Uploaded August 2019 | Updated September 2026, 12 hours ago
Attempting To Break Down The Quantum Many Body Problem | Schrodinger Equation | Dirac Equation
In this video, I try to break apart the multi-electron atom many body problem in quantum mechanics. I attempt to do this using kind of a weird ansatz that I constructed specifically for this application. Let me know what you think of my attempt in the comments!
There is a typo in the spherical harmonics normalization constant. In the second fraction under the square root, the 1's are supposed to be l's.
Also, it's worth noting that if the general solution is right, you could generalize to an arbitrary quantum Coulombic many-body problem, by just including coulomb potential terms with the appropriate sign and charge magnitude, and then propagating that accordingly throughout the solution. One could also include no stationary charges. This is just a change of parameterization. Pairings involving differing masses requires an obvious adjustment of the coordinates to resolve the kinetic terms in the same way.
3/19/2023: I realized recently that I know longer remembered how I justified the formulas I gave in this video for the hatted quantum numbers. In the video, I gave no detailed explanation. I simply said something about normalizability. I therefore decided to sit down and work it out again. I found that the formulas I gave in the video are wrong, and the fully detailed justification (which I meticulously recorded this time) for the justification for replacement formulas leverages much more than normalizability. The following link takes you to a page where I have a new educated guess for the correct formulas, and their justification:
twitter.com/DietterichLabs/status/1638026174969389056
3/20/2023: It is worth noting that I only chose the hydrogen-like solutions for phi as a starting point. I figured that I might find solutions with the right properties within (a generalized set) of them. Had this not worked, I would have looked at other solutions. I did not assume that a solution form from a bound state problem would necessarily be the correct idea for a repulsive potential.
8/7/2023: In the general multi-electron atom solutions that I gave to the Schrodinger and Dirac equations, I included a number of summation and product symbols, which are supposed to generate all of the correct factors and terms. I'm not entirely sure that they do so perfectly, but the intended design should be clear. In the Schrodinger case, there are the usual H-atom factors for each nucleus-electron interaction, as well as a difference-coordinate H-atom factor and a corresponding greens function term for each electron-electron interaction. In the Dirac equation, the greens function terms are unnecessary.
1/26/2024: It turns out that these new eigenvalues are also wrong. Here is a link to an updated video: youtu.be/-xLhrYA6TNU
Here is the link to my video on solving the Dirac equation for the hydrogen atom:
youtube.com/watch?v=jWYtP-rAoYA
Here is my video on solving the Schrodinger equation for the hydrogen atom:
youtube.com/watch?v=MaXFT-c8u1E
In both of these videos, I actually address the general one-electron atom. I don't bother setting Z=1. This, of course, includes the hydrogen atom.
For those who are curious, I also have a video on how to solve the Klein-Gordon equation for the one-electron atom:
youtube.com/watch?v=9_3uQdF-tEs
Superfluid Helium Resonance Experiment video:
youtu.be/unUNQNmuvUQ
Quantum Field Theory Lecture Series:
youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou
Attempting To Break Down The Quantum Many Body Problem | Schrodinger Equation | Dirac Equation
In this video, I try to break apart the multi-electron atom many body problem in quantum mechanics. I attempt to do this using kind of a weird ansatz that I constructed specifically for this application. Let me know what you think of my attempt in the comments!
There is a typo in the spherical harmonics normalization constant. In the second fraction under the square root, the 1's are supposed to be l's.
Also, it's worth noting that if the general solution is right, you could generalize to an arbitrary quantum Coulombic many-body problem, by just including coulomb potential terms with the appropriate sign and charge magnitude, and then propagating that accordingly throughout the solution. One could also include no stationary charges. This is just a change of parameterization. Pairings involving differing masses requires an obvious adjustment of the coordinates to resolve the kinetic terms in the same way.
3/19/2023: I realized recently that I know longer remembered how I justified the formulas I gave in this video for the hatted quantum numbers. In the video, I gave no detailed explanation. I simply said something about normalizability. I therefore decided to sit down and work it out again. I found that the formulas I gave in the video are wrong, and the fully detailed justification (which I meticulously recorded this time) for the justification for replacement formulas leverages much more than normalizability. The following link takes you to a page where I have a new educated guess for the correct formulas, and their justification:
twitter.com/DietterichLabs/status/1638026174969389056
3/20/2023: It is worth noting that I only chose the hydrogen-like solutions for phi as a starting point. I figured that I might find solutions with the right properties within (a generalized set) of them. Had this not worked, I would have looked at other solutions. I did not assume that a solution form from a bound state problem would necessarily be the correct idea for a repulsive potential.
8/7/2023: In the general multi-electron atom solutions that I gave to the Schrodinger and Dirac equations, I included a number of summation and product symbols, which are supposed to generate all of the correct factors and terms. I'm not entirely sure that they do so perfectly, but the intended design should be clear. In the Schrodinger case, there are the usual H-atom factors for each nucleus-electron interaction, as well as a difference-coordinate H-atom factor and a corresponding greens function term for each electron-electron interaction. In the Dirac equation, the greens function terms are unnecessary.
1/26/2024: It turns out that these new eigenvalues are also wrong. Here is a link to an updated video: youtu.be/-xLhrYA6TNU
Here is the link to my video on solving the Dirac equation for the hydrogen atom:
youtube.com/watch?v=jWYtP-rAoYA
Here is my video on solving the Schrodinger equation for the hydrogen atom:
youtube.com/watch?v=MaXFT-c8u1E
In both of these videos, I actually address the general one-electron atom. I don't bother setting Z=1. This, of course, includes the hydrogen atom.
For those who are curious, I also have a video on how to solve the Klein-Gordon equation for the one-electron atom:
youtube.com/watch?v=9_3uQdF-tEs
Superfluid Helium Resonance Experiment video:
youtu.be/unUNQNmuvUQ
Quantum Field Theory Lecture Series:
youtube.com/playlist?list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou
