  In this paper we compute quantum trajectories arising from Bohm's causal
description of quantum mechanics. Our computational methodology is based upon a
finite-element moving least-squares method (MWLS) presented recently by Wyatt
and co-workers (Lopreore and Wyatt, Phys. Rev. Lett. {\bf 82}, 5190 (1999)).
This method treats the "particles" in the quantum Hamilton-Jacobi equation as
Lagrangian fluid elements which carry the phase, $S$, and density,
  $\rho$, required to reconstruct the quantum wavefunctions. Here, we compare
results obtained via the MWLS procedure to exact results obtained either
analytically or by numerical solution of the time dependent Schr\"odinger
equation. Two systems are considered: firstly, dynamics in a harmonic well and
secondly tunneling dynamics in a double well potential. In the case of
tunneling in the double well potential, the quantum potential acts to lower the
barrier separating the right and left hand sides of the well permitting
trajectories to pass from one side to another.
  However, as probability density passes from one side to the other, the
effective barrier begins to rise and eventually will segregate trajectories in
one side from the other. We note that the MWLS trajectories exhibited long time
stability in the purely harmonic cases. However, this stability was not evident
in the barrier crossing dynamics. Comparisons to exact trajectories obtained
via wave packet calculations indicate that the MWLS trajectories tend to
underestimate the effects of constructive and destructive interference effects.
