Quantum Mechanics in Metric Space: Wave Functions and Their Densities

Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.

Figure 1

Sketch of the metric spaces for particle densities (left) and wave functions (right). The onion-shell-like geometry is explicitly displayed for the particle densities. One set of particle densities and one of wave functions are presented: their distance from the reference state—at the north pole—increases with their labeling index, up to maximum distance.

Figure 2

Density distance vs wave function distance. (a) Helium-like atoms. The reference systems have $Z=3$ (circles, decreasing $Z$) and $Z=2$ (triangles for increasing $Z$ and squares for decreasing $Z$). (b) Hooke’s atom. The reference system has $\omega =0.5$ (triangles for increasing $\omega$ and circles for decreasing $\omega$). (c) Hubbard model. The reference system has $\omega =4$, and is compared with smaller $\omega$ systems with 8 sites and different particle numbers (circles for $N=2$, triangles for $N=4$ and squares for $N=8$) and interactions (filled symbols for $U=2$ and unfilled for $U=6$).