The Hydrogen Atom, Part 2 of 3: Solving the Schrodinger Equation

In this video, we explore the solutions of the Schrodinger equation for the hydrogen atom. Thank you to everyone who is continuing on this thrilling adventure into the smallest atom in the universe!

For those of you who want to see even more of the mathematical details, here's a wonderful paper on the topic, which I highly recommend:

More info on the spherical harmonics:

Table of spherical harmonics:

Chapters:
0:00 Intro
1:01 Spherical Harmonics
15:01 Radial Functions
20:04 Energy Eigenstates and Eigenvalues
22:02 Absorption/Emission Spectrum
26:49 Solving the S.E.
44:15 Concluding Remarks

#physics #quantum #math

[RichBehiel]

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! :)

[kaylo]

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.

[stevewhitt]

Best INDEPTH video on the Hydrogen Atom on YouTube. Look forward to Part 3 of 3

[davidgillies]

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.

[warrenchu]

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.
;-)

[December]

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.

[Eloss]

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 YouTube.
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 !

[JH-lesd]

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!

[johnsjarboe]

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.

[MihateLilas]

a friend made me discover your channel, it was a pleasure to watch please continue !

[milk]

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.

[evilotis]

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?

[keldholi]

wow, this is way too advanced for me, but I can't wait to understand all that next year !

[fahadparyani]

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!

[benmcconnell]

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.

[martinsiebert]

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.

[naysay]

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.

[idontexist]

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.

[fabienleguen]

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

[douglasferguson]

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 !