Noncommutative quantum mechanics-a perspective on structure and spatial extent

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 position and its measurement in the sense of a weak measurement (positive operator-valued measure), we find two equivalent pictures in a position representation: 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 the 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.