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I know that it is impossible for real electrons and nuclei. In the Kohn and Sham approach, a system of interacting electrons is approximated by a system of non-interacting Kohn-Sham particles in an effective potential.

My question refers not to a real chemical system. Rather, I am asking about a system of toy particles that don't necessarily correspond to reality.

Is there such a system of multiple interacting quantum particles for which density can be obtained analytically for arbitrary number of particles, at least in terms of some special functions, say, hypergeometric functions?

The particles may be point particles, they may interact not necessarily at contact, may be only pairwise interactions, maybe in 1D, etc.

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  • $\begingroup$ You mean like "2 interacting photons", rather than the particles you mentioned so far (electrons and protons)? $\endgroup$
    – Nike Dattani
    May 20, 2022 at 15:39
  • $\begingroup$ How about the Hubbard chain? $\endgroup$
    – Anyon
    May 20, 2022 at 19:02
  • $\begingroup$ @NikeDattani Is it this one ? I just googled "2 interacting photons" and reading. I didn't want to specify them to be necessarily photons or particles of matter. Just artificial point particles moving in space, like in toy problems in quantum mechanics courses. I am curious if anyone found a system that by chance happens to be analytically solvable for any n interacting particles? $\endgroup$
    – Vladislav Gladkikh
    May 21, 2022 at 3:03
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    $\begingroup$ What about Hooke's atom? It's a 3D system of 2 fully-interacting electrons, which is exactly solvable. $\endgroup$
    – Phil Hasnip
    Aug 5, 2022 at 22:45
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    $\begingroup$ I don't think so, and it's only soluble analytically for certain "magic" values of the interaction strength, but it's still a nice playground for multi-electron physics. $\endgroup$
    – Phil Hasnip
    Aug 6, 2022 at 1:24

1 Answer 1

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Anyon suggested the Hubbard chain:

"How about the Hubbard chain?"

and Phil Hasnip suggested Hooke's atom:

"What about Hooke's atom? It's a 3D system of 2 fully-interacting electrons, which is exactly solvable."

for certain "magic" values of the interaction strength:

"I don't think so, and it's only soluble analytically for certain "magic" values of the interaction strength, but it's still a nice playground for multi-electron physics"

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