Probability operator of a harmonic oscillator

A unique linear rule of constructing quantum operators defined by the probability operator {Mathematical expression} for coordinates and momenta, is considered. {Mathematical expression} is assumed to be a normalized, positive definite operator, establishing a dynamical correspondence between the classical and quantum Poisson brackets. It is shown that such an operator exists in the case of a harmonic oscillator. The principal implications of the suggested rule of constructing the operators of physical quantities are determined, in comparison with the corresponding results of conventional quantum mechanics. © 1983 Plenum Publishing Corporation.