Quantization in generalized coordinates

Abstract

The operator form of the generalized canonical momenta in quantum mechanics is derived by a new, instructive method and the uniqueness of the operator form is proven. If one wishes to find the correct representation of the generalized momentum operator, he finds the Hermitian part of the operator —iħ ∂/∂q, whereq q is the generalized coordinate. There are interesting philosophical implications involved in this: It is like saying that a physical structure is composed of two parts, one which is real (the measurable quantity) and one which is pure imaginary. However, in order to understand the theoretical generation of the physical structure, one must look at the imaginary part as well as the real part since the sum of these two parts gives the simplified physical theory. That is why we can choose the total generalized momentum operator as simply —iħ ∂/∂q, but in order to arrive at the “measurable” momentum operator, we must choose the real (Hermitian) part, the other part being anti-Hermitian (corresponding to pure imaginary eigenvalues). We also discuss the operator form of the generalized Hamiltonian and show that the primary focus in developing fundamental concepts and prescriptions in quantum mechanics should be on the generalized momenta rather than on the Hamiltonian.