Structural Analysis and Uncertainty of Orbital Angular Momentum Operators in Spherical Representation: Projection Approach, Spherical Tensor, and Their Application to Atomic Systems

Abstract

This research is a literature review that comprehensively discusses the properties of the orbital angular momentum operator in spherical representation. The main focus lies in the analysis of the mathematical structure of the operator through the projection operator and spherical tensor
approach, as well as its physical implications in various quantum systems. This study also evaluates the limitations of the uncertainty principle that arises from the commutator relation
between the components of the angular momentum operator, and how this uncertainty depends on the specific quantum state. The results show that the spherical representation provides a natural framework for modeling rotational symmetry in atomic and electromagnetic systems.
The projection operator allows the construction of structured wave functions, while the spherical tensor expands the capabilities of operator decompositions and transformations in Hilbert space. Furthermore, orbital angular momentum is shown to play a crucial role in the formation of multipolar electromagnetic fields and the energy structure of atoms. This study strengthens the conceptual understanding of the orbital angular momentum operator and opens up further research directions in precision quantum control and optical systems engineering.