Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined.
