  We study the dynamics of a particle in continuous time and space, the
displacement of which is governed by an internal degree of freedom (spin). In
one definite limit, the so-called quantum random walk is recovered but,
although quite simple, the model possesses a rich variety of dynamics and goes
far beyond this problem. Generally speaking, our framework can describe the
motion of an electron in a magnetic sea near the Fermi level when linearisation
of the dispersion law is possible, coupled to a transverse magnetic field.
Quite unexpected behaviours are obtained. In particular, we find that when the
initial wave packet is fully localized in space, the $J_{z}$ angular momentum
component is frozen; this is an interesting example of an observable which,
although it is not a constant of motion, has a constant expectation value. For
a non-completely localized wave packet, the effect still occurs although less
pronounced, and the spin keeps for ever memory of its initial state. Generally
speaking, as time goes on, the spatial density profile looks rather complex, as
a consequence of the competition between drift and precession, and displays
various shapes according to the ratio between the Larmor period and the
characteristic time of flight. The density profile gradually changes from a
multimodal quickly moving distribution when the scatttering rate is small, to a
unimodal standing but flattening distribution in the opposite cas case.
