  We present a new semiclassical method that yields an approximation to the
quantum mechanical wavefunction at a fixed, predetermined position. In the
approach, a hierarchy of ODEs are solved along a trajectory with zero velocity.
The new approximation is local, both literally and from a quantum mechanical
point of view, in the sense that neighboring trajectories do not communicate
with each other. The approach is readily extended to imaginary time propagation
and is particularly useful for the calculation of quantities where only local
information is required. We present two applications: the calculation of
tunneling probabilities and the calculation of low energy eigenvalues. In both
applications we obtain excellent agrement with the exact quantum mechanics,
with a single trajectory propagation.
