I have only recently started to learn quantum mechanics and I have only gone so far as to solve the hydrogen atom using Schrödinger's equation and the corresponding Hamiltonian, but all this has me wondering if it is possible to find a wave function for a planet-sun or moon-planet system. I know that this would be amazingly(and needlessly) more complicated than the classical solution. So is this possible, have people done it, and can we learn more from this approach?
Once we get more than one particle in a potential (possibly of the other) quantum mechanics becomes complicated. For macroscopic objects, like crystals, or even superconductors, quantum mechanical models work, assuming collective potentials and using methods of approximations.
The general way to approach quantum mechanical solutions of many body problems is by the use of the density matrix.
A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.
It has the overlap and phases of the individual wave functions in the rows and collumns, which carry the quantum mechanical information.
When "several" becomes larger than 1060 (for the earth), even though in theory one quantum mechanical description should arise from all those individual wavefunctions, the off diagonal elements in the density matrix which show the quantum mechanical correlations between individual atoms are to all effects zero, and the system becomes a classical system.
Even for something as simple as a cup, the same is true. It is the reason why quantum mechanical effects, as tunneling, have zero probability in the macroscopic dimensions.
So even though a quantum mechanical solution might be set up theoretically, to all intents and purposes it is a classical one.
No quantum mechanical effects can be measured between earth and moon as one wave function.
You must at first check if any QM effect can occur in large scale universe. One of the necessary criteria is that the fundamental scales under consideration is comparable with De-Broglie wavelength. For an electron in Hydrogen atom this is the case while for earth orbiting sun it's not so.