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 by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the = 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.

Regularized Born-Oppenheimer molecular dynamics

“…This step is motivated by the fact that, if the gradient was regular, then one could invoke a sharply peaked nuclear density so that ∇ρ φ 2 ∇ρ φ 2 = ∇ρ φ 2 . A similar regularization of the electronic density matrix was recently proposed also within a nonadiabatic context [15]. In the present case, the nuclear trajectory equation reads…”

Momentum maps for mixed states in quantum and classical mechanics

“…It is expected that the momentum maps presented here will open the way to the development of geometric tools for new models in quantum physics and chemistry. An example is provided by the recent work [21] on exact factorization models.…”

“…For example, this type of expression emerges in dynamical models for nonadiabatic molecular dynamics [21,35], where it determines the density operator for the electronic dynamics. As it is shown below, this expression determines the left leg of a dual pair of momentum maps underlying quantum dynamics.…”

Stochastic Variational Formulations of Fluid Wave–Current Interaction

“…This type of coupling is still invoked universally in quantum problems today. For example, one may see J • A used for coupling classical nuclei to quantum electrons in the quantum hydrodynamic theory of molecular chemistry, Foskett et al (2019). Thus, it may be no surprise that minimal coupling might arise here again, as a natural approach for coupling the Lagrangian mean flow of fluid trajectories to the essentially Eulerian field properties of wave propagation.…”

“…microwaves) on the slow dynamics of a fluid plasma (Dewar 1970(Dewar , 1973Littlejohn 1981;Similon et al 1986;Kaufman and Holm 1984). For a modern application of the Frenkel-Dirac phase-space Lagrangian in the classical-quantum interaction for non-adiabatic electron dynamics in molecular chemistry, see Foskett et al (2019). For a recent treatment of phase-space Lagrangians for fast-slow WKB dynamics of high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow, see Burby and Ruiz (2019).…”