Let us assume I wish to simulate a Helium atom, since there does not exist a closed-form solution.
However, I presume I would need to simulate the time-dependent Schrodinger wave equation. I would like to know what the time-dependent hamiltonian looks like for this.
I think I would need the terms for electromagnetic, strong, and weak nuclear forces --- I do not think I need relativistic corrections, since I'm only interested in local information.
I wish to simulate an equation on simulation gives me the evolving probability distributions of the electrons.
I understand that this might be computationally infeasible, but I still wish to know what the exact PDE is that I need to solve --- I'm not looking for approximations!