Tensors in NMR

Interaction Hamiltonians in NMR spectroscopy are typically described as contractions or scalar products of spatial and spin tensors. These expressions arise naturally from expressing the interactions first using vectors and cartesian tensors, and then finding the appropriate spherical tensor formulation. The latter form is used pervasively in NMR literature. Here, we describe how spherical tensors arise from their cartesian counterparts, and how one can arrive at the Hamiltonian expressions in the form of spherical tensor formulations.