Orthonormality and completeness relations are found for scattering states, the states used in treatments of resonant-tunneling double barriers and other short microstructures. Explicit normalized states are used to study the density matrix and Wigner distribution function of some simple but nontrivial structures. Analytic expressions describe the quantum repulsion of a sharp barrier at distances comparable to a de Broglie wavelength. As a function of distance from the step, the zero-current density approaches its constant-potential value asymptotically as a Gaussian. We also observe that, by virtue of the exact orthogonality of scattering states, a proposed correction to Esaki and Tsu's calculation of resonant tunneling current is rigorously zero.