Unique Hamiltonian Operators via Feynman Path Integrals

Abstract

The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well‐defined quantal operator Ĥ is shown to have a clear answer within Feynman's path‐integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born‐Jordan ordering rule for Ĥ in this case.