Very interesting observation!

Thanks for posting that comment . In the book Global Scaling by Hartmut Muller the author , on page 16, says that : ".....in this way , the natural exponential function e[exponential x] of the natural argument x=[n series] generates the set of preferred ratios of quantities which provide the lasting stability of real processes and structures regardless of their complexity. This is a very powerful conclusion . " . This could apply to the human or any other heartbeat . Hartmut explains how Euler's number e prevents resonance in planetary orbits due to e 's unique transcendentalism .

this is amazing!

e and π are fundamentally linked and one can be written in terms of the other.

@@Unmannedair Would that not apply to all transcendental numbers? It may not be a linear or exponential map, but I'd wager that any transcendental can be mapped (with a continuous function on all of the reals), such that f(T1) = T2, where T and T2 are two arbitrarily chosen transcendentals. What's interesting is surely not that they are 'linked', but how they are linked (and all transcendentals are linked, by virtue of them being transcendentals, no?), right?

Also: e to the i: continuously changing cyclic things.

Pi:3 e: 3

@@NukeCloudstalker I think you are misunderstanding the point being made. The point being made is that exponential things and cyclic things are actually just the same thing.

Thanks Lukas, that’s a very kind comment! :) Looking forward to seeing your videos.

Thanks Tim, that’s a very kind comment, and I’m glad you’ve enjoyed these videos! :) Sounds like you’re on a great path with your education.

Thanks! That means a lot :)

Thanks! :)

That’s a great method too! :) I like the elegance of that formula.

KZfaq text is hard to understand and pain for eyes to so is there any alternate method through which I can understand what you written Here

@@Oms-xk2zb sure , i rewrote the equations on paint and i saved the image on drive , here is the drive link of the image : drive.google.com/file/d/1p8Jw9L5OTkJ3Dx-_q272ilK6MPwZx5Xs/view?usp=sharing

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

you have to start somewhere. keep watching

@@jamescollier3 Thanks, I'm starting to learn calculus in my free time. I've learned the rules for differentiation and a few integration tricks. I'm using Khanacademy right now to get a more rigorous approach to learning it instead of just random questions in my head lol.

Thanks! :) I use Python. Matplotlib for 2D things, and plotly for 3D.

@@RichBehiel Interesting, I was looking at using Manim for a video I'm attempting to make (just a overarching introduction to quantum field theory) but I've been having a rough time figuring out how to animate, let alone in 3D. Is there some sort of reference material or course on the methods you use?

That would be really cool, and I’m looking forward to seeing that video! 3D is inherently challenging. I try to avoid it, unless it really adds something. Many physical principles can be communicated effectively in 2+1 dimensions. And I’ve found that 1+1 dimensions, spacetime diagrams in particular, are actually confusingly low-dimensional. So it’s useful to do 2D plots that evolve over time, which you can make by calling plt.contourf in a “while t < tMax” loop and grabbing the frame in each loop iteration. I’ve been meaning to post some example codes, just haven’t gotten around to it yet. Part of the problem is that my animation codes are all spaghetti code, since I just write them quick and dirty to get the animation and move on. It would be more helpful to slow down and write them neatly, but it seems I’m always in a rush these days.

Sure, but those properties can all be derived from the fact that d/dx[e^x] = e^x. As we saw in the video, we can derive the full MacLaurin series for e^x from the fact that it’s equal to its own derivative. That series defines everything there is about e^x. For example, look at the maclaurin series for sine and cosine and you’ll see that each has half the terms of e^ix, with the sin(x) terms needing a multiple of i. You can derive Euler’s identify from that. It has to do with the cyclic nature of the derivatives of sin and cos. For rotations, this is due to the fact that d/dx[e^ix] = i*e^ix which comes from the chain rule. If the rate of change is i times the function, and i rotates a complex number by 90 degrees, then you get circular motion - when the value is 1, the direction of motion is i, when the value is i, the direction of motion is -1, etc. That’s why e^ix rotates.

