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Metric space approach to quantum mechanics

  • Date: Jun 23, 2015
  • Time: 10:00
  • Speaker: Prof. Irene D'Amico
  • York University
  • Location: Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale)
  • Room: Seminarraum A.2.20

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. We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities[1]. All such metric spaces have an "onion-shell" geometry. The related 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 the systems analysed, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities[2]. Similarly, by considering metric spaces associated to many-body systems immersed in a magnetic field, we consider the mapping between wave-function and (current and particle) densities at the core of Current Density functional theory. We find, in the related metric spaces, regions of allowed and forbidden distances, a "band structure", directly arising from the conservation of the z component of the angular momentum[1]. Finally, recent results include appropriate metrics for the external potential which allow us to directly explore the 'third leg' of the Hohenberg-Kohn theorem[3].

[1] P. M. Sharp and I. D'Amico, Phys. Rev. B 89, 115137 (2014)
[2] I. D'Amico, J. P. Coe, V. V. Franca and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)
[3] P. M. Sharp and I. D'Amico, preprint (2015).

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