# How can the effect of a pulse sequence on a strongly coupled spin system be analysed?

In introductory NMR theory there are two models which are commonly used to analyse the effect of a pulse sequence: namely, these are the vector model and the product operator formalism.

The vector model serves as a useful pictorial representation, but is generally considered too simplistic to be used when the system contains coupled spins. Usually at this point the product operator approach is introduced, based on the Liouville–von Neumann equation

$$\frac{\mathrm d\hat{\rho}}{\mathrm dt} = -\mathrm{i}[\hat{H}, \hat{\rho}]$$

and its solution $\hat{\rho}(t) = \exp(-\mathrm{i}\hat{H}t)\hat{\rho}(0)\exp(\mathrm{i}\hat{H}t)$. Then with a knowledge of the initial density matrix $\hat{\rho}(0)$ and the relevant Hamiltonians $\hat{H}$, the density matrix after a pulse sequence $\hat{\rho}(t)$ can be calculated.

Yet, it is also stated that the product operator approach "only applies to weakly coupled spin systems" (Keeler, Understanding NMR Spectroscopy 2nd ed., section 7.1.3). "Weakly coupled", of course, means that the coupling constant $J_\mathrm{IS}$ is much smaller than the difference in resonance frequencies $|\nu_\mathrm I - \nu_\mathrm S|$. This raises two questions:

1. Why does the product operator formalism not work for strongly coupled systems?

2. How can the effect of a pulse sequence on such systems be analysed?

• Full disclosure: the answer is described in Hore, NMR: The Toolkit, How Pulse Sequences Work, 2nd ed., Chpt 10. But I haven't gotten around to reading it fully yet and I just thought I'd post the question here anyway. Aug 8, 2017 at 16:53