TY - JOUR

T1 - Quantum Mechanics in Metric Space

T2 - Wave Functions and Their Densities

AU - D’amico, I.

AU - Coe, J. P.

AU - França, V. V.

AU - Capelle, K.

PY - 2011/2/4

Y1 - 2011/2/4

N2 - 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.

AB - 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.

U2 - 10.1103/PhysRevLett.106.050401

DO - 10.1103/PhysRevLett.106.050401

M3 - Article

VL - 106

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 5

M1 - 050401

ER -