  We analyze the solution of the coined quantum walk on a line. First, we
derive the full solution, for arbitrary unitary transformations, by using a new
approach based on the four "walk fields" which we show determine the dynamics.
The particular way of deriving the solution allows a rigorous derivation of a
long wavelength approximation. This long wavelength approximation is useful as
it provides an approximate analytical expression that captures the basics of
the quantum walk and allows us to gain insight into the physics of the process.
