Non-differentiable Bohmian trajectories

Details Non-differentiable Bohmian trajectories Non-differentiable Bohmian trajectories

2010-11-12

arxiv.org/abs/1011.2852v1

Physics

Quantum Physics

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

Scientific paper

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.

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