Non-differentiable Bohmian trajectories

Abstract

A solution $\psi$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hslash \Im \left(▽\psi /m\psi \right).$ In the case of one specific non-differentiable weak solution $\Psi$ we show how Bohmian trajectories can be obtained for $\Psi$ from the trajectories of a sequence ${\Psi }_n\to \Psi .$ (For any real $t$ the sequence ${\Psi }_n\left(t,\cdot \right)$ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $\Psi$ in the Hilbert space on which the Schrödinger evolution acts.

12 pages, 4 figures; published in the monography 'Quantum Trajectories', Editor P K Chattaraj , Taylor & Francis, Boca Raton, November 1st, 2010