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

Abstract

A generalization of the procedures of Weyl and McCoy for making transition from a classical function to a quantum-mechanical operator function corresponding to it has been carried out, in order to include terms involving more than one pair of conjugate variables. An application of the method is made to the case of two spinless interacting charges—the Darwin interaction. This result is verified by solving directly for the quantum-mechanical Hamiltonian operator, starting from equations which were derived from the classical equations by replacing the classical Poisson brackets by commutators and by requiring the equality of the energy and the Hamiltonian operator. An alternative expression for a quantized Hamiltonian, derived in the Appendix, reduces to the form given originally by Born and Jordan for a single set of conjugate variables.