We consider the transformation of the initial data space for the Schrodinger equation. We study the phenomenon of global existence of a solution of the Cauchy problem the phenomenon of rise of a solution gradient blow up during a finite time. We define a solution extension through the moment of a gradient blow up using the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states.