Evolution of hybrid quantum-classical wavefunctions

Abstract

A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

Joint quantum–classical Hamilton variational principle in the phase space*

“…Thus, the Skodje gauge freedom embedded into the joint Hamilton variational principle is an unexplored resource for improving modern semiclassical approximations to wave packets dynamics, such as developed in [36], and advancing numerical methods used in quantum chemistry [33,[37][38][39] and many-body physics [40]. It would be also interesting to consider the joint Hamilton variational principle as a starting point for developing quantum-classical hybrid models alike recently proposed in references [41,42]. Last but not least, we believe that our results are of a pedagogical interest to remove the conceptual barriers between the classical and quantum worlds.…”