  We study the Poisson geometrical formulation of quantum mechanics for finite
dimensional mixed and pure states. Equivalently, we show that quantum mechanics
can be understood in the language of classical mechanics. We review the
symplectic structure of the Hilbert space and identify its canonical
coordinates. We extend the geometric picture to the space of density matrices
$D_N^+$. We find it is not symplectic but admits a linear $\mathfrak{su}(N)$
Poisson structure. We identify Casimir surfaces of $D_N^+$ and show that the
space of pure states $P_N \equiv \mathbb{C}P^{N-1}$ is one of its symplectic
submanifolds which is an intersection of primitive Casimirs. We identify
generic symplectic submanifolds of $D_N^+$ and calculate their dimensions. We
find that $D_N^+$ is singularly foliated by the symplectic leaves of varying
dimensions, also known as coadjoint orbits. We also find an ascending chain of
Poisson submanifolds $D_N^M \subset D_N^{M+1}$ for $ 1 \leq M \leq N-1$. Each
such Poisson submanifold $D_N^M$ is obtained by tracing out the $\mathbb{C}^M$
states from the bipartite system $\mathbb{C}^N \times \mathbb{C}^M$ and is an
intersection of $N-M$ primitive Casimirs of $D_N^+$. Their Poisson structure is
induced from the symplectic structure of the bipartite system. We also show
their foliations. Finally, we study the positive semi-definite geometry of the
symplectic submanifold $E_N^M$ consisting of the mixed states with maximum
entropy in $D_N^M$.
