I like the way the 1st Hohenberg-Kohn theorem was explained in this video. For me it is more appealing to think about mapping, than to say that “the ground state density uniquely determines the potential”. Because for me the latter phrase has a “flavor” that external potential is caused by electronic density.
But I have a question. Somewhere I saw an counter example about wave function – density mapping. For example, for free electron in a box with periodic boundary conditions the wave function is a plane wave psi=1/sqrt(V)*exp(k.r). So for any wave vector k the density will always be the same rho=1/V.
So, am I right, that the correction is in that it should be ground state. Then we can take only wave function for lowest energy E=0, k=0. And for this case only one wave function corresponds to rho=1/V – psi=1/sqrt(V)?