# How was the quantum three-body problem solved?

What method was used to solve the quantum three-body problem? I heard it was solved and was not sure about the status of the problem.

It never was solved, analytically (which means an exact mathematical solution)

It was solved approximately, using perturbation theories. In the case of the helium atom, an estimate of the energy electron-electron repulsion is added to the calculation. The appropriateness of the perturbation added is based on the comparison of the result to experimentally determined values.

Three-body special cases can have closed solutions. For $\ce{H2+}$, the heavy nuclei are assumed to be stationary while the electron travels. Confocal ellipsoidal coordinates then describe the system to first order, followed by perturbation treatments because the nuclei do move. As stated, there are no general analytical forms for the three-body problem overall.