  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.
