Hydrogen atom: power series solution – YouTube

Hydrogen atom: power series solution



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  • Video Views: 1497
  • Published On: 2022-03-17 17:30:07
  • Video Published/Author: Professor M does Science
  • Video Duration: 00:46:14
  • Source: Watch on YouTube


The mathematical solution of the eigenvalue equation of the Hamiltonian of the hydrogen atom.

📚 The hydrogen atom can be described using a Hamiltonian of a central potential. In this video, we go over the mathematical solution of the eigenvalue equation of the Hamiltonian. This requires the solution of a complex differential equation, and involves interesting concepts such as limiting behaviors and power series expansions.

0:00 Intro
1:00 Hydrogen as a central potential
7:27 Radial equation
9:46 Bound vs unbound states
12:52 Simplifying notation
16:38 Radial equation solution
33:07 Quantized energy eigenvalues
39:56 Energy eigenfunctions
45:47 Wrap-up

🐦 Follow me on Twitter: https://twitter.com/ProfMScience

⏮️ BACKGROUND
Hydrogen atom basics: https://youtu.be/lndTguV0u1g
3D quantum harmonic oscillator: https://youtu.be/jOQThICjLlw
Central potentials: https://youtu.be/Y73ctxnP9gQ
Radial equation of central potentials: https://youtu.be/MsZP7yxpeFg
Orbital angular momentum: https://youtu.be/EyGJ3JE9CgE
Spherical harmonics: https://youtu.be/5PMqf3Hj-Aw
Bound vs unbound states: [COMING SOON]

⏭️ WHAT NEXT?
Hydrogen atom | Eigenfunctions: [COMING SOON] Hydrogen atom | Eigenvalues: [COMING SOON]

~
Director and writer: BM
Producer and designer: MC

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11 comments
  1. 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.)

  2. 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?

  3. 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.

  4. 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?

  5. 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.

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