  A simple one-dimensional model for SNS weak links in the ballistic limit is
presented. In the presence of a bias voltage, the quasiparticle state at any
given instant of time is described as a superposition of that particular set of
phase-dependent Andreev bound states that belongs to the specific phase
difference present at this instant between the superconducting banks. The
treatment -- basically a form of adiabatic perturbation theory -- has a strong
formal similarity to the treatment of the k-space dynamics of an electron in a
periodic potential under perturbation by an external electric field,
sufficiently strong to cause transitions across the energy gaps between bands
(Zener tunneling). It is shown that the quasiparticle wave function retains its
phase information during analogous transitions between Andreev bands. The
experimental observation of Shapiro steps at one-half the canonical voltage
follows naturally from the model, along with some of the experimental
properties of these steps, especially their much weaker temperature dependence,
compared to the canonical steps.
