How to derive or verify Hartree product?

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SUMMARY

The Hartree product is derived from the time-independent Schrödinger equation, particularly when considering non-interacting particles. The key to understanding this derivation lies in recognizing that the Schrödinger equation decouples for each particle's variable, allowing the use of an ansatz involving function products. This approach leads to the conclusion that the Hartree product serves as an eigenfunction of the equation. The derivation is straightforward once the separation of variables is applied correctly.

PREREQUISITES
  • Understanding of the time-independent Schrödinger equation
  • Familiarity with quantum mechanics concepts, particularly eigenfunctions
  • Knowledge of the Hartree product and its implications in quantum systems
  • Basic grasp of separation of variables in differential equations
NEXT STEPS
  • Study the derivation of the Hartree product in quantum mechanics
  • Learn about eigenfunctions and eigenvalues in the context of the Schrödinger equation
  • Explore the implications of non-interacting particles in quantum systems
  • Review separation of variables techniques in solving differential equations
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Students and professionals in quantum mechanics, physicists studying multi-particle systems, and anyone interested in the mathematical foundations of the Hartree product.

gofightwin
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Happy summer...ish!

I am reading the lecture/presentation at:

http://fias.uni-frankfurt.de/~brat/LecturesWS1011/Lecture5.pdf

In slide two, they outline the Hartree product, but can anyone give a hint about how it's derived, or how to verify it?

Put another way, can someone give me a a hint about how to show that that product is an eigenfunction of the time independent Schrödinger equation? I have read that it is 'easy', but I am unsure where to start.
 
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Because the particles do not interact the Scroedinger equation decouples for every single particle's variable ##\vec{r}## and because the vairables separate the ansatz of function products is the right solution.
 

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