I understand that the variational method can give me the "best" approximate ground state wavefunction among the class of the function belongs to. It is the "best" wavefunction in a sense that its energy level is closest to the ground state among its own class. Question: Is it also true that this "best" wavefunction describes the particle density the best also? If not, what the other technique to find one is there? I personally do not think that is true in general. Does anyone happens to know the answer? Thank you very much.