Abstract:

Although it was in fact the study of the Wigner distribution function that suggested to the author the general problem discussed in Part I, a consideration of the question of the formation of quantum mechanical operators corresponding to dynamical variables, being of fundamental importance to quantum theory, should properly precede a discussion of the Wigner distribution function. In both Parts I and II spin and relativistic effects will be ignored.
A word about notation is in order. A symbol such as A will denote the quantum mechanical operator corresponding to the dynamical quantity A. For example, H is the Hamiltonian operator; q is the coordinate operator and p the conjugate momentum operator. Whenever there is a possibility of confusion, O(A) is used to represent the operator corresponding to the dynamical quantity A. For example, O(pq) is the operator corresponding to the product pq, and O(H2) is the operator corresponding to the square of the classical Hamiltonian.
A subscript appearing on a differential operator, such as in 66PH , indicates the function on which the operator acts. Unless otherwise indicated all integrations are to be taken from infinity to infinity. As usual, h denotes Planck's constant divided by 2pi.
Since any physical investigation is directly dependent on the quality of the tools used in the investigation, mathematical rigor is much to be desired in physical theory. That rigor has not been achieved in this work is all too obvious, but it was nonetheless a constant goal. 