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Geometry of nonadiabatic quantum hydrodynamics

Foskett, Michael S., Holm, Darryl D. and Tronci, Cesare (2019) Geometry of nonadiabatic quantum hydrodynamics Acta Applicandae Mathematicae, 162 (1). pp. 63-103.

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The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether's conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called `collective'. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ℏ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ℏ≠0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called 'Bohmions', which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Foskett, Michael
Holm, Darryl D.
Date : August 2019
Funders : Engineering and Physical Sciences Research Council (EPSRC), University of Surrey, Leverhulme Trust, National Science Foundation, Mathematical Society
DOI : 10.1007/s10440-019-00257-1
Copyright Disclaimer : © The Author(s) 2019. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. This is a post-peer-review, pre-copyedited version of an article published in Acta Applicandae Mathematicae. The final authenticated version is available online at:
Related URLs :
Depositing User : Clive Harris
Date Deposited : 09 May 2019 10:15
Last Modified : 19 Dec 2019 12:02

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