# Atomic Hamiltonian, non analytic solution [duplicate]

How can be proved that terms

$$\frac{1}{|r_{i}-r_{j}|}$$

are the ones that avoid the existence of an analytical solution for the many electron atom problem

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## marked as duplicate by Brandon Enright, Waffle's Crazy Peanut, Qmechanic♦Jun 6 '13 at 19:13

Possible duplicates: physics.stackexchange.com/q/252/2451 and physics.stackexchange.com/q/37938/2451 Related classical problem: physics.stackexchange.com/q/1235/2451 –  Qmechanic Jun 6 '13 at 15:15

If you mean "is there some complete set of commuting operators?" then this boils down to the integrability of the many-body problem. This has been a subject of interest for at least 200 years. This leads to a lot of interactions between the KAM theorem, quantum mechanics from a Lie algebraic perspective, regular versus chaotic dynamics, and a lot of really fascinating stuff. But the answer here might be "No, there is not a complete set of operators that have simultaneous eigenvectors with the Hamiltonian", or rather that there is no set $\{\mathcal{J}_i \}_N$ for $2N$ degrees of freedom such that
$[\mathcal{J}_i, \mathcal{H}] = 0 ~ \forall ~ i$