Interaction picture

The interaction picture divides quantum dynamics between an exactly handled reference Hamiltonian and an interaction. Write $H(t)=H_0+V(t)$ with time-independent $H_0$, and definewhere $\hslash$ is the reduced Planck constant and the subscript $S$ denotes the Schrödinger picture. A linear operator transforms as $A_I(t)=U_0^{†}{A}_S(t)U_0$. Differentiating the transformed quantum state and using the Schrödinger equation cancels the $H_0$ terms and givesThe resulting unitary time evolution isThe Dyson series expands this time-ordered expression in powers of the interaction. ^[ref]

This is useful when $H_0$ has a known eigenbasis of energy eigenstates and the interaction is weak enough for perturbation theory. The transformation itself is exact and preserves expectation values: $⟨{\psi }_S|A_S|{\psi }_S⟩=⟨{\psi }_I|A_I|{\psi }_I⟩$. For time-dependent $H_0(t)$, use its exact unitary time evolution in place of the simple matrix exponential.

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.