Quantum states, their properties and dynamics are central to the understanding of the quantum world. They have received even more attention due to the raise of quantum technologies, which exploit the ‘quirks’ of quantum mechanics. Quantum states form Hilbert spaces, which combine 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 conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an “onion-shell” geometry. We will discuss these induced ‘natural’ metrics for some of the quantities fundamental to the understanding of quantum systems, as well as some applications of the metric space approach to quantum mechanics, including applications to further the understanding of the core concepts of Density Functional Theory.
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