  We illustrate an isomorphic representation of the observable algebra for
quantum mechanics in terms of the functions on the projective Hilbert space,
and its Hilbert space analog, with a noncommutative product in terms of
explicit coordinates and discuss the physical and dynamical picture. The
isomorphism is then used as a base for the translation of the differential
symplectic geometry of the infinite dimensional manifolds onto the observable
algebra as a noncommutative geometry. Hence, we obtain the latter from the
physical theory itself. We have essentially an extended formalism of the
Schrodinger versus Heisenberg picture which we describe mathematically as like
a coordinate map from the phase space, for which we have presented argument to
be seen as the quantum model of the physical space, to the noncommutative
geometry coordinated by the six position and momentum operators. The observable
algebra is taken essentially as an algebra of formal functions on the latter
operators. The work formulates the intuitive idea that the noncommutative
geometry can be seen as an alternative, noncommutative coordinate, picture of
familiar quantum phase space, at least so long as the symplectic geometry is
concerned.
