Or quantum mechanics allows to get a simpler exact solution?
Suppose there are 3 particles with uniform potential field. Can their wave functions be found exactly?
The quantum $n$-body problem is considerably simpler than the classical $n$-body problem, since the problems with collisions disappear. Rigorous existence results are much stronger. See, e.g., the book ''Scattering theory of classical and quantum N-particle systems'' by Derezinski and Gérard.
But explicit solutions are about as rare in the quantum case as in the classical case.