@@RichBehiel I just watched the video a second time, and it just shows e to be a solution to that equation. Sorry, I have no idea what I meant when I wrote my previous comment.

All good! :)

Thanks, glad you enjoyed it! :)

yes

Nonsense!

45 90 45!

e = e^1

@@RichBehiel oh ok thanks.

Good point! The math could have been reformulated without using e per se. But the simplest way of writing it involves using e. If you look at the first C in your comment, e^{En hbar}, that C would be just as natural as e, as long as En and hbar were arbitrary constants which we may as well put in the exponential. But the energy eigenvalue and hbar both have real significance in other contexts, not just as things in an exponential. So then when constructing C, one would still have to wonder what’s the deal with this number e.

@@RichBehiel so rewrite "e^{i En t / hbar}" as "i^{2 En t / (pi hbar)" and switch hbar to h to get "i^{4 En t / h}" which has the constants you care about, no e, and hey, even the pi went away...

Ok, that’s messing with my head. I’ll need some time to digest that. You’ve got me rethinking a lot of things about e 😅

@@RichBehiel hah, your schrodinger equation videos messed with my head first. (thank you).

@Tim: Great ideas, got me thinking quite a bit. :) One problem I see with your approach is that i is not only equal to e^(i pi/2), but also so e^(i 5pi/2), e^(-i 3pi/2) and so on. So writing i^{4 En t / h}, you would always additionally have to specify that one has to use i = e^(i pi/2) and not one of the infinitely many other possibilities.

Because when x = 0, all the terms except a cancel out, so we have e^0 = a, and a nonzero number raised to the zeroth power is 1.

And if you do not have i in the exponent, it all comes down to sinh and cosh, as cosh is the even part of e^x and sinh is the odd part and cosh(x) +sinh(x) = e^x

@@ingiford175 Yes, so? There is an i.

Not only are you first, but this is actually my first “first” comment! The ultimate first.

@@RichBehiel oh wow i'm honored lol! your videos have been so helpful for self-teaching and improving my intuition, really appreciate all your work!!

Respectos to the First of the Firsts.

Sorry, my ig is private. But I’m on X, @RBehiel

Yeah, one of the challenges of speaking to a diverse audience is knowing what people know and don’t know. I’ll be doing harder stuff soon, next few vids will be on relativistic QM, building up to applying the Dirac equation to hydrogen. In the meantime, just wanted to a quick standalone thing on e since I noticed it was in all of the equations, and it’s one of my favorite calculations because of the way it feels like we get something for nothing. There are also a lot of people with advanced degrees in STEM who just took e for granted and never saw how it was derived. Anyway, over time I’ll try to become more calibrated to what people want to see. My main goal with this channel is to explore the nature of the electron, using all the tools of modern science, while bringing as many people along for the ride as possible. I might have the occasional short videos that are tangential to that goal, if I think it’s a neat calculation or something. But I respect everyone’s time and I hope I’m not putting totally irrelevant stuff out there. So thanks for your feedback! :)

@@RichBehiel I don’t have any formal higher education background but I have taught myself a lot of math and loved physics in school. I’m about to go to college. I was able to follow your first hydrogen atom video very well, it’s a great video. I did my best with the second video, and it also looks like a great video, but you lost me, it got very very crazy for me. I wanted to ask, what’s the difference between a Hamiltonian and a Lagrangian? I know Lagrangian mechanics and the Hamiltonian operator as you explained it looks almost identical in principle

There are a lot of similarities between the Lagrangian and the Hamiltonian. It’s too much to write up in a comment, but this website does a great job of explaining the similarities and differences: profoundphysics.com/lagrangian-vs-hamiltonian-mechanics/#:~:text=The%20main%20difference%20between%20these,total%20energy%20of%20a%20system.

@@RichBehiel That’s a great resource, thank you