Hydrogen atom: power series solution

[@Hamnafoodfactory]

You are incredible in inculcating the knowledge of QM to the viewers and listeners. God bless you Prof. M.

[@ProfessorMdoesScience]

Glad you like our videos!

[@WilliamDye-willdye]

My favorite chapter is 33:07 -- Quantized energy eigenvalues. I've long been interested in the tension between the implicit infinities in physics equations vs. the need for confirmation through physical measurements that (probably) cannot measure an infinite value even in principle. Therefore it was a real treat to see an analysis of finite vs infinite series in the context of QM. Many thanks to both the M's for taking the time to post this series.

[@ProfessorMdoesScience]

Thanks for your support, and glad you like it!

[@angelmendez-rivera351]

Cannot measure an infinite value even in principle? What led you to that conclusion?

[@ProfessorMdoesScience]

@@angelmendez-rivera351 What do you mean by "measure an infinite value"? What we are doing is looking at the limit of large rho (as it tends to infinity) to understand how the function behaves at that limit.

[@angelmendez-rivera351]

@@ProfessorMdoesScience That part, I understand, but William Dye said that infinite quantities cannot be measured even in principle. I am not entirely sure what William means either, or whether this is relevant to the video, but the confidence which he said it makes me think he has an explanation.

[@nihargupte8032]

Dude are you guys reading my mind literally yesterday I was thinking, man if only Professor M had the video on hydrogen solutions I would be set THEN IT HAPPENED !

[@ProfessorMdoesScience]

This is a nice coincidence! :) Hope you like it!

[@kennethvaten9259]

Thank you!!! I love using a power series for solving differential equations. I used the Griffiths textbook, but the book makes a few big leaps about the time we need to stop the power series. Thank you for such a complete explanation!!

[@ProfessorMdoesScience]

Glad you found it useful! One of our aims is to include every single step in detail, so glad to hear it worked for you :)

[@alessiasambruna9350]

Best explaination of the topic I've found online !

[@ProfessorMdoesScience]

Glad you like it! :)

[@ibrahimsaidali3140]

A: Thank you very much. B: This is very helpful video I have seen so far. C: I will see the other videos for more information and understanding.

[@ProfessorMdoesScience]

Thanks for watching! :)

[@dutonic]

Me at the start of the video: "I'm so excited to finally understand the mathematical nature of the hydrogen atom!" Me 40 minutes into the video: "Why are we still here... just to suffer?"

[@ProfessorMdoesScience]

It is worth doing this in detail at least once! But absolutely agree that it is very long... I think it's our longest video by far!

[@zoubirbecer6529]

Professionals as usual. Good luck.

[@ProfessorMdoesScience]

Thanks for watching!

[@vivekyadav-ft3gz]

Thank you for your time and knowledge sister..

[@ProfessorMdoesScience]

Thanks for watching!

[@viplovedhoke6597]

Would you please also do a video on normal and Anomalous zeeman effect... Really loved your content in this one.. 🤩

[@ProfessorMdoesScience]

These topics are on our list, so hopefully we'll get there soon! :)

[@shankardattamishra]

I just love your each and every videos...

[@ProfessorMdoesScience]

Thanks for watching!

[@Mechmaster_Goku]

Great video!! Do you guys intend to make further videos on Atomic Physics?

[@ProfessorMdoesScience]

Glad you like it! We hope to do more videos on atomic physics and also on condensed matter physics, but it may take a while before we get there!

[@Mechmaster_Goku]

@@ProfessorMdoesScience Sounds great!

[@GeoffryGifari]

I think for this one the technique to get eigenfunctions and eigenvalues is clear enough. The ways to fix the constants need some getting used to (using boundary conditions, symmetries, normalization, etc)... can easily be obscured due to the number of steps involved. I happen to have questions which are related, but not exactly about what is in the video 1. How are ionization energy and binding energy related? equal/sign flipped? sometimes one terminology is used over the other in some cases 2. About the energy eigenvalues of bound states being negative, I also read about negative energies in context of Klein-Gordon and Dirac equations (maybe we'll get there someday, or is the video already there?), which is presented as problematic (which drove Dirac towards his Dirac sea explanation). This got me thinking... in some cases negative energies cause trouble, but in other cases its completely valid. I'm guessing that this is related to the ground state energy of bound states having a fixed (negative) value, that you can't get a lower energy even if its negative. What do you think?

