Momentum and Hamiltonian operators in generalized coordinates

Abstract

The self-adjointness of momentum operators in generalized coordinates, questioned by Domingos and Caldeira is shown. The momentum operators of a particle and the kinetic part of its Hamiltonian operator constructed from them are characterized as self-adjoint operators and geometrical objects in coordinate-free form. Local coordinates of ann-dimensional Riemannian manifold are taken as the generalized coordinates of the particle. As an example the curvilinear coordinates of Euclidean space are treated. The coefficients of connection and curvature are given on the manifold for which the assumed momentum operators exist. It is found that if our momentum operators form a complete set of mutually commuting observables, the manifold is locally Euclidean, i.e., there exists a local coordinate system such that we obtain the usual Schrodinger correspondence rule.