The hydrogen atom is a canonical model in quantum mechanics, yet standard theories have almost exclusively addressed states of the positive kinetic energy, leaving the negative-kinetic-energy (NKE) regime largely unexplored. Using the recently developed NKE-supersymmetric quantum mechanics, the physical properties of hydrogenic atoms with negative kinetic energy are investigated, and the corresponding quantum description is refined. First, the notion of NKE is disentangled from the familiar concept of negative binding energy. Next, the NKE requirement is embedded within a supersymmetric framework, and the Schr?dinger equation governing the NKE hydrogen atom is derived. The equation is solved analytically, and the resulting spectrum and eigenfunctions are contrasted with those of the conventional positive-kinetic-energy case.A closed-form Schr?dinger equation for the NKE hydrogen atom is obtained, and its complete energy spectrum and normalized eigenfunctions are determined. The results confirm that hydrogen atoms with negative kinetic energy are physically admissible within an extended quantum theory, refine the theoretical portrait of hydrogenic systems, and furnish a template for studying atomic models with exotic energy.