Similarities and differencies

One of the similarities between the explanation in the video and in wikipedia is they state that the external potential is unique to the same ground-state density (1st HK theorem)

On the other hand, the approach for the explanations is different. In the video it is explained from the analysis of surjective and injective functions of: the external potential, the wave function and the density. It also elaborates the fact that we can relate the external potential with the density, therefore making the approach to quantum mechanics able without using the complex wave function, just the density.

In wikipedia it further elaborates the implications of the theorem, stating what the HK functional is and explaining that the spectrum of the Hamiltonian is also a unique functional of the ground state charge density