundancy. And thinking in the context of the 2D angular equation, really we just need an orthonormal set of functions to span the solution to that equation. You would have to get into the weeds with the associated Legendre polynomial to really see why we can set B = 0, but I hope this comment points you in the right direction.
@@RichBehielOMG I forgot e^x is its own devrvaite Jesus I feel dumb 😅. But thank you yes it was the first comment you made about how Do we know what the solution to the equation is.
The other comment you made about generality of B and bringing up the U(1) is something I will have to do more reasurch on. I see groups and U(1) group mentioned allot but never got my head around it. Thank you so much for the video can't wait for the third one!!!!!