Wigner Active and Passive Rotation Matrices Applied to Nmr Tensor

NMR Hamiltonians are double contraction of two spherical rank-2 tensors, their space parts are represented by a spherical tensor and their spin parts are composed of spherical tensor operators. The comprehension of modern NMR experiments is very often based on the rotation of these tensors. We present the active and passive rotations in a progressive way from position vector to spherical tensor operator via space function, spherical harmonic, and vector operator. The passive rotation of a physical quantity is described by the rotation of coordinate system. Both the left- and right-handed rotation conventions are applied whereas the right-handed rotation convention is mainly used in the literature. Throughout the article, we explore the equivalence between the active rotation of a physical quantity in one direction and the rotation of the coordinate system in the opposite direction. The article presents redundant mathematical demonstrations between active and passive rotations, but they clarify the meanings of some important expressions not well developed in the literature.