A quick look at PostgreSQL's compliance notes: www.postgresql.org/docs/13/features.html shows the complete utter mess that this standard is. Multiple compliance levels that no one fully implements and optional features everywhere.
Example: nodejs/sequelize/raw/tree.js
- Implementation agnostic
- Postgres
- stackoverflow.com/questions/67848017/simple-recursive-sql-query
- stackoverflow.com/questions/28688264/how-to-traverse-a-hierarchical-tree-structure-structure-backwards-using-recursiv
- stackoverflow.com/questions/51822070/how-can-postgres-represent-a-tree-of-row-ids
- depth first
- uspecified depth first variant
- preorder DFS
- breadth-first stackoverflow.com/questions/3709292/select-rows-from-table-using-tree-order
- MySQL
- Microsoft SQL Server
TODO understand better, mentioned e.g. at Subtle is the Lord by Abraham Pais (1982) page 20, and is something that Einstein worked on.
Leads to the Dirac equation.
Leads to the Proca equation.
Theorized for the graviton.
More interestingly, how is that implied by the Stern-Gerlach experiment?
physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:
- at limit of large
lfor the Schrödinger equation solution for the hydrogen atom the difference between each angular momentum is twice that of the eletron's spin. Not very satisfactory. - it comes directly out of the Dirac equation. This is satisfactory. :-)
Originally done with (neutral) silver atoms in 1921, but even clearer theoretically was the hydrogen reproduction in 1927 by T. E. Phipps and J. B. Taylor.
The hydrogen experiment was apparently harder to do and the result is less visible, TODO why: physics.stackexchange.com/questions/33021/why-silver-atoms-were-used-in-stern-gerlach-experiment
The Stern-Gerlach Experiment by Educational Services, Inc (1967)
Source. Featuring MIT Professor Jerrold R. Zacharias. Amazing experimental setup demonstration, he takes apart much of the experiment to show what's going on.Introduction to Spintronics by Aurélien Manchon (2020)
Source. The Spin on Electronics by Stuart Parkin
. Source. 2013.This is closely linked to the Pauli exclusion principle.
What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.
Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.
As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.
Bibliography:
- www.youtube.com/watch?v=Og13-bSF9kA&list=PLDfPUNusx1Eo60qx3Od2KLUL4b7VDPo9F "Advanced quantum theory" by Tobias J. Osborne says that the course will essentially cover multi-particle quantum mechanics!
- physics.stackexchange.com/questions/54854/equivalence-between-qft-and-many-particle-qm "Equivalence between QFT and many-particle QM"
- Course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) from course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) give a good introduction to non-interacting particles
No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.
Bibliography:
There are unlisted articles, also show them or only show them.