case [*are*]{} well-defined throughout all of configuration space, according to a conventional scattering treatment. Although this is no longer true for continuous potentials, the foundation laid here in paper II can be extended to the continuous (and also time-dependent) case as well, as described in paper III. Additional motivation for the development of a scattering version of the CPWM, vis-a-vis the relevance for chemical physics applications, is provided in paper III. Additional motivation for the consideration of discontinuous potentials is provided in Sec. \[scattering\] of the present paper.

THEORY
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Background
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### Bohmian mechanics {#Bohmian}

According to the Bohmian formulation,[@wyatt; @holland] the QHEM are derived via substitution of the 1D (unipolar) wavefunction ansatz, (x,t) = R(x,t) e\^[i S(x,t)/]{} into the time-dependent  equation. For the 1D Hamiltonian, H = -[\^2 2 m]{} [\^2 x\^2]{} + V(x), \[hameqn\] this results in the coupled pair of nonlinear partial differential equations, where $m$ is the mass, $V(x)$ is the system potential, and primes denote spatial partial differentiation.

The first of the two equations above is the quantum Hamilton-Jacobi equation (QHJE), whose last term is equal to $-Q(x,t)$, i.e. comprises the quantum potential correction. The second equation is a continuity equation. When combined with 