Variational method, is the wavefunction the best for all

[cheong]

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.

 

[DrClaude]

I imagine that you could find pathological cases where the energy of the trial wave function is low while its projection on the rea ground state wave function is small, but that would mean that your trial wave function is badly chosen (or isn't "flexible" enough with respect to some parameters defining the function). For instance, if you take the harmonic oscillator and impose that the trial wave function is anti-symmetric, you will of course get something that is not at all like the actual ground state.

If the trial wave function is a reasonable guess, then minimizing the energy and maximizing the projection on the ground state go hand in hand. And if you somehow know that the energy you get is close to the energy of the actual ground state, then the wave function itself has also to be close.