On the Function in Quantum Mechanics Which Corresponds to a Given Function in Classical Mechanics

It is the purpose of this note to obtain an explicit expression for F(P,Q), and although we confine our statements to the case in which f(p,q) is a polynomial, the results remain formally correct for infinite series. Any polynomial G(P,Q) may, by means of relation (1), be written in a form in which all of the Q-factors occur on the left in each term. This form of the function G(P,Q) will be denoted by GQ(P,Q). Let GQ(P,q) indicate the function of the commutative variables p,q obtained from GQ(P,Q) by replacing P,Q by p,q, respectively. In a similar manner Gp(P,Q) and Gp(p,q) may be defined. For example, if G(P,Q) = PQP, we find GQ(P,Q) = QP2 + yP, Gp(P,Q) = p2Q -yP, GQ(P,q) = p2q + 'ypp Gp(p,q) = pq yP. Our principal result may now be stated as follows. Let f(p,q) be a polynomial in the canonical variables p,q of classical mechanics, and F(P,Q) the corresponding function of the Hermitian operators P,Q in quantum mechanics. Then