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?

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    $\begingroup$ 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. $\endgroup$ Aug 8, 2017 at 16:53

1 Answer 1


The product operator formalism still applies to strongly coupled systems. The simple "magic formula" for calculating a closed-form solution to the step-wise time evolution only applies when the terms of the Hamiltonian commute with each other, which is not the case in a strongly-coupled system. Maybe more accurately stated: in a weakly-coupled system, the non-commuting terms can be neglected. The other assumption that allows the removal of non-commuting terms is that the pulses are very short compared to other evolution times, so during pulse periods the free-evolution terms are omitted from the Hamiltonian.

The effects of a pulse sequence without these on such a system can still be found by computing the exponential of the matrix rather than using the shortcut.

Now a shameless plug: the SpinDrops tool (http://spindrops.org) can be used to simulate and visualize the effects of pulse sequences with arbitrary Hamiltonians. It shows an exact 1-1 mapping of coupled-spin operators to the visualization which can not be done with the vector model. The setting 'Pure Pulses' can be disabled to include the free-evolution Hamiltonian during pulses, but incorporating strong coupling would currently need to be done by defining the Hamiltonian explicitly.


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