Deriving the generalized momentum operators from covariant derivatives

The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.