How to derive or verify Hartree product?

[gofightwin]

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.

 

[LayMuon]

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.