arXiv subjects

Michael S. Foskett

Publications and source records attributed to Michael S. Foskett.

3 recordsLinked to original sources

arXiv2023

Holonomy and vortex structures in quantum hydrodynamics

We consider a new geometric approach to Madelung's quantum hydrodynamics (QHD) based on the theory of gauge connections. In particular, our treatment comprises a constant curvature thereby endowing QHD with intrinsic non-zero holonomy. In the hydrodynamic context, this leads to a fluid velocity which no longer is constrained to be irrotational and allows instead for vortex filaments solutions. After exploiting the Rasetti-Regge method to couple the Schrödinger equation to vortex filament dynamics, the latter is then considered as a source of geometric phase in the context of Born-Oppenheimer molecular dynamics. Similarly, we consider the Pauli equation for the motion of spin particles in electromagnetic fields and we exploit its underlying hydrodynamic picture to include vortex dynamics.

math-ph↗

arXiv2020

Geometry of quantum hydrodynamics in theoretical chemistry

This thesis investigates geometric approaches to quantum hydrodynamics (QHD) in order to develop applications in theoretical quantum chemistry. Based upon the momentum map geometric structure of QHD and the associated Lie-Poisson and Euler-Poincaré equations, alternative geometric approaches to the classical limit in QHD are presented. These include a new regularised Lagrangian which allows for singular solutions called 'Bohmions' as well as a 'cold fluid' classical closure quantum mixed states. The momentum map approach to QHD is then applied to the nuclear dynamics in a chemistry model known as exact factorization. The geometric treatment extends existing approaches to include unitary electronic evolution in the frame of the nuclear flow, with the resulting dynamics carrying both Euler-Poincaré and Lie-Poisson structures. A new mixed quantum-classical model is then derived by considering a generalised factorisation ansatz at the level of the molecular density matrix. A new alternative geometric formulation of QHD is then constructed. Introducing a $\mathfrak{u}(1)$ connection as the new fundamental variable provides a new method for incorporating holonomy in QHD, which follows from its constant non-zero curvature. The fluid flow is no longer irrotational and carries a non-trivial circulation theorem, allowing for vortex filament solutions. Finally, non-Abelian connections are then considered in quantum mechanics. The dynamics of the spin vector in the Pauli equation allows for the introduction of an $\mathfrak{so}(3)$ connection whilst a more general $\mathfrak{u}(\mathscr{H})$ connection is introduced from the unitary evolution of a quantum system. This is used to provide a new geometric picture for the Berry connection and quantum geometric tensor, whilst relevant applications to quantum chemistry are then considered.

math-ph↗

arXiv2019

Geometry of nonadiabatic quantum hydrodynamics

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.

math-ph↗