Why L 2 Topology in Quantum Physics
Chris Culellar, Evan Longpre, Владик Крейнович · scholarworks - UTEP (The University of Texas at El Paso) · 2012
It is known that in quantum mechanics, the set S of all possible states coincides with the set of all the complex-valued functions f(x) for which the integral of |f(x)|2 is 1. From the mathematical viewpoint, this set is a unit sphere in the space L2 of all the functions for which this integralis finite. Because of this mathematical fact, usually the set S is considered with the topology induced by L2, i.e., topology in which the basis of open neighborhood of a state f is formed by the open balls. This topology seem to work fine, but since this is a purely mathematical definition, a natural question appears: does this topology have a physical meaning? In this paper, we show that a natural physical definition of closeness indeed leads to the usual L2-topology.