A density functional perspective for one-particle systems

Density functional theory is discussed in the context of one-particle systems. We show that the ground state density $\rho_0(x)$ and energy $E_0$ are simply related to a family of external potential energy functions with ground state wave functions $\psi_n(x) \propto \rho_0(x)^n$ and energies $E_n=2nE_0$ for certain integer values of $n$.