Geometry of nonadiabatic quantum hydrodynamics

Abstract

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 $\hslash$ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the $\hslash \ne 0$ 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.

arXiv comment41 pages, no figures

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Cited by in corpus (15)

• Madelung transform and probability densities in hybrid classical-quantum dynamics
• Evolution of hybrid quantum-classical wavefunctions
• Geometry of the Madelung transform
• Hybrid quantum-classical dynamics of pure-dephasing systems
• The bohmion method in nonadiabatic quantum hydrodynamics
• Regularized Born-Oppenheimer molecular dynamics
• Stochastic Variational Formulations of Fluid Wave-Current Interaction
• Lagrangian reduction and wave mean flow interaction
• Dynamics of mixed quantum-classical spin systems
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• Koopmon trajectories in nonadiabatic quantum-classical dynamics
• Madelung transform and variational asymptotics in Born-Oppenheimer molecular dynamics
• Holonomy and vortex structures in quantum hydrodynamics