Quantum Mechanical Measurements
The theory of quantum mechanical measurements was formulated most precisely by von Neumann. According to his ideas, the theory consists in the establishment of a statistical correlation between the state of the object on which the measurement is taking place and the state of the measuring apparatus. The “reading” of the apparatus consists in determining in which of several possible states the apparatus is; the statistical correlation between apparatus and object after the measurement is such that the state of the apparatus determines the state of the object, and that this latter state is one of the eigenfunctions of the operator which is being measured. It has been shown some time ago that only those operators can be measured precisely in this way which commute with all additive conserved quantities, such as the components of the angular momentum. As a result, we find that there are, from the point of view of measurability, three types of quantities. The precisely measurable quantities’s operators commute with all additive conserved quantities; the collision matrix belongs to this category. Most quantities can only be measured approximately, the possible degree of approximation increasing with the measuring instrument’s content of the conserved quantities. Finally, there are quantities which cannot be measured at all; this gives rise to the so-called superselection rules.