  We consider the time evolution of the density matrix $\rho$ in a
2-dimensional complex Hilbert space. We allow for dissipation by adding to the
von Neumann equation a term $D[\rho]$, which is of Lindblad type in order to
assure complete positivity of the time evolution. We present five equivalent
forms of $D[\rho]$. In particular, we connect the familiar dissipation matrix
$L$ with a geometric version of $D[\rho]$, where $L$ consists of a positive sum
of projectors onto planes in $\mathbf{R}^3$. We also study the minimal number
of Lindblad terms needed to describe the most general case of $D[\rho]$. All
proofs are worked out comprehensively, as they present at the same time a
practical procedure how to determine explicitly the different forms of
$D[\rho]$. Finally, we perform a general discussion of the asymptotic behaviour
$t \to \infty$ of the density matrix and we relate the two types of asymptotic
behaviour with our geometric version of $D[\rho]$.
