ABSTRACT
This chapter shows that the correspondence between the elements of a linear vector space and the pure states of an ensemble emerges quite naturally from such an analysis. It introduces the term pure state to describe the situation where one wave function represented a state. The chapter discusses the arbitrary pairs of incompatible observables. It is clear that an essential task of quantum theory is the evaluation of the transformation functions. In fact, the time dependence of these quantities determines the dynamical properties of the system. In classical mechanics the addition of two dynamically independent angular momenta is carried out by applying the elementary laws of vector algebra. The analogous quantum mechanical problem is considerably more involved, and the machinery of transformation theory as developed in ideally suited to its solution. The square of the angular momentum is a familiar example of an observable.