[@ProfessorMdoesScience]

Glad you found it clear! Here are some thoughts for your questions: 1. I think ionization energy is specific to atomic systems, and refers to the energy necessary to remove the most weakly bound electron, the one electron in the case of hydrogen. Binding energy is typically used in a wider context to refer to the energy that binds together a collection of particles that are not necessarily electrons to nuclei. But may be wrong on this terminology! 2. If we add a constant potential to the Hamiltonian it will not change the eigenstates, and all eigenvalues will shift by the value of the constant potential (feel free to check this!). From this point of view, having negative eigenvalues is somewhat arbitrary, we could always make them positive by adding an appropriate constant. What is important is that the eigenvalue spectrum is bounded by below. What really matters for bound states is that their energy is below the potential at infinity, and in our case this potential goes to zero at infinity, so bound states have negative energies. Adding a positive constant would make the potential tend to a finite value at infinity, and bound states would be states whose energy is below that value. As to the Diract concept, very briefly as this would require more than a comment, the "issue" arises because the negative energies are not bounded below. We then identify the vacuum as the state in which all negative energy states are occupied. We will hopefully cover these ideas in more detail in the future. I hope this helps!

[@GeoffryGifari]

@@ProfessorMdoesScience ah yes! just as i thought, even if we try to construct a Dirac hamiltonian, because the energy levels drop indefinitely we can't just set a ground state as reference, and Dirac had to postulate all the filled states. I imagine if somehow we can add a potential to Dirac Hamiltonian that set a lower bound on energy eigenvalues, this will be a problem no more. thanks for the reply!

[@DrDeuteron]

@@GeoffryGifarinaw, Dirac needed those so called negative energy solutions. So do we.

[@GeoffryGifari]

@@DrDeuteron btw, do you also make content (maybe nuclear related)?

[@DrDeuteron]

@@GeoffryGifari ha ha, no. I used to publish papers in Phys Rev C and the like, but have moved on.

[@matiullahkhan6327]

thank you ma'am ❤

[@ProfessorMdoesScience]

Thanks for watching!

[@martinarmoa9898]

Great video!

[@ProfessorMdoesScience]

Glad you like it!

[@andrewdirr824]

Yet another brilliant video making QM fun to learn. May I ask what software and digital writing pad you guys use to write your equations on the screen? (I am looking for a setup that allows me to maintain my penmanship and that feels natural.)

[@ProfessorMdoesScience]

Glad you like the videos! We use an iPad, and the app "Explain Everything". We have not done a thorough comparison of alternative setups, but this seems to work well for us!

[@andrewdirr824]

@@ProfessorMdoesScience Thank you! (Unfortunately, I don't have an iPad, but knowing the app's name and the setup might be helpful in finding a similar alternative. Thanks again; and I look forward to more videos. I have passed around some your videos' links to my fellow quantum mechanics classmates, both undergraduate and graduate, and everyone is delighted by how concise and helpful they are.)

[@ProfessorMdoesScience]

@@andrewdirr824 Thanks for your support!

[@stranger3944]

Why does c_q =0 for q=k ? I understood that the expansion of v shouldn't diverge but why does it have to terminate at k itself? Can't it be some value other than k?

[@ProfessorMdoesScience]

Note that k can take any integer value, so the expansion will terminate at different values of k depending on the energy eigenvalue we are considering. We look at a few explicit examples here: kzbin.info/www/bejne/imibf56kmcaAmJo I hope this helps!

[@stranger3944]

@@ProfessorMdoesScience how can k take any integer value? Isn't it's value determined by the energy E_k? You mean to say the k associated with E_k and the k associated with c_k are different ?

