  We explore the notion of spatial extent and structure, already alluded to in
earlier literature, within the formulation of quantum mechanics on the
noncommutative plane. Introducing the notion of average position and its
measurement, we find two equivalent pictures: a constrained local description
in position containing additional degrees of freedom, and an unconstrained
nonlocal description in terms of the position without any other degrees of
freedom. Both these descriptions have a corresponding classical theory which
shows that the concept of extended, structured objects emerges quite naturally
and unavoidably there. It is explicitly demonstrated that the conserved energy
and angular momentum contain corrections to those of a point particle. We argue
that these notions also extend naturally to the quantum level. The local
description is found to be the most convenient as it manifestly displays
additional information about structure of quantum states that is more subtly
encoded in the nonlocal, unconstrained description. Subsequently we use this
picture to discuss the free particle and harmonic oscillator as examples.
