Time-independent, paraxial and time-dependent Madelung trajectories near zeros

Abstract

The Madelung trajectories associated with a wavefunction are the integral curves (streamlines) of its phase gradient, interpretable in terms of the local velocity (momentum) vector field. The pattern of trajectories provides an immediately visualisable representation of the wave. The patterns can be completely different when the same wave equation describes different physical contexts. For the time-independent Schrödinger or Helmholtz equation, trajectories circulate around the phase singularities (zeros) of the wavefunction; and in the paraxially approximate wave, streamlines spiral slowly in or out of the zeros as well as circulating. But if the paraxial wave equation is reinterpreted as the time-dependent Schrödinger equation, its Madelung trajectories do not circulate around the zeros in spacetime: they undulate while avoiding them, except for isolated trajectories that encounter each zero in a cusp singularity. The different local trajectory geometries are illustrated with two examples; a local model explains the spacetime cusps.