= Weak coupling condition for two NMR spins
{title2=$|\omega_1-\omega_2|\gg|J|$}
For the two-<spin> <Hamiltonian operator> $H=(\omega_1ZI+\omega_2IZ+J(XX+YY+ZZ))/2$, juxtaposition of two <Pauli matrices> denotes their <tensor product>. Put $\Delta=\omega_1-\omega_2$. The exchange matrix element between $|01\rangle$ and $|10\rangle$ is $J$, whereas their diagonal separation is $\Delta$. <Nondegenerate perturbation theory> therefore gives admixtures $\pm J/\Delta$ and second-order frequency shifts $\pm J^2/\Delta$. Neglecting the exchange terms is justified to leading order when $|J/\Delta|\ll1$. A large common <Zeeman energy> is insufficient. To approximate accumulated phases by this leading-order <Hamiltonian operator>, one also needs $J^2t/|\Delta|\ll1$ over the observation time, or must include the second-order shifts.
For fixed nonidentical <chemical shifts>, increasing the static <magnetic field> increases $|\Delta|$ in angular-frequency units while the <NMR scalar coupling> changes little. If $\Delta=0$ remains exact, the approximation fails at every nonzero $J$.
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