  We investigate the Brownian motion of a charged particle in a magnetic field.
We study this in the high temperature classical and low temperature quantum
domains. In both domains, we observe a transition of the mean square
displacement from a monotonic behaviour to a damped oscillatory behaviour as
one increases the strength of the magnetic field. When the strength of the
magnetic field is negligible, the mean square displacement grows linearly with
time in the classical domain and logarithmically with time in the quantum
domain. We notice that these features of the mean square displacement are
robust and remain essentially the same for an Ohmic dissipation model and a
single relaxation time model for the memory kernel. The predictions stemming
from our analysis can be tested against experiments in trapped cold ions.
