The vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates
European Journal of Physics, vol.47, no.3, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 47 Issue: 3
- Publication Date: 2026
- Doi Number: 10.1088/1361-6404/ae68c4
- Journal Name: European Journal of Physics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, INSPEC, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO)
- Keywords: canonical momentum in curvilinear coordinates, Hermitian momentum operator, momentum in curvilinear coordinates
- Middle East Technical University Affiliated: No
Abstract
We construct the Hermitian vector and canonical components of the momentum operator in 3D Euclidean space spanned by general curvilinear coordinates (GCCs) using a simple, natural and unified approach based on identifying the momentum operator in any coordinate system as mass times the velocity operator. When this latter is calculated by applying the Heisenberg equation of motion, it returns ( (Formula presented) (Formula presented) times) the gradient operator plus an additional zero-valued sum, which when distributed among the components of the gradient, it makes each the Hermitian vector component of the momentum operator in GCCs. The canonical components follow immediately upon symmetrizing each of these vector components in the corresponding base vector. For accessability by wider audiences, we first develop the formalism for the simple polar coordinates (PCs) and then we develop the case for GCCs.