Non-differentiable Bohmian trajectories

TL;DR

This paper demonstrates how Bohmian trajectories can be constructed for non-differentiable solutions of Schrödinger's equation by approximating with differentiable solutions, potentially extending Bohmian mechanics to all Hilbert space vectors.

Contribution

It introduces a method to define Bohmian trajectories for non-differentiable wave functions via limits of trajectories from smooth approximations.

Findings

Bohmian trajectories can be obtained from non-differentiable solutions as limits of smooth solutions.

The approach extends Bohmian mechanics to arbitrary initial states in the Hilbert space.

Trajectories for non-differentiable solutions need not be differentiable themselves.

Abstract

A solution $\psi $ to Schr\"odinger'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 $\hbar \Im \left( \bigtriangledown \psi /m\psi \right) .$ 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 $\Psi_{n}\rightarrow \Psi.$ (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 Schr\"{o}dinger evolution $% \Psi \mapsto e^{-iht}\Psi $ acts.