ABSTRACT
Many selfadjoint operators important to physical applications, e.g., number operators and some first order and second order differential operators, possess a discrete spectrum. For a selfadjoint operator with a discrete spectrum the spectral function reduces to a sum in the form. An arbitrary symmetric operator does not possess a spectral function and the spectral theorem does not apply to such an operator. Hence the relation between selfadjoint operators and probability distributions presented above does not apply to symmetric operators. This is the reason that the probabilistic nature of quantum mechanics is formulated in terms of selfadjoint operators, not symmetric operators. In the momentum representation the momentum appears as a multiplication operator in the same way that the position acts as a multiplication operator in the coordinate representation. It follows that we can treat the momentum in the momentum representation in the same way as position in the coordinate representation.
