Momentum operators for curvilinear coordinate systems

Expressions for the covariant and contravariant components of the quantum momentum operator (h//i) ∇ are formulated for curvilinear coordinates in three-dimensional space. The classical formalism is modified to take account of the noncommutivity of coordinates and momenta in the quantum case. These components are shown to be simply related to canonical momentum operators conjugate to the curvilinear coordinates. Expressions for components of angular momenta are also obtained. These results are specialized for orthogonal curvilinear coordinates and illustrated for spherical polar coordinates.