The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws. We develop this approach to show that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law. All of these metric spaces have an "onion-shell" geometry consisting of concentric spheres, with functions conserved to the same value lying on the same sphere. We apply this approach to generate metrics for wavefunctions, particle densities, paramagnetic current densities, and external scalar potentials. In addition, we demonstrate the extensions to our approach that ensure that the metrics for wavefunctions and paramagnetic current densities are gauge invariant.
We use our metric space approach to explore the unique relationship between ground-state wavefunctions, particle densities and paramagnetic current densities in Current Density Functional Theory (CDFT). We study how this relationship is affected by variations in the external scalar potential, pairwise electronic interaction strength, and magnetic field strength. We find that all of the metric spaces exhibit a "band structure", consisting of "bands" of points characterised by the value of the angular momentum quantum number, m. These "bands" were found to either be separated by "gaps" of forbidden distances, or be "overlapping". We also extend this analysis beyond CDFT to explore excited states.
We apply our metrics in order to gain new insight into the Hohenberg-Kohn theorem and the Kohn-Sham scheme of Density Functional Theory. For the Hohenberg-Kohn theorem, we find that the relationship between potential and wavefunction metrics, and between potential and density metrics, is monotonic and includes a linear region. Comparing Kohn-Sham quantities to many-body quantities, we find that the distance between them increases as the electron interaction dominates over the external potential.
