 Quantum rotating frame

A time-dependent unitary operator $R(t)$ defines transformed quantum states by $|{\psi }_{\mathrm{r}}⟩=R^{†}|\psi ⟩$. Substituting $|\psi ⟩=R|{\psi }_{\mathrm{r}}⟩$ into the Schrödinger equation gives the displayed transformed Hamiltonian operator, in units $\hslash =1$. If $R=e^{-it{H}_0}$ for constant $H_0$, this becomes $H_{\mathrm{r}}=R^{†}HR-H_0$. Removing a common nuclear Zeeman energy with $H_0=\omega (ZI+IZ)/2$ leaves an isotropic Heisenberg model coupling unchanged, because it commutes with the generator of the common rotation. Removing two unequal nuclear frequencies instead makes their exchange terms oscillate at the difference frequency.