The Product Operator Formalism: A Physical and Graphical Interpretation

The product-operator formalism is the most commonly used tool for describing and designing multidimensional NMR experiments. In spite of its relative simplicity and sound theoretical underpinnings, however, students and practitioners often find it difficult to relate the mathematical manipulations to a physical picture. In an effort to address this pedagogical challenge, the present paper begins with a quantum-mechanical treatment of pure populations of scalar-coupled spin-pairs, rather than the equilibrium population of spin-pairs in different quantum states, which is the usual starting point for treatments based on the density matrix and product operators. In the context of pure populations, the product operators are shown to represent quantum correlations between the nuclei in individual molecules, and a new variation on the classical vector diagram is introduced to represent these correlations. The treatment is extended to mixed populations that begin at thermal equilibrium, and the density matrix is introduced as an efficient means of carrying out quantum calculations for a mixed population. Finally, it is shown that the operators for observable magnetization and correlations can be used as a basis set for the density matrix, providing the formal justification for the widely-used rules of the product-operator treatment. Throughout the discussion, the vector diagrams are used to help maintain a connection between the mathematics and the sometimes subtle physical principles. An electronic supplement created with the Mathematica computer program is used to provide additional mathematical details and the means to carry out further calculations.