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Incompleteness of trajectory-based interpretations of quantum mechanics

Michael J. W. Hall
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Abstract

Trajectory-based approaches to quantum mechanics include the de Broglie-Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schroedinger equation into a continuity equation and a modified Hamilton-Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction initially uniform over the interior of the box. For this example there is no corresponding continuity or modified Hamilton-Jacobi equation, and the spacetime dependence of the wavefunction has a known fractal structure. Examples with finite average energies are also constructed.

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Michael J. W. Hall. 2004-08-29. Incompleteness of trajectory-based interpretations of quantum mechanics. https://doi.org/10.1088/0305-4470/37/40/015

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