all spatial derivatives, must be specified. Fortunately, the unique correspondence between $\Psi_+(x)$ and $\Psi_-(x)$ described above, enables one to do just that. In particular, specifies the correct transformations for $r$ and $s$, as transported by the quantum trajectories. All spatial derivatives of arbitrary orders can then be obtained via spatial differentiation of —although in the hard wall case, only the $s'$ condition, $p_-(0) = - p_+(0)$ is relevant, because all higher order derivatives are identically zero.

Since the magnitudes of the $p$ and $r$ fields associated with a given quantum trajectory are unchanged as a result of the trajectory hop, implies that the incident and reflected flux values are the same (apart from sign), and so the scattering system exhibits 100% reflection and zero transmission (along each LM, the flux is invariant[@poirier04bohmI]). These basic facts of the hard wall system are of course well understood. The point, though, is that we have now obtained the information in a time-[*dependent*]{} quantum trajectory manner, rather than through the usual route of applying boundary conditions to time-[*independent*]{} piecewise component functions. In other words, now refers to individual [*quantum trajectories*]{}, rather than to wavefunctions.

This shift of emphasis is very important, and leads to 