Geometry of nonadiabatic quantum hydrodynamics

TL;DR

This paper develops a geometric framework for nonadiabatic quantum hydrodynamics, deriving collective Hamiltonians and fluid models that describe quantum molecular dynamics, including regularized finite-dimensional systems with Bohmian trajectories.

Contribution

It introduces a geometric approach to derive quantum fluid models and regularizes their Hamiltonians for finite 6, leading to finite-dimensional systems involving Bohmian trajectories and electronic states.

Findings

Derivation of collective Hamiltonians for quantum molecular models.

Introduction of regularization via spatial smoothing for finite 6.

Finite-dimensional Hamiltonian systems for Bohmions and electronic states.

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 $\hbar$ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the $\hbar\ne0$ 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.