By W.M. de Muynck
Taking a brand new standpoint supplied by way of a generalization of the mathematical formalism encompassing optimistic operator-valued measures, this publication perspectives previous and new difficulties of the principles of quantum mechanics. It demonstrates the the most important position of the generalized formalism in primary concerns and useful applications.
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Additional info for Foundations of Quantum Mechanics, an Empiricist Approach
36) it follows that Hence, can be considered as a vector in Hilbert-Schmidt space. 38), they constitute only a subset of this space. Since a convex combination of two density operators and is another density operator, this subset is convex (cf. 3). Thus, represents a possible quantum mechanical state that is a mixture of the two states represented by and with relative weights and respectively. The convexity of the set of density operators reflects the convexity of the set of probability measures on a linear space (cf.
Starting from the initial situation given above there are different possible final situations, depending on the angle at which the photon is scattered. 71) in a discontinuous way. Yet, the projection postulate is controversial (see, for instance, [16, 17]). We shall see in the following that von Neumann’s reasoning is not entirely cogent. 70) is generally valid can immediately be seen by considering observables, like position and momentum, having a continuous spectrum. Such observables do not even have normalizable eigenvectors; hence, it is impossible that the final state of the measurement process is an eigenvector of such an observable.
38), they constitute only a subset of this space. Since a convex combination of two density operators and is another density operator, this subset is convex (cf. 3). Thus, represents a possible quantum mechanical state that is a mixture of the two states represented by and with relative weights and respectively. The convexity of the set of density operators reflects the convexity of the set of probability measures on a linear space (cf. 2). 33). 96) of extreme elements. 36) of we see that every density operator can be written as a convex combination of extreme elements.