linear combinations of $\Psi_+^\sc(x)$ and $\Psi_-^\sc(x)$ are to be allowed, in order to construct arbitrary scattering solutions $\Psi(x)$. For convenience, the linear combination coefficients will from here on out be directly incorporated into the amplitude functions, $r_\pm(x)$, and phase functions, $s_\pm(x)$, so that is still correct.

If from all solutions $\Psi(x)$ one considers only that which satisfies the usual scattering boundary conditions (i.e. incident wave incoming from the left) then the negative momentum wave $\Psi_-$ vanishes, and $\Psi(x)=\Psi_+(x)$. There is zero reflection, and $100\%$ transmission. Put another way, the incident flux, $\lim_{x\ra -\infty} j_+(x)$, is equal to the transmitted flux, $\lim_{x\ra +\infty} j_+(x)$, where \[both flux values are equal to $F$, as in \].

In the quantum trajectory description, flux manifests as probability-transporting trajectories, which move along the LMs. For the boundary conditions described above, there are only positive momentum trajectories, moving uniformly from left to right with momentum $p_+(x) = \sqrt{2mE}$. If a $\Psi_-(x)$ contribution were present, its trajectories would move uniformly in the opposite direction \[$p_-(x)=-\sqrt{2mE}$.\] Since the two components $\Ppm(x)$ are in this case uncoupled, the positive and negative momentum trajectories would have no interaction with each other.

### hard wall system {#hardwall}

We next consider the 