Quantum mechanics in metric space: wave functions and their densities

TL;DR

This paper explores the metric space structure of quantum wave functions and densities, revealing simplified and nearly linear relationships in metric space that contrast with traditional Hilbert space analysis.

Contribution

It introduces a metric-based analysis of Fock space and ground-state densities, providing new geometric insights and simplifying the Hohenberg-Kohn mapping in quantum mechanics.

Findings

Metric stratifies Fock space into concentric spheres.

Hohenberg-Kohn mapping is simple and nearly linear in metric space.

Maximum and minimum distances between wave functions are derived and interpreted geometrically.

Abstract

Hilbert space combines the properties of two fundamentally different types of mathematical spaces: vector space and metric space. While the vector-space aspects of Hilbert space, such as formation of linear combinations of state vectors, are routinely used in quantum mechanics, the metric-space aspects of Hilbert space are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres. Maximum and minimum distances between wave functions are derived and geometrically interpreted in terms of this metric. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground-state wave functions and to that of particle densities, each of which forms a metric space but not a Hilbert space. The Hohenberg-Kohn mapping between densities and ground-state wave functions, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be very simple in metric space, where it is represented by a monotonic mapping of vicinities onto vicinities. Surprisingly, it is also found to be nearly linear over a wide range of parameters.