Geometric phase of quantum wave function and singularities of Bohm dynamics in a one-dimensional system

Abstract The phase of a single valued wave function with local zeros (nodes), is globally represented by a function connecting different branches of Riemann sheets. We investigate the mathematical limitations and the loss of regularity associated to such a global representation of the quantum phase. Our study is based on a geometrical description of the dynamics of one-dimensional quantum systems. We develop a mathematical model based on the Bohm formalism of quantum mechanics where the quantum dynamics is described in terms of horizontal lifts associated to a Ehresmann connection. We express the non triviality of the bundle in terms of holonomy along loops delimited by Bohm trajectories. We assume that the nodes of the wave function form a discrete set of points in the spacetime and we show that the connection becomes singular on this set. We perform a local study of the behaviour of the Bohm trajectories around the singularities and we provide a control on the Bohm potential by deriving semi-analytical expressions for the particle trajectories. We express the total phase rotation along a loop in terms of the derivative of the quantum current evaluated on the set of nodes inside the loop.