Product operator formalism

In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism.

In this model, for a single spin, four base operators exist: ${\displaystyle I_{x}}$, ${\displaystyle I_{y}}$, ${\displaystyle I_{z}}$ and ${\displaystyle E/2}$ which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for coupled systems. Using this approach, the evolution of the magnetization under free precession is represented by ${\displaystyle I_{z}}$ and corresponds to a rotation about the z-axis with a phase angle proportional to the chemical shift of the spin in question:

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