[@ProfessorMdoesScience]

@@stranger3944 The precise value that k takes will determine the energy eigenvalue Ek, as you correctly say. What I meant with my statement was that the possible values of k are any (positive) integer, and each will have an associated energy eigenvalue. I hope this is clearer!

[@Handelsbilanzdefizit]

Interesting. Maybe the electrostatic potential is the most relevant example. But just for fun. I'm wondering, if I compress my car to a sub-atomic scale black hole, do the particles in vicinity also have discrete energylevels? Did someone made this calculations with very strong gravity fields? (catchword: "Quantumgravity") For particles that are far enough from the center, in the shell, newtonian gravity could be a good approximation.

[@ProfessorMdoesScience]

Gravity is a completely different beast to quantum mechanics and there is not yet a quantum theory of gravity. Very interesting questions at the forefront of research!

[@ahsanhayat8035]

Thanks

[@ProfessorMdoesScience]

Glad you like it!

[@kisho2679]

Which are the Legendre Polynomials for Lanthanum *element 57, La)?

[@saliabdo5853]

Thanks for every thing but , there is no videos in the description 😅

[@ProfessorMdoesScience]

Glad you like it! And I just checked and the list of videos does appear for us, some under "Background" and some under "What next?". Can you find those?

[@cocoabutter5888]

Oh my god thank you so much

[@ProfessorMdoesScience]

Hope you like it!

[@shyamaugustine7272]

Anyone know of such lectures in Electrodynamics offered by any channel?

[@ProfessorMdoesScience]

We do hope to treat more advanced topics in the future, but it would certainly be interesting to find other channels that already treat them :)

[@shyamaugustine7272]

@@ProfessorMdoesScience oh I thought you will be sticking to quantum mechanics. It would be really cool if you will be doing such videos on quantum electrodynamics and quantum field theory. I know the audience for that will be rather limited. But great work!! You are doing a lot of people a favour by giving such content for free.

[@زينالعابدينماجدمحمد]

❤

[@quantum4everyone]

I think your final answer is in error, or at least easily confused. The summation should run only up to q=k. Now, of course, the higher q’s vanish, but it is a bit misleading not to show the finite summation in the summary, especially since you worked so hard to establish that it is a finite series. Also, it is common to relate this to the Associated Laguerre polynomial, but you did not. Maybe that is coming up later?

[@ProfessorMdoesScience]

Thanks for the feedback! We could probably have added a reminder in the final slide about the fact that the series terminates at ck, but hopefully this will still be clear from the earlier derivation. We will explicitly build a few eigenfunctions in a future video where this will also become clear. As to the associated Laguerre polynomial, we had a long discussion about whether we should mention them but in the end decided against it as the video is already very long as it is. We may do a separate video on that topic as we've done in the past in similar situations, see for example the video on Hermite polynomials in the context of the quantum harmonic oscillator: kzbin.info/www/bejne/pmOVhqWLq56Emq8

[@barrybai5123]

What do you do if Professor M uploads more videos? Immediately click in and watch.

[@ProfessorMdoesScience]

:) Glad you like them!

[@mattphillips538]

Is this your actual handwriting??

[@ProfessorMdoesScience]

This is indeed the handwriting of one of the Profs. M :)

[@mattphillips538]

@@ProfessorMdoesScience A minor point but precision and repeatability in letter formation is a huge help to your students when the nomenclature gets complex; it really helps to be able to tell when someone wrote, say, Rho and not t. That said, as a math major with no training in physics I was able to follow and understand this outstanding presentation. My only discomfort is coming from the unstated assumption the solutions are never sensitive to all these approximations. I know that's how Physicists like to roll, but I'd feel better if it at least stated out in the open as a premise.

[@ProfessorMdoesScience]

@@mattphillips538 Thanks for your feedback! We do have to make choices as to how deeply to go into every detail, and indeed our target audience is a standard physics/chemistry student, not a maths student! :)

[@ahsanhayat8035]

Let me promote you on my social media accounts

[@ProfessorMdoesScience]

Thanks!

[@nikhilmalik5178]

Please get a nice mic

[@ProfessorMdoesScience]

Thanks for the suggestion!